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12,856	by Evan Heimlich Acadians are the descendants of a group of French-speaking settlers who migrated from coastal France in the late sixteenth century to establish a French colony called Acadia in the maritime provinces of Canada and part of what is now the state of Maine. Forced out by the British in the mid-sixteenth century, a few settlers remained in Maine, but most resettled in southern Louisiana and are popularly known as Cajuns. Before 1713, Acadia was a French colony pioneered mostly by settlers from the coastal provinces of Brittany, Normandy, Picardy, and Poitou—a region that suffered great hardships in the late sixteenth and early seventeenth centuries. In 1628, famine and plague followed the end of a series of religious wars between Catholics and Protestants. When social tensions in coastal France ripened, more than 10,000 people left for the colony founded by Samuel Champlain in 1604 known as "La Cadie" or Acadia. The area, which included what is now Nova Scotia, New Brunswick, Prince Edward Island, and part of Maine, was one of the first European colonies in North America. The Company of New France recruited colonists from coastal France as indentured servants. Fishermen, farmers, and trappers served for five years to repay the company with their labor for the transportation and materials it had provided. In the New World, colonists forged alliances with local Indians, who generally preferred the settlers from France over those from Britain because, unlike the British who took all the land they could, the coastal French in Acadia did not invade Indian hunting grounds inland. The early French settlers called themselves "Acadiens" or "Cadiens" (which eventually became Anglicized as "Cajuns") and were among the first Old World settlers to identify themselves as North Americans. The New World offered them relative freedom and independence from the French upper class. When French owners of Acadian lands tried to collect seignorial rents from settlers who were farming, many Acadians simply moved away from the colonial centers. When France tried legally to control their profit from their trade in furs or grain, Acadians traded illegally; they even traded with New England while France and England waged war against each other. As French colonial power waned, Great Britain captured Acadia in 1647; the French got it back in 1670 only to lose it again to the British in the 1690s. Acadians adapted to political changes as their region repeatedly changed hands. Before the British took the Nova Scotia region, they waged the Hundred Year War against French colonial forces in a struggle over the region's territory. The Treaty of Utrecht in 1713, which failed to define realistic boundaries for the French and English territories after Queen Anne's War, converted most of the peninsula into a British colony. Despite British attempts to impose its language and culture, Acadian culture persisted. Large families increased their numbers and new settlers spoke French. The British tried to settle Scottish and other Protestant colonists in Acadia to change the region's French-Catholic culture to a British-Protestant one. The French-speaking Acadians, however, held onto their own culture. In 1745 the British threatened to expel the Acadians unless they pledged allegiance to the King of England. Unwilling to subject themselves to any king (especially the King of England who opposed the French and Catholics), Acadians refused, claiming that they were not allied with France. They also did not want to join the British in fights against the Indians, who were their allies and relatives. To dominate the region militarily, culturally, and agriculturally without interference, the British expelled the Acadians, dispersing them to colonies such as Georgia and South Carolina. This eventually led the British to deport Acadians in what became known as Le Grand Dèrangement, or the Expulsion of 1755. The roundup and mass deportation of Acadians, which presaged British domination of much of North America, involved much cruelty, as indicated by letters from British governor, Major Charles Lawrence. In an attempt to eliminate the Acadians from Acadia, the British packed them by the hundreds into the cargo holds of ships, where many died from the cold and smallpox. At the time, Acadians numbered about 15,000, however, the Expulsion killed almost half the population. Of the survivors and those who escaped expulsion, some found their way back to the region, and many drifted through England, France, the Caribbean, and other colonies. Small pockets of descendants of Acadians can still be found in France. In 1763 there were more than 6,000 Acadians in New England. Of the thousands sent to Massachusetts, 700 reached Connecticut and then escaped to Montreal. Many reached the Carolinas; some in Georgia were sold as slaves; many eventually were taken to the West Indies as indentured servants. Most, however, made their way down the Mississippi River to Louisiana. At New Orleans and other southern Louisiana ports, about 2,400 Acadians arrived between 1763 and 1776 from the American colonies, the West Indies, St. Pierre and Miquelon islands, and Acadia/Nova Scotia. To this day, many Acadians have strong sentiments about the expulsion 225 years ago. In 1997, Warren A. Perrin, an attorney from Lafayette, Louisiana, filed a lawsuit against the British Crown for the expulsion in 1775. Perrin is not seeking monetary compensation. Instead, he wants the British government to formally apologize for the suffering it caused Acadians and build a memorial to honor them. The British Foreign Office is fighting the lawsuit, arguing it cannot be held responsible for something that happened more than two centuries ago. According to Cajun Country, after Spain gained control of Louisiana in the mid-1760s, Acadian exiles "who had been repatriated to France volunteered to the king of Spain to help settle his newly acquired colony." The Spanish government accepted their offer and paid for the transport of 1,600 settlers. When they arrived in Louisiana in 1785, colonial forts continued Spain's services to Acadian pioneers (which officially began with a proclamation by Governor Galvez in February of 1778). Forts employed and otherwise sponsored the settlers in starting their new lives by providing tools, seed corn, livestock, guns, medical services, and a church. A second group of Acadians came 20 years later. Louisiana attracted Acadians who wanted to rejoin their kin and Acadian culture. After decades of exile, immigrants came from many different regions. The making of "Acadiana" in southern Louisiana occurred amid a broader context of French-speaking immigration to the region, including the arrival of European and American whites, African and Caribbean slaves, and free Blacks. Like others, such as Mexicans who lived in annexed territory of the United States, Cajuns and other Louisianans became citizens when the United States acquired Louisiana from Napoleon through the Louisiana Purchase in 1803. The diaspora of Acadians in the United States interweaves with the diaspora of French Canadians. In 1990, one-third as many Americans (668,000) reported to the U.S. Census Bureau as "Acadian/Cajun" as did Americans reporting "French Canadian" (2,167,000). Louisiana became the new Acadian homeland and "creolized," or formed a cultural and ethnic hybrid, as cultures mixed. French settlers in Louisiana adapted to the subtropics. Local Indians taught them, as did the slaves brought from Africa by settlers to work their plantations. When French settlers raised a generation of sons and daughters who grew up knowing the ways of the region—unlike the immigrants— Louisianans called these native-born, locally adapted people "Creoles." Louisianans similarly categorized slaves—those born locally were also "Creoles." By the time the Acadians arrived, Creoles had established themselves economically and socially. French Creoles dominated Louisiana, even after Spain officially took over the colony in the mid-eighteenth century and some Spanish settled there. Louisiana also absorbed immigrants from Germany, England, and New England, in addition to those from Acadia. Spanish administrators welcomed the Acadians to Louisiana. Their large families increased the colony's population and they could serve the capital, New Orleans, as a supplier of produce. The Spanish expected the Acadians, who were generally poor, small-scale farmers who tended to keep to themselves, not to resist their administration. At first, Spanish administrators regulated Acadians toward the fringes of Louisiana's non-Indian settlement. As Louisiana grew, some Cajuns were pushed and some voluntarily moved with the frontier. Beginning in 1764, Cajun settlements spread above New Orleans in undeveloped regions along the Mississippi River. This area later became known as the Acadian coast. Cajun settlements spread upriver, then down the Bayou Lafourche, then along other rivers and bayous. People settled along the waterways in lines, as they had done in Acadia/Nova Scotia. Their houses sat on narrow plots of land that extended from the riverbank into the swamps. The settlers boated from house to house, and later built a road parallel to the bayou, extending the levees as long as 150 miles. The settlement also spread to the prairies, swamps, and the Gulf Coast. There is still a small colony of Acadians in the St. John Valley of northeastern Maine, however. Soon after the Louisiana Purchase, the Creoles pushed many Acadians westward, off the prime farmland of the Mississippi levees, mainly by buying their lands. Besides wanting the land, many Creole sugar-planters wanted the Cajuns to leave the vicinity so that the slaves on their plantations would not see Cajun examples of freedom and self-support. After the Cajuns had reconsolidated their society, a second exodus, on a much smaller scale, spread the Cajuns culturally and geographically. For example, a few Acadians joined wealthy Creoles as owners of plantations, rejecting their Cajun identity for one with higher social standing. Although some Cajuns stayed on the rivers and bayous or in the swamps, many others headed west to the prairies where they settled not in lines but in small, dispersed coves. As early as 1780, Cajuns headed westward into frontier lands and befriended Indians whom others feared. By the end of the nineteenth century, Cajuns had established settlements in the Louisiana-Texas border region. Texans refer to the triangle of the Acadian colonies of Beaumont, Port Arthur, and Orange as Cajun Lapland because that is where Louisiana "laps over" into Texas. Heading westward, Cajuns first reached the eastern, then the western prairie. In the first region, densely settled by Cajuns, farmers grew corn and cotton. On the western prairie, farmers grew rice and ranchers raised cattle. This second region was thinly settled until the late 1800s when the railroad companies lured Midwesterners to the Louisiana prairies to grow rice. The arrival of Midwesterners again displaced many Cajuns; however, some remained on the prairies in clusters of small farms. A third region of Cajun settlement, to the south of the prairies and their waterways, were the coastal wetlands—one of the most distinctive regions in North America and one central to the Cajun image. The culture and seafood cuisine of these Cajuns has represented Cajuns to the world. Life for Cajuns in swamps, which periodically flood, demanded adaptations such as building houses on stilts. When floods wrecked their houses, Cajuns Cajuns have always been considered a marginal group, a minority culture. Language, culture, and kinship patterns have kept them separate, and they have maintained their sense of group identity despite difficulties. Cajun settlement patterns have isolated them and Cajun French has tended to keep its speakers out of the English-speaking mainstream. Acadians brought a solidarity with them to Louisiana. As one of the first groups to cross the Atlantic and adopt a new identity, they felt connected to each other by their common experience. Differences in backgrounds separated the Acadians from those who were more established Americans. Creole Louisianans, with years of established communities in Louisiana, often looked down on Acadians as peasants. Some Cajuns left their rural Cajun communities and found acceptance, either as Cajuns or by passing as some other ethnicity. Some Cajuns became gentleman planters, repudiated their origins, and joined the upper-class (white) Creoles. Others learned the ways of local Indians, as Creoles before them had done, and as the Cajuns themselves had done earlier in Acadia/Nova Scotia. Because Cajuns usually married among themselves, as a group they do not have many surnames; however, the original population of Acadian exiles in Louisiana grew, especially by incorporating other people into their group. Colonists of Spanish, German, and Italian origins, as well as Americans of English-Scotch-Irish stock, became thoroughly acculturated and today claim Acadian descent. Black Creoles and white Cajuns mingled their bloodlines and cultures; more recently, Louisiana Cajuns include Yugoslavs and Filipinos. Economics helped Cajuns stay somewhat separate. The majority of Cajuns farmed, hunted, and/or fished; their livelihoods hardly required them to assimilate. Moreover, until the beginning of the twentieth century, U.S. corporate culture had relatively little impact on southern Louisiana. The majority of Cajuns did not begin to Americanize until the turn of the twentieth century, when several factors combined to quicken the pace. These factors included the nationalistic fervor of the early 1900s, followed by World War I. Perhaps the most substantial change for Cajuns occurred when big business came to extract and sell southern Louisiana's oil. The discovery of oil in 1901 in Jennings, Louisiana, brought in outsiders and created salaried jobs. Although the oil industry is the region's main employer, it is also a source of economic and ecological concern because it represents the region's main polluter, threatening fragile ecosystems and finite resources. Although the speaking of Cajun French has been crucial to the survival of Cajun traditions, it has also represented resistance to assimilation. Whereas Cajuns in the oilfields spoke French to each other at work (and still do), Cajuns in public schools were forced to abandon French because the compulsory Education Act of 1922 banned the speaking of any other language but English at school or on school grounds. While some teachers labeled Cajun French as a low-class and ignorant mode of speech, other Louisianans ridiculed the Cajuns as uneducable. As late as 1939, reports called the Cajuns "North America's last unassimilated [white] minority;" Cajuns referred to themselves, even as late as World War II, as " le français, " and all English-speaking outsiders as " les Americains. " The 1930s and 1940s witnessed the education and acculturation of Cajuns into the American mainstream. Other factors affecting the assimilation of the Cajuns were the improvement of transportation, the leveling effects of the Great Depression, and the development of radio and motion pictures, which introduced young Cajuns to other cultures. Yet Cajun culture survived and resurged. After World War II, Cajun culture boomed as soldiers returned home and danced to Cajun bands, thereby renewing Cajun identity. Cajuns rallied around their traditional music in the 1950s, and in the 1960s this music gained attention and acceptance from the American mainstream. On the whole, though, the 1950s and 1960s were times of further mainstreaming for the Cajuns. As network television and other mass media came to dominate American culture, the nation's regional, ethnic cultures began to weaken. Since the 1970s, Cajuns have exhibited renewed pride in their heritage and consider themselves a national resource. By the 1980s, ethnicities first marginalized by the American mainstream became valuable as regional flavors; however, while Cajuns may be proud of the place that versions of their music and food occupy in the mainstream, they—especially the swamp Cajuns—are also proud of their physical and social marginality. Cajun society closely knits family members and neighbors who tend to depend on each other socially and economically, and this cooperation helps to maintain their culture. According to Cajun Country, "The survival—indeed the domination— of Acadian culture was a direct result of the strength of traditional social institutions and agricultural practices that promoted economic self-sufficiency and group solidarity." Cajuns developed customs to bring themselves together. For example, before roads, people visited by boat; before electrical amplification and telephones, people sang loudly in large halls, and passed news by shouting from house to house. And when Cajuns follow their customs, their culture focuses inwardly on the group and maintains itself. Cajuns maintain distinctive values that predate the industrial age. Foremost among these, perhaps, is a traditional rejection of protocols of social hierarchy. When speaking Cajun French, for instance, Cajuns use the French familiar form of address, tu, rather than vous (except in jest) and do not address anyone as monsieur. Their joie de vivre is legendary (manifested in spicy food and lively dancing), as is their combativeness. Cajun traditions help make Cajuns formidable, mobile adversaries when fighting, trapping, hunting, or fishing. Cajun boaters invented a flatboat called the bateau, to pass through shallow swamps. They also built European-style luggers and skiffs, and the pirogue, based on Indian dugout canoes. Cajuns often race pirogues; or, two competitors stand at opposite ends on one and try to make each other fall in the water first. Fishers hold their own competitions, sometimes called "fishing rodeos." Cajuns value horses, too. American cowboy culture itself evolved partly out of one of its earliest ranching frontiers on Louisiana's Cajun prairies. Cajun ranchers developed a tradition called the barrel or buddy pickup, which evolved into a rodeo event. Today, Cajuns enjoy horse racing, trail-riding clubs, and Mardi Gras processions, called courses, on horseback. Cajuns also enjoy telling stories and jokes during their abundant socializing. White Cajuns have many folktales in common with black Creoles—for example, stories about buried treasure abound in Louisiana. One reason for this proliferation was Louisiana's early and close ties to the Caribbean where piracy was rampant. Also, many people actually did bury treasure in Louisiana to keep it from banks or—during the Civil War—from invading Yankees. Typically, the stories describe buried treasure guarded by ghosts. Cajuns relish telling stories about moonshiners, smugglers, and contraband runners who successfully fool and evade federal agents. Many Cajun beliefs fall into the mainstream's category of superstition, such as spells ( gris-gris, to both Cajuns and Creoles) and faith healing. In legends, Madame Grandsdoigts uses her long fingers to pull the toes of naughty children at night, and the werewolf, known as loup garou, prowls. Omens appear in the form of blackbirds, cows, and the moon. For example, according to Cajun Country: "When the tips of a crescent moon point upward, [the weather] is supposed to be dry for a week. A halo of light around a full moon supposedly means clear weather for as many days as there are stars visible inside the ring." Cajun cuisine, perhaps best known for its hot, redpepper seasoning, is a blend of styles. Acadians brought with them provincial cooking styles from France. Availability of ingredients determined much of Cajun cuisine. Frontier Cajuns borrowed or invented recipes for cooking turtle, alligator, raccoon, possum, and armadillo, which some people still eat. Louisianans' basic ingredients of bean and rice dishes—milled rice, dried beans, and cured ham or smoked sausage—were easy to store over relatively long periods. Beans and rice, like gumbo and crawfish, have become fashionable cuisine in recent times. They are still often served with cornbread, thus duplicating typical nineteenth-century poor Southern fare. Cajun cooking is influenced by the cuisine of the French, Acadian, Spanish, German, Anglo-American, Afro-Caribbean, and Native American cultures. Gumbo, a main Cajun dish, is a prime metaphor for creolization because it draws from several cultures. Its main ingredient, okra, also gave the dish its name; the vegetable, called " guingombo, " was first imported from western Africa. Cayenne, a spicy seasoning used in subtropical cuisines, represents Spanish and Afro-Caribbean influences. Today Louisianans who eat gumbo with rice, usually call gumbo made with okra gumbo fèvi, to distinguish it from gumbo filè, which draws on French culinary tradition for its base, a roux. Just before serving, gumbo filè (also called filè gumbo ) is thickened by the addition of powdered sassafras leaves, one of the Native American contributions to Louisiana cooking. Cajuns thriftily made use of a variety of animals in their cuisine. Gratons , also known as cracklings, were made of pig skin. Internal organs were used in the sausages and boudin . White boudin is a spicy rice and pork sausage; red boudin, which is made from the same rice dressing but is flavored and colored with blood, can still be found in neighborhood boucheries. Edible pig guts not made into boudin were cooked in a sauce piquante de dèbris or entrail stew. The intestines were cleaned and used for sausage casings. Meat was carefully removed from the head and congealed for a spicy fromage de tête de cochon (hogshead cheese). Brains were cooked in a pungent brown sauce. Other Cajun specialties include tasso, a spicy Cajun version of jerky, smoked beef and pork sausages (such as andouille made from the large intestines), chourice (made from the small intestines), and chaudin (stuffed stomach). Perhaps the most representative food of Cajun culture is crawfish, or mudbug. Its popularity is a relatively recent tradition. It was not until the Cooking is considered a performance, and invited guests often gather around the kitchen stove or around the barbecue pit (more recently, the butane grill) to observe the cooking and comment on it. Guests also help, tell jokes and stories, and sing songs at events such as outdoor crawfish, crab, and shrimp boils in the spring and summer, and indoor gumbos in winter. The history of Cajun music goes back to Acadia/ Nova Scotia, and to France. Acadian exiles, who had no instruments such as those in Santo Domingo, danced to reels á bouche, wordless dance music made by only their voices at stopping places on their way to Louisiana. After they arrived in Louisiana, Anglo-American immigrants to Louisiana contributed new fiddle tunes and dances, such as reels, jigs, and hoedowns. Singers also translated English songs into French and made them their own. Accordi to Cajun Country, "Native Americans contributed a wailing, terraced singing style in which vocal lines descend progressively in steps." Moreover, Cajun music owes much to the music of black Creoles, who contributed to Cajun music as they developed their own similiar music, which became zydeco. Since the nineteenth century, Cajuns and black Creoles have performed together. Not only the songs, but also the instruments constitute an intercultural gumbo. Traditional Cajun and Creole instruments are French fiddles, German accordians, Spanish guitars, and an assortment of percussion instruments (triangles, washboards, and spoons), which share European and Afro-Caribbean origins. German-American Jewish merchants imported diatonic accordians (shortly after they were invented in Austria early in the nineteenth century), which soon took over the lead instrumental role from the violin. Cajuns improvised and improved the instruments first by bending rake tines, replacing rasps and notched gourds used in Afro-Caribbean music with washboards, and eventually producing their own masterful accordians. During the rise of the record industry, to sell record players in southern Louisiana, companies released records of Cajun music. Its high-pitched and emotionally charged style of singing, which evolved so that the noise of frontier dance halls could be pierced, filled the airwaves. Cajun music influenced country music; moreover, for a period, Harry Choates's string band defined Western swing music. Beginning in 1948, Iry Lejeune recorded country music and renditions of Amèe Ardoin's Creole blues, which Ardoin recorded in the late 1920s. Lejeune prompted "a new wave of old music" and a postwar revival of Cajun culture. Southern Louisiana's music influenced Hank Williams— whose own music, in turn, has been extremely influential. "Jambalaya" was one of his most successful recordings and was based on a lively but unassuming Cajun two-step called "Grand Texas" or "L'Anse Couche-Couche." In the 1950s, "swamp pop" developed as essentially Cajun rhythm and blues or rock and roll. In the 1960s, national organizations began to try to preserve traditional Cajun music. Mardi Gras, which occurs on the day before Ash Wednesday, the beginning of Lent, is the carnival that precedes Lent's denial. French for "Fat Tuesday," Mardi Gras (pre-Christian Europe's New Year's Eve) is based on medieval European adaptations of even older rituals, particularly those including reversals of the social order, in which the lower classes parody the elite. Men dress as women, women as men; the poor dress as rich, the rich as poor; the old as young, the young as old; black as white, white as black. While most Americans know Mardi Gras as the city of New Orleans celebrates it, rural Cajun Mardi Gras stems from a medieval European procession in which revelers traveled through the countryside performing in exchange for gifts. Those in a Cajun procession, called a course (which traditionally did not openly include women), masquerade across lines of gender, age, race, and class. They also play at crossing the line of life and death with a ritual skit, "The Dead Man Revived," in which the companions of a fallen actor revive him by dripping wine or beer into his mouth. Participants in a Cajun Mardi Gras course cross from house to house, storming into the yard in a mock-pillage of the inhabitant's food. Like a trick-or-treat gang, they travel from house to house and customarily get a series of chickens, from which their cooks will make a communal gumbo that night. The celebration continues as a rite of passage in many communities. Carnival, as celebrated by Afro-Caribbeans (and as a ritual of ethnic impersonation whereby Euro-and Afro-Caribbean Americans in New Orleans chant, sing, dance, name themselves, and dress as Indians), also influences Mardi Gras as celebrated in southern Louisiana. On one hand, the mainstream Mardi Gras celebration retains some Cajun folkloric elements, but the influence of New Orleans invariably supplants the country customs. Conversely, Mardi Gras of white, rural Cajuns differs in its geographic origins from Mardi Gras of Creole New Orleans; some organizers of Cajun Mardi Gras attempt to maintain its cultural specificity. Cajun Mardi Gras participants traditionally wear masks, the anonymity of which enables the wearers to cross social boundaries; at one time, masks also provided an opportunity for retaliation without punishment. Course riders, who may be accompanied by musicians riding in their own vehicle, might surround a person's front yard, dismount and begin a ritualistic song and dance. The silent penitence of Lent, however, follows the boisterous transgression of Mardi Gras. A masked ball, as described in Cajun Country, "marks the final hours of revelry before the beginning of Lent the next day. All festivities stop abruptly at midnight, and many of Tuesday's rowdiest riders can be found on their knees receiving the penitential ashes on their fore-heads on Wednesday." Good Friday, which signals the approaching end of Lent, is celebrated with a traditional procession called "Way of the Cross" between the towns of Catahoula and St. Martinville. The stations of the cross, which usually hang on the walls of a church, are mounted on large oak trees between the two towns. On Christmas Eve, bonfires dot the levees along the Mississippi River between New Orleans and Baton Rouge. This celebration, according to Cajun Country, has European roots: "The huge bonfires ... are descendants of the bonfires lit by ancient European civilizations, particularly along the Rhine and Seine rivers, to encourage and reinforce the sun at the winter solstice, its 'weakest' moment." Other holidays are uniquely Cajun and reflect the Catholic church's involvement in harvests. Priests bless the fields of sugar cane and the fleets of decorated shrimp boats by reciting prayers and sprinkling holy water upon them. Professional doctors were rare in rural Louisiana and only the most serious of conditions were treated by them. Although the expense of professional medical care was prohibitive even when it was available, rural Cajuns preferred to use folk cures and administered them themselves, or relied on someone adept at such cures. These healers, who did not make their living from curing other Cajuns, were called traiteurs, or treaters, and were found in every community. They also believed that folk practitioners, unlike their professional counterparts, dealt with the spiritual and emotional—not just the physiological— needs of the individual. Each traiteur typically specializes in only a few types of treatment and has his or her own cures, which may involve the laying-on of hands or making the sign of the cross and reciting of prayers drawn from passages of the Bible. Of their practices—some of which have been legitimated today as holistic medicine—some are pre-Christian, some Christian, and some modern. Residual pre-Christian traditions include roles of the full moon in healing, and left-handedness of the treaters themselves. Christian components of Cajun healing draw on faith by making use of Catholic prayers, candles, prayer beads, and crosses. Cajuns' herbal medicine derives from post-medieval French homeopathic medicine. A more recent category of Cajun cures consists of patent medicines and certain other commercial products. Some Cajun cures were learned from Indians, such as the application of a poultice of chewing tobacco on bee stings, snakebites, boils, and headaches. Other cures came from French doctors or folk cures, such as treating stomach pains by putting a warm plate on the stomach, treating ring-worm with vinegar, and treating headaches with a treater's prayers. Some Cajun cures are unique to Louisiana: for example, holding an infection over a burning cane reed, or putting a necklace of garlic on a baby with worms. Cajuns have a higher-than-average incidence of cystic fibrosis, muscular dystrophy, albinism, and other inherited, recessive disorders, perhaps due to intermarriage with relatives who have recessive genes in common. Other problems, generally attributed to a high-fat diet and inadequate medical care, include diabetes, hypertension (high blood pressure), obesity, stroke, and heart disease. Cajun French, for the most part, is a spoken, unwritten language filled with colloquialisms and slang. Although the French spoken by Cajuns in different parts of Louisiana varies little, it differs from the standard French of Paris as well as the French of Quebec; it also differs from the French of both white and black Creoles. Cajun French-speakers hold their lips more loosely than do the Parisians. They tend to shorten phrases, words, and names, and to simplify some verb conjugations. Nicknames are ubiquitous, such as " 'tit joe" or " 'tit black," where " 'tit " is slang for " petite " or "little." Cajun French simplifies the tenses of verbs by making them more regular. It forms the present participle of verbs—e.g., "is singing"— in a way that would translate directly as "is after to sing." So, "Marie is singing," in Cajun French is " Marie est apres chanter. " Another distinguishing feature of Cajun French is that it retains nautical usages, which reflects the history of Acadians as boaters. For example, the word for tying a shoelace is amerrer (to moor [a boat]), and the phrase for making a U-turn in a car is virer de bord (to come about [with a sailboat]). Generally, Cajun French shows the influence of its specific history in Louisiana and Acadia/Nova Scotia, as well as its roots in coastal France. Since Brittany, in northern coastal France, is heavily Celtic, Cajun French bears "grammatical and other linguistic evidences of Celtic influence." Some scattered Indian words survive in Cajun French, such as "bayou," which came from the Muskhogean Indian word, " bay-uk, " through Cajun French, and into English. Louisiana, which had already made school attendance compulsory, implemented a law in the 1920s that constitutionally forbade the speaking of French in public schools and on school grounds. The state expected Cajuns to come to school and to leave their language at home. This attempt to assimilate the Cajuns met with some success; young Cajuns appeared to be losing their language. In an attempt to redress this situation, the Council for the Development of French in Louisiana (CODOFIL) recently reintroduced French into many Louisianan schools. However, the French is the standard French of Parisians, not that of Cajuns. Although French is generally not spoken by the younger generation in Maine, New England schools are beginning to emphasize it and efforts to repeal the law that made English the sole language in Maine schools have been successful. In addition, secondary schools have begun to offer classes in Acadian and French history. In 1976, Revon Reed wrote in a mix of Cajun and standard French for his book about Cajun Louisiana, Lâche pas la patate, which translates as, "Don't drop the potato" (a Cajun idiom for "Don't neglect to pass on the tradition"). Anthologies of stories and series of other writings have been published in the wake of Reed's book. However, Cajun French was essentially a spoken language until the publication of Randall Whatley's Cajun French textbook ( Conversational Cajun French I [Baton Rouge: Louisiana State University Press, 1978]). In the oilfields, on fishing boats, and other places where Cajuns work together, though, they have continued to speak Cajun French. Storytellers, joke tellers, and singers use Cajun French for its expressiveness, and for its value as in-group communication. Cajun politicians and businessmen find it useful to identify themselves as fellow insiders to Cajun constituents and patrons by speaking their language. Cajuns learned to rely on their families and communities when they had little else. Traditionally they have lived close to their families and villages. Daily visits were usual, as were frequent parties and dances, including the traditional Cajun house-party called the fais-dodo, which is Cajun baby talk for "go to sleep," as in "put all the small kids in a back bedroom to sleep" during the party. Traditionally, almost everyone who would come to a party would be a neighbor from the same community or a family member. Cajuns of all ages and abilities participated in music-making and dancing since almost everyone was a dancer or a player. In the 1970s, 76 percent of the surnames accounted for 86 percent of all Cajuns; each of those surnames reflected an extended family which functioned historically as a Cajun subcommunity. In addition to socializing together, a community gathered to do a job for someone in need, such as building a house or harvesting a field. Members of Cajun communities traditionally took turns butchering animals and distributing shares of the meat. Although boucheries were essentially social events, they were a useful way to get fresh meat to participating families. Today, boucheries are unnecessary because of modern refrigeration methods and the advent of supermarkets, but a few families still hold boucheries for the fun of it, and a few local festivals feature boucheries as a folk craft. This cooperation, called coups de main (literally, "strokes of the hand"), was especially crucial in the era before worker's compensation, welfare, social security, and the like. Today such cooperation is still important, notably for the way it binds together members of a community. A challenge to a group's cohesiveness, however, was infighting. Fighting could divide a community, yet, on the other hand, as a spectator sport, it brought communities together for an activity. The bataille au mouchoir, as described in Cajun Country, was a ritualized fight "in which the challenger offered his opponent a corner of his handkerchief and the two went at each other with fists or knives, each holding a corner, until one gave up." Organized bare-knuckle fights persisted at least until the late 1960s. More recently, many Cajuns have joined boxing teams. Neighboring communities maintain rivalries in which violence has historically been common. A practice called casser le bal ("breaking up the dance") or prendre la place ("taking over the place") involved gangs starting fights with others or among themselves with the purpose of ending a dance. Threats of violence and other difficulties of travel hardly kept Cajuns at home, though. According to Cajun Country, "As late as 1932, Saturday night dances were attended by families within a radius of fifty miles, despite the fact that less than a third of the families owned automobiles at that time." Traditionally, Cajun family relations are important to all family members. Cajun fathers, uncles, and grandfathers join mothers, aunts, and grandmothers in raising children; and children participate in family matters. Godfathering and godmothering are still very important in Cajun country. Even non-French-speaking youth usually refer to their godparents as parrain and marraine, and consider them family. Nevertheless, traditionally it has been the mother who has transmitted values and culture to the children. Cajuns have often devalued formal education, viewing it as a function of the Catholic church—not the state. Families needed children's labor; and, until the oil boom, few jobs awaited educated Cajuns. During the 1920s many Cajuns attended school not only because law required it and jobs awaited them, but also because an agricultural slump meant that farming was less successful then. Although today Cajuns tend to date like other Americans, historically, pre-modern traditions were the rule. Females usually married before the age of 20 or risked being considered "an old maid." A young girl required a chaperon—usually a parent or an older brother or uncle, to protect her honor and prevent premarital pregnancy, which could result in banishment until her marriage. If a courtship seemed to be indefinitely prolonged, the suitor might receive an envelope from his intended containing a coat, which signified that the engagement was over. Proposals were formally made on Thursday evenings to the parents, rather than to the fiancee herself. Couples who wanted to marry did not make the final decision; rather, this often required the approval of the entire extended family. Because Cajuns traditionally marry within their own community where a high proportion of residents are related to one another, marriages between cousins are not unusual. Pairs of siblings frequently married pairs of siblings from another family. Although forbidden by law, first-cousin marriages have occurred as well. Financial concerns influenced such a choice because intermarriage kept property within family groupings. One result of such marriages is that a single town might be dominated by a handful of surnames. Cajun marriage customs are frequently similar to those of other Europeans. Customarily, older unmarried siblings may be required to dance barefoot, often in a tub, at the reception or wedding dance. This may be to remind them of the poverty awaiting them in old age if they do not begin families of their own. Guests contribute to the new household by pinning money to the bride's veil in exchange for a dance with her or a kiss. Before the wedding dance is over, the bride will often be wearing a headdress of money. Today, wedding guests have extended this practice to the groom as well, covering his suit jacket with bills. One rural custom involved holding the wedding reception in a commercial dance hall and giving the entrance fees to the newlyweds. Another Cajun wedding custom, "flocking the bride," involved the community's women bringing a young chick from each of their flocks so that the new bride could start her own brood. These gifts helped a bride establish a small measure of independence, in that wives could could sell their surplus eggs for extra money over which their husbands had no control. Roman Catholicism is a major element of Cajun culture and history. Some pre-Christian traditions seem to influence or reside in Cajun Catholicism. Historians partly account for Cajun Catholicism's variation from Rome's edicts by noting that historically Acadians often lacked contact with orthodox clergymen. Baptism of Cajun children occurs in infancy. Cajun homes often feature altars, or shrines with lawn statues, such as those of Our Lady of the Assumption—whom Pope Pius XI in 1938 declared the patroness of Acadians worldwide—in homemade grottoes made of pieces of bathtubs or oil drums. Some Cajun communal customs also revolve around Catholicism. For decades, it was customary for men to race their horses around the church during the sermon. Wakes call for mourners to keep company with each other around the deceased so that the body is never left alone. Restaurants and school cafeterias cater to Cajuns by providing alternatives to meat for south Louisiana's predominantly Catholic students during Ash Wednesday and Lenten Fridays. Some uniquely Cajun beliefs surround their Catholicism. For example, legends say that "the Virgin will slap children who whistle at the dinner table;" another taboo forbids any digging on Good Friday, which is, on the other hand, believed to be the best day to plant parsley. Coastal Louisiana is home to one of America's most extensive wetlands in which trapping and hunting have been important occupations. In the 1910s extensive alligator hunting allowed huge increases in rat musquè (muskrat) populations. Muskrat over-grazing promoted marsh erosion. At first the muskrats were trapped mainly to reduce their numbers, but cheap Louisiana muskrat pelts hastened New York's capture of America's fur industry from St. Louis, and spurred the rage for muskrat and raccoon coats that typified the 1920s. Cajuns helped Louisiana achieve its long-standing reputation as America's primary fur producer. Since the 1960s, Cajuns in the fur business have raised mostly nutria. The original Acadians and Cajuns were farmers, herders, and ranchers, but they also worked as carpenters, coopers, blacksmiths, fishermen, shipbuilders, trappers, and sealers. They learned trapping, trading, and other skills for survival from regional Indians. Industrialization has not ended such traditions. Workers in oil fields and on oil rigs have schedules whereby they work for one or two weeks and are then off work for the same amount of time, which allows them time to pursue traditional occupations like trapping and fishing. Because present-day laws ban commercial hunting, this activity has remained a recreation, but an intensely popular one. Louisiana is located at the southern end of one of the world's major flyways, providing an abundance of migratory birds like dove, woodcock, and a wide variety of ducks and geese. A wide range of folk practice is associated with hunting—how to build blinds, how to call game, how to handle, call and drive packs of hunting dogs, and how to make decoys. Cajun custom holds that if you hunt or fish a certain area, you have the clear-cut folk right to defend it from trespassers. Shooting a trespasser is "trapper's justice." Certain animals are always illegal to hunt, and some others are illegal to hunt during their off-season. Cajuns sometimes circumvent restrictions on hunting illegal game, which is a practice called "outlawing." According to some claims, the modern American cattle industry began on the Cajun prairie almost a full century before Anglo-Americans even began to move to Texas. Learning from the Spanish and the Indians, Cajuns and black Creoles were among the first cowboys in America, and they took part in some of this country's earliest cattle drives. Cattle rearing remains part of prairie Cajun life today, but the spread of agriculture, especially rice, has reduced both its economic importance and much of its flamboyant ways. In the nonagricultural coastal marshes, however, much of the old-style of cattle rearing remains. Cajuns catch a large proportion of American seafood. In addition to catching their own food, many Cajuns are employees of shrimp companies, which own both boats and factories, with their own brand name. Some fisherman and froggers catch large catfish, turtles, and bullfrogs by hand, thus preserving an ancient art. And families frequently go crawfishing together in the spring. The gathering and curing of Spanish moss, which was widely employed for stuffing of mattresses and automobile seats until after World War II, was an industry found only in the area. Cajun fishermen invented or modified numerous devices: nets and seines, crab traps, shrimp boxes, bait boxes, trotlines, and frog grabs. Moss picking, once an important part-time occupation for many wetlands Cajuns, faded with the loss of the natural resource and changes in technology. Dried moss was replaced by synthetic materials used in stuffing car seats and furniture. Now there is a mild resurgence in the tradition as moss is making a comeback from the virus which once threatened it and as catfish and crawfish farmers have found that it makes a perfect breeding nest. Cajuns learned to be economically self-reliant, if not completely self-sufficient. They learned many of southern Louisiana's ways from local Indians, who taught them about native edible foods and the cultivation of a variety of melons, gourds, and root crops. The French and black Creoles taught the Cajuns how to grow cotton, sugarcane, and okra; they learned rice and soybean production from Anglo-Americans. As a result, Cajuns were able to establish small farms and produce an array of various vegetables and livestock. Such crops also provided the cash they needed to buy such items as coffee, flour, salt, and tobacco, in addition to cloth and farming tools. A result of such Cajun agricultural success is that today Cajuns and Creoles alike still earn their livelihood by farming. Cajuns traded with whomever they wanted to trade, regardless of legal restrictions. Soon after their arrival in Louisiana, they were directed by the administration to sell their excess crops to the government. Many Cajuns became bootleggers. One of their proudest historical roles was assisting the pirate-smuggler Jean Lafitte in an early and successful smuggling operation. In the twentieth century, the Cajuns' trading system has declined as many Cajuns work for wages in the oil industry. In the view of some Cajuns, moreover, outside oilmen from Texas—or "Takesus"—have been depriving them of control over their own region's resource, by taking it literally out from under them and reaping the profits. Some Cajun traders have capitalized on economic change by selling what resources they can control to outside markets: for example, fur trappers have done so, as have fishermen, and farmers such as those who sell their rice to the Budweiser brewery in Houston. Cajuns, many of whom are conservative Democrats today, have been involved at all levels of Louisiana politics. Louisiana's first elected governor, as well as the state's first Cajun governor, was Alexander Mouton, who took office in 1843. Yet perhaps the most well known of Louisiana's politicians is Cajun governor Edwin Edwards (1927-), who served for four terms in that office—the first French-speaking Catholic to do so in almost half a century. In recent decades, more Cajuns have entered electoral politics to regain some control from powerful oil companies. Historically, Cajuns have been drafted and named for symbolic roles in pivotal fights over North America. In the mid-1700s in Acadia/Nova Scotia, when the French colonial army drafted Acadians, they weakened the Acadians' identity to the British as "French Neutrals," and prompted the British to try to expel all Acadians from the region. In 1778, when France joined the American Revolutionary War against the British, the Marquis de Lafayette declared that the plight of the Acadians helped bring the French into the fight. The following year, 600 Cajun volunteers joined Galvez and fought the British. In 1815, Cajuns joined Andrew Jackson in preventing the British from retaking the United States. Cajuns were also active in the American Civil War; General Alfred Mouton (1829–1864), the son of Alexander Mouton, commanded the Eighteenth Louisiana Regiment in the Battle of Pittsburgh Landing (1862), the Battle of Shiloh (1863), and the Battle of Mansfield (1864), where he was killed by a sniper's bullet. Thomas J. Arceneaux, who was Dean Emeritus of the College of Agriculture at the University of Southwestern Louisiana, conducted extensive research in weed control, training numerous Cajun rice and cattle farmers in the process. A descendent of Louis Arceneaux, who was the model for the hero in Longfellow's Evangeline, Arceneaux also designed the Louisiana Cajun flag. Tulane University of Louisiana professor Alcè Fortier was Louisiana's first folklore scholar and one of the founders of the American Folklore Society (AFS). Author of Lâche pas la patate (1976), a book describing Cajun Louisiana life, Revon Reed has also launched a small Cajun newspaper called Mamou Prairie. Lulu Olivier's traveling "Acadian Exhibit" of Cajun weaving led to the founding of the Council for the Development of French in Louisiana (CODOFIL), and generally fostered Cajun cultural pride. Chef Paul Prudhomme's name graces a line of Cajun-style supermarket food, "Chef Paul's." Dewey Balfa (1927– ), Gladius Thibodeaux, and Louis Vinesse Lejeune performed at the 1964 Newport Folk Festival and inspired a renewed pride in Cajun music. Dennis McGee performed and recorded regularly with black Creole accordionist and singer Amèdè Ardoin in the 1920s and 1930s; together they improvised much of what was to become the core repertoire of Cajun music. Cajun jockeys Kent Desormeaux and Eddie Delahoussaye became famous, as did Ron Guidry, the fastballer who led the New York Yankees to win the 1978 World Series, and that year won the Cy Young Award for his pitching. Guidry's nicknames were "Louisiana Lightnin "' and "The Ragin' Cajun." Formerly The Morning Star, it was founded in 1954 and is primarily a religious monthly. Contact: Barbara Gutierrez, Editor. Address: 1408 Carmel Avenue, Lafayette, Louisiana 70501-5215. Telephone: (318) 261-5511. Fax: (318) 261-5603. Acadian Genealogy Exchange. Devoted to Acadians, French Canadian families sent into exile in 1755. Carries family genealogies, historical notes, cemetery lists, census records, and church and civil registers. Recurring features include inquiries and answers, book reviews, and news of research. Contact: Janet B. Jehn. Address: 863 Wayman Branch Road, Covington, Kentucky 41015. Telephone: (606) 356-9825. Published by the Acadian News Agency since 1969, this is a magazine for bilingual Louisiana. Contact: Trent Angers, Editor. Address: Acadian House Publishing, Inc., Box 52247, Oil Center Station, Lafayette, Louisiana 70505. Telephone: (800) 200-7919. Cajun Country Guide. Covers Cajun and Zydeco dance halls, Creole and Caju restaurants, swamp tours, and other sites in the southern Louisiana region. Contact: Macon Fry or Julie Posner, Editors. Address: Pelican Publishing Co., 1101 Monroe Street, P.O. Box 3110, Gretna, Louisiana 70054. Telephone: (504) 368-1175; or, (800) 843-1724. Fax: (504) 368-1195. Mamou Acadian Press. Founded in 1955, publishes weekly. Contact: Bernice Ardion, Editor. Address: P.O. Box 360, Mamou, Louisiana 70554. Telephone: (318) 363-3939. Fax: (318) 363-2841. Rayne Acadian Tribune. A newspaper with a Democratic orientation; founded in 1894. Contact: Steven Bandy, Editor. Address: 108 North Adams Avenue, P.O. Box 260, Rayne, Louisiana 70578. Telephone: (318) 334-3186. Fax: (318) 334-2069. The Times of Acadiana. Weekly newspaper covering politics, lifestyle, entertainment, and general news with a circulation of 32,000; founded in 1980. Contact: James Edmonds, Editor. Address: 201 Jefferson Street, P.O. Box 3528, Lafayette, Louisiana 70502. Telephone: (318) 237-3560. Fax: (318) 233-7484. This station, which has a country format, plays "Cajun and Zydeco Music" from 6:00 a.m. to 9:00 a.m. on Saturdays. Contact: Johnny Bordelon, Station Manager. Address: 100 Chester, Box 7, Marksville, Louisiana 71351. Telephone: (318) 253-5272. Country, ethnic, and French-language format. Contact: Paul J. Cook. Address: P.O. Box 847, Morgan City, Louisiana 70381. Telephone: (504) 395-2853. KJEF-AM (1290), FM (92.9). Country, ethnic, and French-language format. Contact: Bill Bailey, General Manager. Address: 122 North Market Street, Jennings, Louisiana 70545. Telephone: (318) 824-2934. Fax: (318) 824-1384. Country, ethnic, and French-language format. Contact: Paul J. Cook. Address: P.O. Box 847, Morgan City, Louisiana 70380. Telephone: (504) 395-2853. Fax: (504) 395-5094. Contact: Garland Bernard, General Manager. Address: Highway 167 North, Box 610, Abbeville, Louisiana 70511-0610. Telephone: (318) 893-2531. Fax: (318) 893-2569. National Public Radio; features bilingual newscasts, Cajun and Zydeco music, and Acadian cultural programs. Contact: Dave Spizale, General Manager. Address: P.O. Box 42171, Lafayette, Louisiana 70504. Telephone: (318) 482-6991. KVOL-AM (1330), FM (105.9). Blues, ethnic format. Contact: Roger Cavaness, General Manager. Address: 202 Galbert Road, Lafayette, Louisiana 70506. Telephone: (318) 233-1330. Fax: (318) 237-7733. Country, ethnic, and French-language format. Contact: Jim Soileau, General Manager. Address: 809 West LaSalle Street, P.O. Drawer J, Ville Platte, Louisiana 70586. Telephone: (318) 363-2124. Fax: (318) 363-3574. Acadian Cultural Society. Dedicated to helping Acadian Americans better understand their history, culture, and heritage. Founded in 1985; publishes quarterly magazine Le Reveil Acadien. Contact: P. A. Cyr, President. Address: P.O. Box 2304, Fitchburg, Massachusetts 01420-8804. Telephone: (978) 342-7173. Those interested in maintaining links among individuals of Acadian descent and their relatives in New England. Conducts seminars and workshops on Acadian history, culture, and traditions. Contact: Richard L. Fortin. Address: P.O. Box 556, Manchester, New Hampshire 03105. Telephone: (603) 641-3450 The Center for Acadian and Creole Folklore. Located at the University of Southwestern Louisiana ( Universitè des Acadiens ), the center organizes festivals, special performances, and television and radio programs; it offers classes and workshops through the French and Francophone Studies Program; it also sponsors musicians as adjunct professors at the university. The Council for the Development of French in Louisiana (CODOFIL). A proponent of the standard French language, this council arranges visits, exchanges, scholarships, and conferences; it also publishes a free bilingual newsletter. Address: Louisiane Française, Boite Postale 3936, Lafayette, Louisiana 70502. The International Relations Association of Acadiana (TIRAA). This private-sector economic development group funds various French Renaissance activities in Cajun country. The Madawaska Historical Society. Promotes local historical projects and celebrates events important in the history of Acadians in Maine. Visitors can see preservations and reconstructions of many nineteenth-century buildings at the Acadian Village and Vermilionville in Lafayette; the Louisiana State University, Rural Life Museum in Baton Rouge, and at the Village Historique Acadien at Caraquet. Researchers can find sources at Nichols State University Library in Thibodaux; at the Center for Acadian and Creole Folklore of the University of Southwestern Louisiana; and at the Center for Louisiana Studies at the University of Southwestern Louisiana. Offers on-site reference assistance to its Acadian archives, and to regional history, folklore and Acadian life. Contact: Lisa Ornstein, Director. Address: Univerity of Maine at Fort Kent, 25 Pleasant Street, Fort Kent, Maine 04743. Telephone: (207) 834-7535. Fax: (207) 834-7518. Ancelet, Barry, Jay D. Edwards, and Glen Pitre (with additional material by Carl Brasseaux, Fred B. Kniffen, Maida Bergeron, Janet Shoemaker, and Mathe Allain). Cajun Country. Jackson: University of Mississippi Press, 1991. Brasseaux, Carl. Founding of New Acadia, 1765-1803; In Search of Evangeline: Birth and Evolution of the Myth. Thibodaux, Louisiana: Blue Heron Press, 1988. The First Franco-Americans: New England Life Histories From the Federal Writers Project, 1938-1939, edited by C. Stewart Doty. Orono: University of Maine at Orono Press, 1985.<\|endoftext\|>	4.1875
1,563	The capacitor is the most common component in electronics and used in almost every electronics application. There are many types of capacitor available in the market for serving different purposes in any electronic circuit. They are available in many different values from 1 Pico-Farad to 1 Farad capacitor and Supercapacitor. Capacitor also have a different types of ratings, such as working voltage, working temperature, tolerance of the rated value and leakage current. The leakage current of capacitor is a crucial factor for the application, especially if used in Power electronics or Audio Electronics. Different types of capacitors provide different leakage current ratings. Apart from selecting the perfect capacitor with proper leakage, circuit should also have the ability to control the leakage current. So first we should have a clear understanding of capacitor leakage current. Relation with Dielectric Layer The leakage current of a capacitor has a direct relationship with the dielectric of the capacitor. Let's see the below image - The above image is an internal construction of the Aluminum Electrolytic Capacitor. An Aluminum Electrolytic Capacitor has few parts which are encapsulated in a compact tight packaging. The parts are Anode, Cathode, Electrolyte, Dielectric layer Insulator, etc. The dielectric insulator provides insulation of the conductive plate inside the capacitor. But as there is nothing perfect in this world, the insulator is not an ideal insulator and has an insulation tolerance. Due to this, a very low amount of current will flow through the insulator. This current is called as Leakage current. The Insulator and the flow of current can be demonstrated by using a simple capacitor and resistor. The resistor has a very high value of resistance, which can be identified as an insulator resistance and the capacitor is used to replicate the actual capacitor. Since the resistor has a very high value of resistance, the current flowing through the resistor is very low, typically in a number of nano-amperes. Insulation resistance is dependent on the type of dielectric insulator as different type of materials changes the leakage current. The low dielectric constant provides very good insulation resistance, resulting in a very low leakage current. For example, polypropylene, plastic or teflon type capacitors are the example of low dielectric constant. But for those capacitors, the capacitance is very less. Increasing the capacitance also increases the dielectric constant. Electrolytic capacitors typically have very high capacitance, and the leakage current is also high. Dependent Factors for Capacitor Leakage Current Capacitor Leakage Current generally depends on below four factors: - Dielectric Layer - Ambient Temperature - Storing Temperature - Applied Voltage Capacitor construction requires a chemical process. The dielectric material is the main separation between the conductive plates. As the dielectric is the main insulator, the leakage current has major dependencies with it. Therefore, if the dielectric is tempered during the manufacturing process, it will directly contribute to the increase of leakage current. Sometimes, the dielectric layers have impurities, resulting in a weakness in the layer. A weaker dielectric decreases the flow of current which is further contributed to the slow oxidation process. Not only this, but improper mechanical stress also contribute to the dielectric weakness in a capacitor. The capacitor has a rating of the working temperature. The working temperature can be ranged from 85 degree Celsius to the 125 degree Celsius or even more. As the capacitor is a chemically composed device, the temperature has a direct relationship with the chemical process inside the capacitor. The leakage current generally increases when the ambient temperature is high enough. Storing a capacitor for a long time without voltage is not good for the capacitor. The storing temperature is also a important factor for leakage current. When the capacitors are stored, the oxide layer is attacked by the electrolyte material. The oxide layer starts to dissolve in the electrolyte material. The chemical process is different for different type of electrolyte material. The water-based electrolyte is not stable whereas inert solvent-based electrolyte contributes less leakage current due to the reduction of the oxidation layer. However, this leakage current is temporary as the capacitor has self-healing properties when applied to a voltage. During the exposure to a voltage, the oxidation layer starts to regenerate. Each capacitor has a voltage rating. Therefore, using a capacitor above the rated voltage is a bad thing. If the voltage increases, the leakage current also increases. If the voltage across the capacitor is higher than the rated voltage, the chemical reaction inside a capacitor creates Gases and degrade the Electrolyte. If the capacitor is stored for a long time such as for years, the capacitor is needed to be restored into the working state by providing rated voltage for a few minutes. During this stage, the oxidation layer built up again and restores the capacitor in a functional stage. How to reduce Capacitor Leakage Current to improve the Capacitor Life As discussed above a capacitor has dependencies with many factors. The first question is how the capacitor life is calculated? The answer is by calculating the time until the electrolyte is run out. The electrolyte is consumed by the oxidation layer. Leakage current is the primary component for the measurement of how much the oxidation layer is hampered. Therefore, the reduction of leakage current in the capacitor is a major key component for the life of a capacitor. 1. Manufacturing or the production plant is the first place of a capacitor life cycle where capacitors are carefully manufactured for low leakage current. The precaution needs to be taken that the dielectric layer is not damaged or hampered. 2. The second stage is the storage. Capacitors need to be stored in proper temperature. Improper temperature affects the capacitor electrolyte which further downgrades the oxidation layer quality. Make sure to operate the capacitors in proper ambient temperature, less than the maximum value. 3. In the third stage, when the capacitor is soldered on the board, the soldering temperature is a key factor. Because for the electrolytic capacitors, the soldering temperature can become high enough, more than the boiling point of the capacitor. The soldering temperature affects the dielectric layers across the lead pins and weakens the oxidation layer resulting in high leakage current. To overcome this, each capacitor comes with a data sheet where the manufacturer provides a safe soldering temperature rating and maximum exposure time. One needs to be careful about those ratings for the safe operation of the respective capacitor. This is also applicable for the Surface Mount Device (SMD) capacitors too, the peak temperature of reflow soldering or wave soldering should not exceed than the maximum allowable rating. 4. As the voltage of the capacitor is an important factor, the capacitor voltage should not exceed the rated voltage. 5. Balancing the capacitor in Series connection. The capacitor series connection is a bit complex job to balance the leakage current. This is due to the imbalance of leakage current divide the voltage and split between the capacitors. The split voltage can be different for each capacitor and there can be a chance that the voltage across a particular capacitor could be excess than the rated voltage and the capacitor start to malfunction. To overcome this situation, two high-value resistors are added across the individual capacitor to reduce the leakage current. In the below image, the balancing technique is shown where two capacitors in series are balanced using high-value resistors. By using the balancing technique, the voltage difference influenced by leakage current can be controlled.<\|endoftext\|>	3.828125
1,156	// Numbas version: exam_results_page_options {"name": "Factorise the quadratic expression - Refresh", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "preventleave": false, "showfrontpage": false}, "question_groups": [{"questions": [{"preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a", "b", "c", "d"], "rulesets": {}, "extensions": [], "metadata": {"description": " Testing factorisation of quadratics. ", "licence": "Creative Commons Attribution 4.0 International"}, "advice": " The first step is to find two numbers that add together to give $\\var{ad+cb}$ and multiply to give $\\var{ac} \\times \\var{bd} = \\var{abcd}$. \n \n These are $\\var{ad}$ and $\\var{cb}$. \n \n We split the $x$-term using $\\simplify{{a + b}x = {a} x + {b} x}$ and work as follows: \n \\\begin{aligned}\\simplify{{ac}x^2 + {ad + cb} x + {bd}} &= \\simplify{{ac}x^2 + {ad} x + {cb} x + {cd}} \\\\ &= \\simplify{{a} x({c} x + {d}) + {b}({c}x + {d})} \\\\ &= \\simplify{({a}x + {b})({c}x + {d})}\\end{aligned}\ ", "variable_groups": [], "variablesTest": {"condition": "(b/a <> d/c)\nand \n(a <> 0)\nand\n(b <> 0)\nand\n(c <> 0)\nand\n(d <> 0)\nand\n(gcd(a,b) = 1)\nand\n(gcd(c,d) = 1)", "maxRuns": 100}, "tags": [], "variables": {"b": {"definition": "random(-5 .. 5#1)", "name": "b", "group": "Ungrouped variables", "description": " constant in first factor ", "templateType": "randrange"}, "c": {"definition": "random(-5 .. 5#1)", "name": "c", "group": "Ungrouped variables", "description": " x-coefficient in second factor ", "templateType": "randrange"}, "d": {"definition": "random(-5 .. 5#1)", "name": "d", "group": "Ungrouped variables", "description": " constant in second factor ", "templateType": "randrange"}, "a": {"definition": "random(-5 .. 5#1)", "name": "a", "group": "Ungrouped variables", "description": " x-coefficient in first factor ", "templateType": "randrange"}}, "statement": " Factorise the following quadratic. ", "parts": [{"valuegenerators": [{"name": "x", "value": ""}], "adaptiveMarkingPenalty": 0, "scripts": {}, "marks": "2", "prompt": " $\\simplify{{ac}x^2 + {ad + cb}x + {b*d}}$ ", "checkVariableNames": false, "variableReplacements": [], "useCustomName": false, "musthave": {"message": "", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "extendBaseMarkingAlgorithm": true, "showPreview": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "type": "jme", "customName": "", "failureRate": 1, "unitTests": [], "checkingType": "absdiff", "answer": "({a}x+{b})({c}x+{d})", "vsetRangePoints": 5, "showFeedbackIcon": true, "vsetRange": [0, 1], "notallowed": {"message": "", "showStrings": false, "strings": ["^"], "partialCredit": 0}}], "name": "Factorise the quadratic expression - Refresh", "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1605/", "name": "Andrew Stacey"}, {"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/", "name": "Xiaodan Leng"}, {"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3492/", "name": "Maria Pickett"}], "functions": {}}], "pickingStrategy": "all-ordered"}], "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1605/", "name": "Andrew Stacey"}, {"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/", "name": "Xiaodan Leng"}, {"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3492/", "name": "Maria Pickett"}]}<\|endoftext\|>	4.40625
2,492	In this article, we will learn how to find the highest power of a number in a factorial. We will look at the three different variations of questions based on this concept that you can come across on the GMAT. So, let us get started. The first variety of question on this concept is – ## 1. How to find the highest power of a prime number in a factorial. Let us take an example to understand this. Say, we need to find the highest power of 3 in 20! In the exam, they can ask you this question in two ways: ### Question 1. A If 20! contains 3k, where k is a positive integer, what is the highest value of k? Or they can ask the question as show below: ### Question 1.B What is the highest power of 3 in 20!? The solution is the same for either of the above questions and there are two ways to solve it. We will first solve it using method 1 which is Brute Force method, where we simply count the number of 3s. We’ll then analyse the advantages and disadvantages of this method and then move to a better method (method 2). ## Solution ### Method 1 • We need to find the highest power of 3 in 20! #### Step 1 • Firstly, we will jot down all the multiples of 3 which are less or equal to 20. • Multiples of 3 which are less than or equal to 20 are 3, 6, 9, 12, 15, and 18. #### Step 2 • We will prime factorize the multiples of 3 to get the greatest power of 3 in each of them. So, • 3 = 31 • 6 = 2131 • 9 = 32 • 12 = 2231 • 15 = 3151 • 18 = 2132 #### Step 3 • We will add up all the highest powers of 3 obtained from each of its multiple. • So, the highest power of 3 in 20! = 1 + 1+ 2+ 1+ 1+ 2 = 8 • And thus, k = 8 In this question, it was easy to find the highest power of 3, using method 1, because the factorial value was small. However, this method becomes tedious when the factorial numbers are high. For example, instead of 20! If we had 200! then solving the question using the above method would have taken considerable time. So, we’ll use another method to solve this type of questions and we would recommend you use the same. ### Method 2 • In this method, we take number whose factorial is given. And we keep on dividing it by the powers of the prime number whose highest power we are looking for. On each division, we are simply looking for the quotient. • The highest power of 3 in 20! = $(\frac{20}{3^1})_Q + (\frac{20}{3^2})_Q$ = 6 + 2 = 8 • Here, ( )_Q denotes the quotient of the division operation. • We know that 20 = 6 * 3 + 2, where 6 is the quotient and 2 is the remainder. So, we’ll just take the quotient 6. • Similarly, 20 = 2 * 9 + 2. In this case the quotient is 2, so we’ll just take that. • We’ll continue dividing the factorial number until the quotient, becomes 0. • Like in the above case the quotient of 20/33 is 0. So, we are not finding it and stopping before that only. • Here’s another example, let’s say we need to find the highest power of 3 in 60! • Then, the highest power of 3 in 60! = $(\frac{60}{3^1}_Q) + (\frac{60}{3^2})_Q + (\frac{60}{3^3}_Q)$= 20 + 6 + 2 = 28 • Notice, we have not considered $(\frac{60}{3^4}_Q)$ and terms involving other higher powers of 3 in the denominator, because in all those cases the quotient is 0. • Once you get all the quotients, as you can see, we just need to add up all the quotients and we’ll have the highest power of that prime number in that factorial. Now, let us look at the second variety of questions. ## 2. How to find the highest power of a power of prime number in afactorial In the last section, we learned how to find the highest power of a prime number in a factorial. In this section, we will extend the same concept to find the highest power of a power of prime number( i.e. a number in the form of $p^q$, where p is a prime number and q is a positive integer greater than 1) in a factorial. Let us understand with an example. ### Question 2 What is the highest power of 8 in 70! ? #### Common Mistake: On seeing this question, a lot of students follow the approach shown below: • The highest power of 8 in 70! = $(\frac {70}{8^1})_Q + (\frac {70}{8^2})_Q$ = 8 + 1 = 9 However, this is INCORRECT, because 8 is not a prime number and we cannot directly divide by a non-prime number to find the highest power of it in a factorial. The following section explains the correct step-by-step procedure of solving such type of questions. ## Solution #### Step 1 • Prime factorize the given number to find the prime factor and its highest power. • Now, prime factorization of 8 = $2^3$ • Prime factor of 8 = 2 • And the highest power of its prime factor (i.e. 2) in 8 = 3 #### Step 2 • Find the highest power of the prime factor of the given number in the given factorial • So, we’ll first find the highest power of 2 in case of 70! • So, the highest power of 2 in 70! = $(\frac {70}{2^1})_Q + (\frac {70}{2^2})_Q + (\frac {70}{2^3})_Q + (\frac {70}{2^4})_Q + (\frac {70}{2^5})_Q + (\frac {70}{2^6})_Q$= 35 +17+8 + 4 +2 +1= 67 #### Step 3 • Now that we know the highest power of 2, can be written as 267. But we need the highest power of 8 and we know that we need three twos to make an 8 (since 8 = 23) • So, we’ll simply divide 67 by 3 in this case and see how many 8’s we can make. In this division, all we need to do is to take the quotient. • So, the highest power of 8 in 70! = $(\frac {67}{3})_Q = 22$ So, one major learning here is that don’t divide the factorial by a non-prime number. First break the number down into its prime factorized form and then find the highest power of that number. And after that figure out what will be the highest power of that non-prime number. Till now, we have seen how to find the highest power of a prime number or a power of a prime number, in the next section, we will see how to find the highest power of a number that has two distinct prime number. Recommended: 3 Important Properties of Prime Numbers ## 3. How to find the highest power of a number that has two distinct prime numbers Finding the highest power in this case is only a little bit different from the last section as instead of one prime factor there will be more than one prime factor i.e. the number will be in the form of $p_1 * p_2$, where $p_1$ and $p_2$ are prime numbers. So, with the help of an example, let us understand how to solve this type of question. ### Question 3 What is the highest power of 10 in 100!? ## Solution #### Step 1 • Prime factorize the given number. • So prime factorization of 10 = 2*5 So, now we know that to make one 10 we need one 2 and one 5. So, in our last step, let’s see how many 10s we can make in 100! #### Step 2 • The highest power of 2 is 100! = $(\frac {100}{2^1})_Q + (\frac {100}{2^2})_Q + (\frac {100}{2^3})_Q + (\frac {100}{2^4})_Q + (\frac {100}{2^5})_Q + (\frac {100}{2^6})_Q$ = 50 + 25 + 12 + 6 + 3 +1= 97 • And, the highest power of 5 in 100! = $(\frac {100}{5^1})_Q + (\frac {100}{5^2})_Q$ = 20 + 4 = 24 • Hence, the highest power of 2 in 100! is 97 i.e. 100! contains 97 twos or $2^{97}$ and the highest power of 5 in 100! is 24 i.e. 100! Contains 24 fives or $5^{24}$. • We know that we need one 2 and one 5 to make one 10. • So, the highest number of 10s that we can make from 97 twos $(2^{97})$ and 24 fives $(5^{24})$ is 24. • That is, the highest power of 10 in 100! = 24 • If you have noticed the highest power of 10 in 100! is equal to the highest power of 5. This is because the factorial of any number is always going to have more 2s than 5s as 5 > 2 and we need equal number of 2 and 5 to make one 10. So, if you are able to make this inference beforehand, you can save time by finding the highest power of only one number instead of two numbers. • Therefore, only find the highest power of the greatest prime factor in the factorial. • So, the highest power of 10 in 100! = the highest power of 5 in 100! = $(\frac{100}{5^1})_Q + (\frac {100}{5^2})_Q$ = 20 + 4 = 24 ## Takeaway • In questions where you have to find the highest power in a factorial, • If the number, whose factorial is given, is small, you can count manually to find the highest power. • If the number, whose factorial is given, is large, then use the division method. • While using the division method, keep the following points in mind: • Always prime factorize the number first. • After that decide, out of the above three, the question falls in which category. And then solve it accordingly to find the highest power of the given number.<\|endoftext\|>	4.90625
567	Q: # What are some tips for learning multiplication tables? A: Tips for learning multiplication tables include Skip Counting, which students can use to multiply any numbers. Students can learn two- and four-fact families using the Doubling and Double Double methods, respectively, while they can learn the nine-fact family through a trick called Handy-Nines. ## Keep Learning Students can memorize all multiplication tables through the use of Skip Counting, wherein the user determines the numbers that they should skip. For example, if the problem was 2 x 2, the first number indicates that the user should count every second number, while the second indicates that he should repeat the counting twice. When looking at numbers in sequence, apply the Skip Counting method to arrive at the answer that 2 x 2 is 4. The Doubling method helps students to concentrate on learning the two-fact family. To obtain the answer, double the number that is to be multiplied by 2. For the problem 2 x 3, the user would add 3 + 3 to achieve 6, which serves as the answer to both problems. Similarly, the Double Double method helps reinforce the four-fact family. In a problem such as 4 x 4, the user would first add 4 + 4 to equal 8, and then add 8 + 8 to achieve the final answer of 16. Handy-Nines visually teaches the nine-fact family. Place both hands side by side. Starting with the left pinkie, number the fingers by tens. For a problem such as 2 x 9, drop the left ring finger, as it is the second finger from left to right. Look to the left of the curled finger to find the 10s place; in this case, it is 10. Count the remaining fingers to the right of the curled finger to determine the ones place and reach the correct answer of 18. Sources: ## Related Questions • A: To perform partial product multiplication, you use the distributive property of numbers, multiplying each digit of a number by each digit of the other numb... Full Answer > Filed Under: • A: One can teach fourth-grade students how to estimate quotients by providing instruction on substituting the original problem with compatible numbers. These ... Full Answer > Filed Under: • A: Teaching methods that feature games, music, physical activity and art make multiplication fun for both students and teachers. To create new and engaging le... Full Answer > Filed Under: • A: The solution to a multiplication problem is called the "product." For example, the product of 2 and 3 is 6. When the word "product" appears in a mathematic... Full Answer > Filed Under: PEOPLE SEARCH FOR<\|endoftext\|>	4.4375
806	Bean plants are a favorite food of certain caterpillars. These pests eat the leaves and can, in severe infestations, defoliate the bean plants. Bean-eating caterpillars are larvae of butterflies, moths or beetles. Two types of bean-eating caterpillars can be identified by their black or dark-brown heads and greenish-yellow bodies. Both types are the larvae of a butterfly belonging to the skipper family. The caterpillar of the long-tailed skipper butterfly (Urbanus proteus) hatches in late summer. It starts out tiny, about a quarter-inch long, growing to more than 1 inch as it feeds on bean leaves. Its black head appears disproportionately large in relation to its striped, greenish-yellow body. The head has two orange spots directly behind the mandibles. This caterpillar cuts flaps from the edges of bean leaves, rolling them over and sticking the flaps down with silk to form a sheltering tent. This habit gives this pest the common name of bean leafroller. In its late life stages, the caterpillar sticks two leaves together to form a shelter where it will pupate. The long-tailed skipper inhabits the southeastern United States. The adult is a brown butterfly with five to seven white spots on the top side of each forewing. It has a wingspan of about two inches. Each hind wing has a tail-like extension trailing backward, which gives the species its common name. The caterpillars feed for about three weeks and pupate for about 10 days before emerging as adults. The adults feed on nectar until autumn sends them migrating south to Florida, Mexico and Central America where they spend the winter before returning north to lay eggs and start the life cycle over again. Silver Spotted Skipper Another dark-headed caterpillar that feeds on bean plants is the larva of the silver-spotted skipper butterfly (Epargyreus clarus). The caterpillars have plump greenish-yellow bodies with faint transverse stripes, dark brown heads that look almost black, and yellow eye patches near the mandibles. These caterpillars also roll leaves to make tents. When the caterpillar is small it only uses the edges of leaves, but when it is bigger -- the caterpillar can grow to 2 inches -- it uses an entire leaf for its tent. The caterpillars are also feed on the foliage, including a variety of cultivated and wild plants in the bean and pea family. The silver-spotted skipper ranges through states along the East, West and Gulf Coasts, the Great Lakes and the Great Plains. The adult butterflies are brown with a two-inch average wingspans. The forewings have a row of yellow-gold spots and dashes of white on the fringe. The hindwings have a large irregularly-shaped silvery-white patch and a rounded edge. The life cycle varies with the climate. In areas with mild or no winters, this species will go through multiple generations. In areas with severe winters, there may be only one generation until the insect develops a hardened pupa for spending the winter. Bean leafrollers can be controlled by hand-picking or with insecticides. For minor infestations, inspect your bean plants frequently. Pick off and destroy any caterpillars you see and crush their leaf tents. They won't bite or sting you. Also look for orange or cream-colored spherical eggs laid on the underside of the leaves and crush them. If yours is a large bean patch or you have a major infestation, you can spray or dust with a biological or chemical insecticide. Follow the application directions and cautions of the product you choose. When using any insecticide you should minimize direct contact, wash hands and change clothes after application. You should thoroughly wash treated produce before eating. And if you have a child in the house, keep insecticides under lock and key.<\|endoftext\|>	3.875
2,845	## 25 Liters to Gallons: Converting Your Tank Capacity When it comes to measuring liquids, different countries use different units. If you come across liters, but you’re more familiar with gallons, it’s important to know how to convert between the two. In this blog post, we’ll dive into the conversion from 25 liters to gallons, along with some other related questions. So, whether you’re wondering about the tank capacity of 25 liters or if 4 liters really does equal 1 gallon, keep reading to find out! ## Converting 25 Liters to Gallons: A Lighthearted Guide ### Breaking it Down: From Liters to Gallons So, you’ve found yourself in a situation where you need to convert 25 liters to gallons. Don’t worry, friend, I’ve got your back! Let’s dive right into the world of measurements, gallons, and liters – but fear not, we’re going to make this journey lighthearted and fun! ### The Basics: What’s the Difference? Before we begin the conversion extravaganza, let’s quickly refresh our memories on what exactly liters and gallons are. Liters are a metric unit of volume, commonly used in most parts of the world except for the United States (because why make things easy, right?). On the other hand, we have gallons, which are those beloved units of volume used by our friends across the pond – in the US, that is. ### Step 1: The Conversion Formula Alright, now that we know our units, let’s get to the nitty-gritty of converting 25 liters to gallons. The conversion formula is pretty straightforward: divide the number of liters by 3.7854. Why 3.7854, you ask? Well, that’s the conversion factor we’ll use to go from liters to gallons. It’s like magic, but without the sparkles. ### Step 2: The Math Magic Okay, let’s put our formula to work and convert those 25 liters into gallons. Buckle up, because this is where the real excitement begins. Ready? Drumroll, please! 25 liters divided by 3.7854 equals approximately 6.604 gallons. ### Step 3: The Result Ta-da! We did it! We’ve successfully converted 25 liters to gallons. Pat yourself on the back, my friend, because you are now a master of unit conversion. Our original 25 liters magically transformed into approximately 6.604 gallons. ### Wrap-Up: From Liters to Gallons and Beyond Now that you know how to convert liters to gallons, the sky’s the limit! Impress your friends, conquer unit conversion questions on trivia night, and navigate the international grocery store with ease. So whether you’re talking milk, gasoline, or swimming pool sizes, remember the simple formula, and you’ll always have gallons at your fingertips. That’s a wrap on our adventure through liters and gallons. I hope you found both entertainment and enlightenment in this lighthearted guide. Remember, my friend, knowledge is power, and now you’ve unlocked the power to convert liters to gallons! ## 25 Liters to Pounds: The Quirky Conversion Journey ### Converting Liters to Pounds: An Unexpected Detour So, you’ve mastered the art of converting liters to gallons, and now you’re seeking the next thrilling adventure in the land of conversions. Fear not, fellow traveler, for today we embark on a whimsical journey into the realm of liters to pounds conversion. Get ready for a bumpy yet entertaining ride as we uncover the quirky secrets behind this peculiar transformation. ### The Offbeat Connection: Liters and Pounds Before we dive headfirst into the conversion madness, let’s take a moment to appreciate the sheer peculiarity of connecting volumes with masses. Liters, those charming units used to quantify the capacity of our favorite beverages and cooking ingredients, are about to collide with the weighty pounds – a match made in conversion heaven or sheer madness? We shall soon find out! ### Wiggling Through the Conversion Maze As we venture into converting 25 liters to pounds, we might find ourselves slightly perplexed, wondering how liquids and mass can be so effortlessly intertwined. Well, hold on tight, my friend, because this journey promises surprises galore! Now, to begin our expedition, we must first understand that liters measure volume, while pounds measure weight. To convert between the two, we need a magical bridge called density. In simpler terms, density tells us how much mass is squeezed into a given volume. It’s like trying to fit an entire circus troupe into a small car – it’s all about packing efficiently! ### The Secret of the Density Dance To conquer the 25 liters to pounds challenge, we must uncover the density of the substance we’re dealing with. Different liquids have different densities, just like humans have different dance moves. Whether it’s the elegantly swaying olive oil or the slightly jiggling honey, each liquid has its own unique density. Once we have the density in pounds per liter, it’s time for the final showdown – multiplying that density by our original volume of 25 liters. This calculation will orchestrate a harmonious union between liters and pounds, resulting in a delightful numerical answer. ### Tip-toeing into the Final Conversion Now, you might be wondering which substances have the specific density needed to convert liters to pounds. Well, my curious friend, it varies. For water, the density is approximately 2.2046 pounds per liter. But if you’re juggling with other liquids or substances, their densities might differ. Researching the specific density of your desired substance will ensure a successful and accurate conversion. ### In Conclusion: Celebrating the Quirkiness Congratulations, brave souls! You have braved yet another conversion odyssey, this time venturing from liters to pounds. As we bid farewell to this fantastical realm of conversions, don’t forget to cherish the sheer quirkiness of connecting volumes and weights. May this newfound knowledge serve you well on your future journeys into the fascinating world of conversions! So, until we meet again, keep those conversions flowing, and never stop exploring the delightful eccentricities of the measurement universe. Happy converting, my fellow adventurers! P.S. Rumor has it that the gallons and pounds have a secret dance routine they perform together – but that’s a tale for another day! ## 25 Liters to Gallons: A Mind-Boggling Conversion! ### From 25 Liters to 6.60430131 US Gallons Are you ready for a dose of mind-boggling conversions? Buckle up, because we’re about to dive into the world of liters and gallons! Today, our quirky journey takes us from 25 liters to a mind-bending 6.60430131 US gallons. Get ready to have your mind blown! ### Let’s Get the Party Started! So, you’ve got 25 liters on one side of the conversion fence and a dizzying number like 6.60430131 US gallons on the other side. How on earth do we make sense of this madness? Fear not, fellow conversion explorers, for I shall be your guide on this hilarious adventure! ### Unraveling the Mystery To unravel this puzzling mystery, we need to understand the relationship between liters and gallons. Think of liters as the sleek, sophisticated cousins who prefer the metric system, while gallons are the funky, flamboyant relatives who groove to the beat of the imperial system. They’re so different, yet we’ll find a way to bring them together! ### The Conversion Dance To convert liters to gallons, we need the perfect dance moves. Picture liters as the party animals who can’t resist shaking their “L” hips to the left and right. In comparison, gallons are all about the smooth, swirling moves that take up a lot of space. But fear not, my witty friends, for with the right moves, we can find our way! ### The Conversion Magic So, you’ve got your 25 liters on the dancefloor. Now, imagine gathering them up into one of those iconic red Solo cups – like the ones you see at college parties. That lively gathering of liters amounts to a whopping 6.60430131 US gallons. Quite the party, huh? So be sure to sip your gallon-sized humorously! ### Conclusion: From One World to Another! In the world of conversions, going from 25 liters to 6.60430131 US gallons may seem like a leap of faith. But armed with a touch of humor and a little dance floor magic, you’ve successfully bridged the gap between two different measurement systems. Cheers to you, my conversion connoisseur! So there you have it, folks! The mind-bending journey from 25 liters to 6.60430131 US gallons has come to an end. We hope you enjoyed this whimsical adventure and gained a deeper appreciation for the wacky world of conversions. Until next time, keep rocking those party-sized conversions! ## Does 4 Liters Equal 1 Gallon? Let’s dive into the age-old question: does 4 liters really equal 1 gallon? Well, buckle up, because we’re about to take a joyride through the fascinating world of unit conversions. ### The Metric versus the Imperial Before we delve into conversions, let’s take a moment to appreciate the fact that the metric system and the imperial system just can’t seem to get along. The metric system, used by most of the world, relies on nice decimal-based units, making calculations oh-so-easy. On the other hand, we have the imperial system, which seems to enjoy throwing random numbers and measurements our way just to keep things interesting. Thanks, imperial system… ### Converting Liters to Gallons Now, back to the million-dollar question: 4 liters to gallons? The short answer is no, they’re not equal. In fact, 4 liters is roughly equivalent to 1.06 gallons. Yes, you read that right. Approximately. ### How did We Get Here? To fully appreciate this conversion, we have to understand the reasoning behind it. You see, the liter is a metric unit of volume, while the gallon is an imperial unit. They just can’t seem to see eye to eye. Maybe a trip to the therapist would help? Anyway, because the two systems are different, a liter will never be the same as a gallon. It’s like trying to compare apples to oranges, or cats to dogs – they’re just not meant to be the same. ### The Math Behind the Madness So, how did we arrive at the conversion factor of 1.06? Well, it all comes down to the magic of numbers. One gallon is equal to approximately 3.78541 liters. Impressive precision, isn’t it? Therefore, if we divide 4 liters by 3.78541, we get our delightful answer of 1.06 gallons. And voila! We’ve solved the mystery. In conclusion, 4 liters is not equal to 1 gallon, but it’s close. So close, yet so far. The metric and imperial systems will forever dance around each other, with conversions like this serving as a constant reminder of their differences. One can only dream of a world where these two systems can peacefully coexist. But until then, let’s embrace the quirkiness of the conversions and make sure we have a calculator handy just in case. After all, you never know when you might need to convert 857 milliliters to fluid ounces. Cheers to the magical world of endless conversions! ## How Many Gallons is a 25 Litre Tank? If you’ve ever wondered how many gallons are in a 25 litre tank, you’ve come to the right place. Sit back, relax, and let’s dive into this riveting topic. ### Converting litres to gallons First things first, let’s talk conversions. Converting litres to gallons might sound like a math headache, but fear not, it’s actually quite simple. It’s like translating from English to French, but without all the accents and confusion. So here’s the lowdown: 1. Start with the number of litres you have (in this case, 25 litres). 2. Multiply that number by the conversion factor. And what’s the magical conversion factor, you ask? Well, it’s approximately 0.264172 gallons per litre. 3. Bust out your trusty calculator (or utilize the power of mental math if you’re feeling adventurous) and multiply 25 litres by 0.264172. ### The big reveal Are you ready for the grand reveal? Drumroll, please! 25 litres is equivalent to approximately 6.604 gallons. Ta-da! Now you can impress your friends and family with your newfound knowledge of liquid volume conversions. Just make sure you don’t bring it up during dinner, or you might end up in a heated debate about the metric system. ### What can a 25-litre tank hold? Now that we know how many gallons a 25-litre tank contains, let’s take a moment to ponder what you can actually do with that amount of liquid. It’s time for a little imagination session. Picture this: You could fill up a medium-sized fish tank, or make a whole lot of delicious lemonade. Feeling extra adventurous? Why not try bathing in 6.604 gallons of bubbles? The possibilities are endless! In conclusion, the answer to the burning question “how many gallons is a 25-litre tank” is a resounding 6.604 gallons. Armed with this newfound knowledge, you can now conquer the world (or at least impress your friends at the next trivia night). So go forth, embrace your inner conversion wizard, and let the gallons flow! 🌊<\|endoftext\|>	4.46875
6,221	Download Presentation KS3 Mathematics Loading in 2 Seconds... 1 / 69 # KS3 Mathematics - PowerPoint PPT Presentation KS3 Mathematics. A4 Sequences. A4 Sequences. Contents. A4.2 Describing and continuing sequences. A4.1 Introducing sequences. A4.3 Generating sequences. A4.4 Finding the n th term. A4.5 Sequences from practical contexts. Introducing sequences. 4, 8, 12, 16, 20, 24, 28, 32,. 1 st term. I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described. Download Presentation ## PowerPoint Slideshow about 'KS3 Mathematics' - robert-larson An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - Presentation Transcript A4 Sequences A4 Sequences ### Contents A4.2 Describing and continuing sequences A4.1 Introducing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 Sequences from practical contexts Introducing sequences 4, 8, 12, 16, 20, 24, 28, 32, . . . 1st term 6th term In maths, we call a list of numbers in order a sequence. Each number in a sequence is called a term. If terms are next to each other they are referred to as consecutive terms. When we write out sequences, consecutive terms are usually separated by commas. Infinite and finite sequences A sequence can be infinite. That means it continues forever. For example, the sequence of multiples of 10, 10, 20 ,30, 40, 50, 60, 70, 80, 90 . . . is infinite. We show this by adding three dots at the end. If a sequence has a fixed number of terms it is called a finite sequence. For example, the sequence of two-digit square numbers 16, 25 ,36, 49, 64, 81 is finite. Sequences and rules Some sequences follow a simple rule that is easy to describe. For example, this sequence 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, … continues by adding 3 each time. Each number in this sequence is one less than a multiple of three. Other sequences are completely random. For example, the sequence of winning raffle tickets in a prize draw. In maths we are mainly concerned with sequences of numbers that follow a rule. Naming sequences Here are the names of some sequences which you may know already: 2, 4, 6, 8, 10, . . . Even Numbers (or multiples of 2) 1, 3, 5, 7, 9, . . . Odd numbers 3, 6, 9, 12, 15, . . . Multiples of 3 5, 10, 15, 20, 25 . . . Multiples of 5 1, 4, 9, 16, 25, . . . Square numbers 1, 3, 6, 10,15, . . . Triangular numbers Ascending sequences ×2 +5 +5 +5 +5 +5 +5 +5 ×2 ×2 ×2 ×2 ×2 ×2 When each term in a sequence is bigger than the one before the sequence is called an ascending sequence. For example, The terms in this ascending sequence increase in equal steps by adding 5 each time. 2, 7, 12, 17, 22, 27, 32, 37, . . . The terms in this ascending sequence increase in unequal steps by starting at 0.1 and doubling each time. 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, . . . Descending sequences –7 –7 –7 –7 –7 –7 –7 –7 –1 –2 –3 –4 –5 –6 When each term in a sequence is smaller than the one before the sequence is called a descending sequence. For example, The terms in this descending sequence decrease in equal steps by starting at 24 and subtracting 7 each time. 24, 17, 10, 3, –4, –11, –18, –25, . . . The terms in this descending sequence decrease in unequal steps by starting at 100 and subtracting 1, 2, 3, … 100, 99, 97, 94, 90, 85, 79, 72, . . . Sequences from real-life Some sequences are completely random, like the sequence of numbers drawn in the lottery. Number sequences are all around us. Some sequences, like the ones we have looked at today follow a simple rule. Some sequences follow more complex rules, for example, the time the sun sets each day. What other number sequences can be made from real-life situations? A4 Sequences ### Contents A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 Sequences from practical contexts Sequences from geometrical patterns 2 4 6 8 10 1 3 5 7 9 We can show many well-known sequences using geometrical patterns of counters. Even Numbers Odd Numbers Sequences from geometrical patterns 3 6 9 12 15 5 10 15 20 25 Multiples of Three Multiples of Five Sequences from geometrical patterns 4 1 9 16 25 1 3 6 10 15 Square Numbers Triangular Numbers Sequences with geometrical patterns 2 × 3 = 6 3 × 4 = 12 4 × 5 = 20 5 × 6 = 30 1 × 2 = 2 How could we arrange counters to represent the sequence 2, 6, 12, 20, 30, . . .? The numbers in this sequence can be written as: 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, . . . We can show this sequence using a sequence of rectangles: Powers of two 22 = 4 23 = 8 21 = 2 24 = 16 25 = 32 26 = 64 We can show powers of two like this: Each term in this sequence is double the term before it. Powers of three 31 = 3 32 = 9 33 = 27 34 = 81 35 = 243 36 = 729 We can show powers of three like this: Each term in this sequence is three times the term before it. Sequences that increase in equal steps +4 +4 +4 +4 +4 +4 +4 We can describe sequences by finding a rule that tells us how the sequence continues. To work out a rule it is often helpful to find the difference between consecutive terms. For example, look at the difference between each term in this sequence: 3, 7, 11, 15 19, 23, 27, 31, . . . This sequence starts with 3 and increases by 4 each time. Every term in this sequence is one less than a multiple of 4. Sequences that decrease in equal steps –6 –6 –6 –6 –6 –6 –6 Can you work out the next three terms in this sequence? 22, 16, 10, 4, –2, –8, –14, –20, . . . How did you work these out? This sequence starts with 22 and decreases by 6 each time. Each term in the sequence is two less than a multiple of 6. Sequences that increase or decrease in equal steps are called linear or arithmetic sequences. Sequences that increase in increasing steps +1 +2 +3 +4 +5 +6 +7 Some sequences increase or decrease in unequal steps. For example, look at the differences between terms in this sequence: 2, 6, 8, 11, 15, 20, 26, 33, . . . This sequence starts with 5 and increases by 1, 2, 3, 4, … The differences between the terms form a linear sequence. Sequences that decrease in decreasing steps –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 Can you work out the next three terms in this sequence? 7, 6.9, 6.7, 6.4, 6, 5.5, 4.9, 4.2, . . . How did you work these out? This sequence starts with 7 and decreases by 0.1, 0.2, 0.3, 0.4, 0.5, … With sequences of this type it is often helpful to find a second row of differences. Using a second row of differences +20 +17 +14 +8 +5 +2 +3 +3 +3 +3 +3 +11 +3 Can you work out the next three terms in this sequence? 1, 3, 8, 16, 27, 41, 58, 78, . . . Look at the differences between terms. A sequence is formed by the differences so we look at the second row of differences. This shows that the differences increase by 3 each time. Sequences that increase by multiplying ×2 ×2 ×2 ×2 ×2 ×2 ×2 Some sequences increase or decrease by multiplying or dividing each term by a constant factor. For example, look at this sequence: 2, 4, 8, 16, 32, 64, 128, 256, . . . This sequence starts with 2 and increases by multiplying the previous term by 2. All of the terms in this sequence are powers of 2. Sequences that decrease by dividing ÷4 ÷4 ÷4 ÷4 ÷4 ÷4 ÷4 We could also continue this sequence by multiplying by each time. 1 4 Can you work out the next three terms in this sequence? 512, 256, 64, 16, 4, 1, 0.25, 0.125, . . . How did you work these out? This sequence starts with 512 and decreases by dividing by 4 each time. Fibonacci-type sequences 21+13 1+1 1+2 3+5 5+8 8+13 13+21 21+34 Can you work out the next three terms in this sequence? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . How did you work these out? This sequence starts 1, 1 and each term is found by adding together the two previous terms. This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it. Describing and continuing sequences Here are some of the types of sequence you may come across: • Sequences that increase or decrease in equal steps. • These are calledlinearorarithmetic sequences. • Sequences that increase or decrease in unequal steps • by multiplying or dividing by a constant factor. • Sequences that increase or decrease in unequal steps • by adding or subtracting increasing or decreasing numbers. • Sequences that increase or decrease by adding together • the two previous terms. Continuing sequences A number sequence starts as follows 1, 2, . . . How many ways can you think of continuing the sequence? Give the next three terms and the rule for each one. A4 Sequences ### Contents A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 Sequences from practical contexts Generating sequences from flow charts No Yes A sequence can be given by a flow chart. For example, START This flow chart generates the sequence 3, 4.5, 6, 6.5, 9. Write down 3. Add on 1.5. Write down the answer. This sequence has only five terms. Is the answer more than 10? It is finite. STOP Generating sequences from flow charts No Yes START Write down 5. This flow chart generates the sequence 5, 2.9, 0.8, –1.3, –3.4. Subtract 2.1. Write down the answer. Is the answer less than -5? STOP Generating sequences from flow charts No Yes START Write down 200. This flow chart generates the sequence 200, 100, 50, 25, 12.5, 6.25. Divide by 2. Write down the answer. Is the answer less than 4? STOP Generating sequences from flow charts No Yes START Write down 3 and 4. This flow chart generates the sequence 3, 4, 7, 11, 18, 29, 47, 76. Add together the two previous numbers. Write down the answer. Is the answer more than 100? STOP Predicting terms in a sequence For example, 87, 84, 81, 78, . . . Usually, we can predict how a sequence will continue by looking for patterns. We can predict that this sequence continues by subtracting 3 each time. However, sequences do not always continue as we would expect. For example, A sequence starts with the numbers 1, 2, 4, . . . How could this sequence continue? Continuing sequences +1 +2 +3 +4 +5 +6 ×2 ×2 ×2 ×2 ×2 ×2 Here are some different ways in which the sequence might continue: 1 2 4 7 11 16 22 1 2 4 8 16 32 64 We can never be certain how a sequence will continue unless we are given a rule or we can justify a rule from a practical context. Continuing sequences +3 +3 +3 +3 +3 +3 This sequence continues by adding 3 each time. 1 4 7 10 13 16 19 We can say that rule for getting from one term to the next term is add 3. This is called the term-to-term rule. The term-to-term rule for this sequence is +3. Using a term-to-term rule Does the rule +3 always produce the same sequence? No, it depends on the starting number. For example, if we start with 2 and add on 3 each time we have, 2, 5, 8, 11, 14, 17, 20, 23, . . . If we start with 0.4 and add on 3 each time we have, 0.4, 3.4, 6.4, 9.4, 12.4, 15.4, 18.4, 21.4, . . . Writing sequences from term-to-term-rules +2 +4 +6 +10 +12 +14 A term-to-term rule gives a rule for finding each term of a sequence from the previous term or terms. To generate a sequence from a term-to-term rule we must also be given the first number in the sequence. For example, 1st term Term-to-term rule 5 Add consecutive even numbers starting with 2. This gives us the sequence, 5 7 11 17 27 39 53 . . . Sequences from a term-to-term rule Write the first five terms of each sequence given the first term and the term-to-term rule. 1st term Term-to-term rule 10 Add 3 10, 13, 16, 19, 21 Subtract 5 100, 95, 90, 85, 80 100 3 Double 3, 6, 12, 24, 48 Multiply by 10 5, 50, 500, 5000, 50000 5 7 Subtract 2 7, 5, 3, 1, –1 0.8 Add 0.1 0.8, 0.9, 1.0, 1.1, 1.2 Sequences from position-to-term rules Sometimes sequences are arranged in a table like this: We can say that each term can be found by multiplying the position of the term by 3. This is called a position-to-term rule. For this sequence we can say that the nth term is 3n, where n is a term’s position in the sequence. What is the 100th term in this sequence? 3 × 100 = 300 Writing sequences from position-to-term rules The position-to-term rule for a sequence is very useful because it allows us to work out any term in the sequence without having to work out any other terms. We can use algebraic shorthand to do this. We call the first term T(1), for Term number 1, we call the second term T(2), we call the third term T(3), . . . we call the nth term T(n). T(n) is called the the nth term or the general term. Writing sequences from position-to-term rules For example, suppose the nth term of a sequence is 4n + 1. We can write this rule as: T(n) = 4n + 1 Find the first 5 terms. T(1) = 4 ×1 + 1 = 5 T(2) = 4 ×2 + 1 = 9 T(3) = 4 ×3 + 1 = 13 T(4) = 4 ×4 + 1 = 17 T(5) = 4 ×5 + 1 = 21 The first 5 terms in the sequence are: 5, 9, 13, 17 and 21. Writing sequences from position-to-term rules If the nth term of a sequence is 2n2 + 3. We can write this rule as: T(n) = 2n2 + 3 Find the first 4 terms. T(1) = 2 ×12 + 3 = 5 T(2) = 2 ×22 + 3 = 11 T(3) = 2 ×32 + 3 = 21 T(4) = 2 ×42 + 3 = 35 The first 4 terms in the sequence are: 5, 11, 21, and 35. This sequence is a quadratic sequence. Sequences and rules Which rule is best? The term-to-term rule? The position-to-term rule? A4 Sequences ### Contents A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.4 Finding the nth term A4.3 Generating sequences A4.5 Sequences from practical contexts Sequences of multiples +5 +5 +5 +5 +5 +5 +5 × 5 × 5 × 5 × 5 × 5 × 5 All sequences of multiples can be generated by adding the same amount each time. They are linear sequences. For example, the sequence of multiples of 5: 5, 10, 15, 20, 25, 30 35 40 … can be found by adding 5 each time. Compare the terms in the sequence of multiples of 5 to their position in the sequence: 2 10 3 15 4 20 5 25 n Position 1 5 5n Term Sequences of multiples +3 +3 +3 +3 +3 +3 +3 ×3 ×3 ×3 ×3 ×3 ×3 The sequence of multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, … can be found by adding 3 each time. Compare the terms in the sequence of multiples of 3 to their position in the sequence: 2 6 3 9 4 12 5 15 n Position 1 3 3n Term The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms. Sequences of multiples The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms. For example, The nth term of 4, 8, 12, 16, 20, 24 … is 4n The 10th term of this sequence is 4 × 10 = 40 The 25th term of this sequence is 4 × 25 = 100 The 47th term of this sequence is 4 × 47 = 188 Finding the nth term of a linear sequence +3 +3 +3 +3 +3 +3 +3 × 3 × 3 × 3 × 3 × 3 + 1 + 1 + 1 × 3 + 1 + 1 + 1 The terms in this sequence 4, 7, 10, 13, 16, 19, 22, 25 … can be found by adding 3 each time. Compare the terms in the sequence to the multiples of 3. 2 3 4 5 n Position 1 Multiples of 3 3 6 9 12 15 3n Term 4 7 10 13 16 3n + 1 Each term is one more than a multiple of 3. Finding the nth term of a linear sequence +5 +5 +5 +5 +5 +5 +5 × 5 × 5 × 5 × 5 × 5 – 4 – 4 – 4 × 5 – 4 – 4 – 4 The terms in this sequence 1, 6, 11, 16, 21, 26, 31, 36 … can be found by adding 5 each time. Compare the terms in the sequence to the multiples of 5. 2 3 4 5 n Position 1 Multiples of 5 5 10 15 20 25 5n Term 1 6 11 16 21 5n– 4 Each term is four less than a multiple of 5. Finding the nth term of a linear sequence –2 –2 –2 –2 –2 –2 –2 × –2 × –2 × –2 × –2 × –2 + 7 + 7 + 7 × –2 + 7 + 7 + 7 The terms in this sequence 5, 3, 1, –1, –3, –5, –7, –9 … can be found by subtracting 2 each time. Compare the terms in the sequence to the multiples of –2. 2 3 4 5 n Position 1 Multiples of –2 –2 –4 –6 –8 –10 –2n Term 5 3 1 –1 –3 7 – 2n Each term is seven more than a multiple of –2. Arithmetic sequences Sequences that increase (or decrease) in equal steps are called linear or arithmetic sequences. The difference between any two consecutive terms in an arithmetic sequence is a constant number. When we describe arithmetic sequences we call the difference between consecutive terms, d. We call the first term in an arithmetic sequence, a. For example, if an arithmetic sequence has a = 5 and d = -2, We have the sequence: 5, 3, 1, -1, -3, -5, . . . The nth term of an arithmetic sequence The rule for the nth term of any arithmetic sequence is of the form: T(n) = an + b a and b can be any number, including fractions and negative numbers. For example, Generates odd numbers starting at 3. T(n) = 2n + 1 Generates even numbers starting at 6. T(n) = 2n + 4 Generates even numbers starting at –2. T(n) = 2n– 4 Generates multiples of 3 starting at 9. T(n) = 3n + 6 Generates descending integers starting at 3. T(n) = 4 –n A4 Sequences ### Contents A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.5 Sequences from practical contexts A4.3 Generating sequences A4.4 Finding the nth term Sequences from practical contexts The following sequence of patterns is made from L-shaped tiles: Number of Tiles 4 8 12 16 The number of tiles in each pattern form a sequence. How many tiles will be needed for the next pattern? We add on four tiles each time. This is a term-to-term rule. Sequences from practical contexts A possible justification of this rule is that each shape has four ‘arms’ each increasing by one tile in the next arrangement. The pattern give us multiples of 4: 1 lot of 4 2 lots of 4 3 lots of 4 4 lots of 4 The nth term is 4 ×n or 4n. Justification: This follows because the 10th term would be 10 lots of 4. Sequences from practical contexts Now, look at this pattern of blocks: Number of Blocks 4 7 10 13 How many blocks will there be in the next shape? We add on 3 blocks each time. This is the term-to term rule. Justification: The shapes have three ‘arms’ each increasing by one block each time. Sequences from practical contexts How many blocks will there be in the 100th arrangement? We need a rule for the nth term. Look at pattern again: 1st pattern 2nd pattern 3rd pattern 4th pattern The nth pattern has 3n + 1 blocks in it. Justification: The patterns have 3 ‘arms’ each increasing by one block each time. So the nth pattern has 3n blocks in the arms, plus one more in the centre. Sequences from practical contexts So, how many blocks will there be in the 100th pattern? Number of blocks in the nth pattern = 3n + 1 When n is 100, Number of blocks = (3 ×100) + 1 = 301 How many blocks will there be in: a) Pattern 10? (3 ×10) + 1 = 31 b) Pattern 25? (3 ×25) + 1 = 76 c) Pattern 52? (3 ×52) + 1 = 156 Paving slabs 2 The number of blue tiles form the sequence 8, 13, 18, 32, . . . Pattern number 1 2 3 Number of blue tiles 8 13 18 The rule for the nth term of this sequence is T(n) = 5n + 3 Justification: Each time we add another yellow tile we add 5 blue tiles. The +3 comes from the 3 tiles at the start of each pattern.<\|endoftext\|>	4.71875
1,349	Many of today’s developmental psychologists defend the hypothesis that “babies are smarter than we think” — a lot smarter than we think, explained Nora Newcombe of Temple University during her APS William James Fellow Award Address at the 2014 APS Convention in San Francisco. But Newcombe’s work on mental rotation and human spatial perception has made her question the complex cognitive abilities that many researchers ascribe to infants. “They’re smarter than you think if you think they don’t know anything,” she said, noting that no serious contemporary scientist sees babies as “blank slates” devoid of mental ability. Referencing the concept of “experience expectancy” proposed by neuroscientist William Greenough, also an APS William James Fellow, Newcombe acknowledged that human infants benefit from “strong starting points” — brains capable of learning from the physical world. The empirical questions that remain, then — questions that science has yet to answer — are whether babies are also born with strong representational knowledge of the world and, if so, how much knowledge they are born with. Infants can’t tell scientists what they think or understand. So developmental psychologists have long relied on “looking time” as a clue in the study of infant cognition. In 1997, for example, Sue Hespos and Philippe Rochat designed a study in which 2- to 8-month-old infants were shown a two-dimensional animation of an object (shaped more or less like a two-tined fork) turning as it fell behind a larger occluding square. Then, the “fork” was either removed from behind the occluder in the position in which it would be expected to have fallen considering its trajectory, or it was removed from the occluder in a position that would be totally unexpected considering the trajectory the babies witnessed. Seeing the second, physically improbable scenario caused infants to look longer at the object. Apparently, Hespos and Rochat concluded, these infants understood that something about the animation didn’t mesh with the way the physical world works. Subsequent studies have used looking time to suggest that babies as young as 3 to 4 months can distinguish between pairs of objects that mirror each other and pairs of objects that do not mirror each other. Newcombe urges developmental psychologists to think twice about interpreting such results as evidence of incredible spatial insight in babies. Work conducted in her own lab and by Wenke Möhring and Andrea Frick used looking time to examine how well infants could predict the position of a p-shaped object that was completely blue on one side and yellow with a red bull’s-eye shape on the opposite side. Infants watched the p fall straight down behind an occluder (blue side facing forward and with no rotation at all). Only when 6-month-old infants had been allowed to handle the p before watching it fall (as opposed to just being shown both sides of the shape) did they stare (i.e., display an “elevated looking time”) when the p emerged from behind an occluder with the yellow side facing forward — a surprising event considering the p’s trajectory. Ten-month-olds seemed puzzled by unexplained object rotation even without having handled the p figure; however, this effect was carried by the 10-month-olds in the study who had learned to walk. Ten-month-olds who had not yet learned to walk showed shorter looking times than their mobile age-mates when the p shape was pulled from behind the occluder in what should have seemed like an “impossible” position. These results contradict the dominant “nativist” view of development, which attributes spatial learning to innate intellectual abilities: Experiential learning — exploring and learning from one’s environment as the babies did when they handled the p shape — may be much more important than many researchers acknowledge. Even mobile 10-month-olds like the ones from Möhring and Frick’s study, who seemed capable of making predictions about a simple object rotation task, have a very limited understanding of the geometry of their surroundings, Newcombe said. The scientists who design object-rotation tasks may get infants to stare longer at “impossible” scenarios. However, Newcombe warned, the babies “don’t think, ‘That’s impossible.’ They just think it looks a little weird; that’s all it takes to drive a looking-time finding.” “As you interact [with your environment] through grasping and crawling,” she asserted, “that propels developmental change.” According to the evidence that Newcombe and her colleagues have gathered from their work with older children, it takes years of interactive experience before children develop truly complex spatial reasoning capacities. A 2013 study led by Andrea Frick tested 4- and 5-year-olds’ ability to play a simplified version of the game Tetris, which required the player to look at a figure and determine how it could fit into a larger puzzle. The team asked only that the children touch the place where the blocks would fit; they didn’t actually have to rotate the blocks or fit them into the puzzle. Four-year-olds struggled with the activity; many 5-year-olds could do it. When these results were combined with those from an experiment that tested how well children ages 3–5 could fit various ghost-shaped pieces into a puzzle board, the researchers concluded that it isn’t until about 4.5 years of age that most children are able to make accurate, active predictions about object rotation. “I think a lot of cognitive development needs to be rethought … in light of these doubts about what looking time is telling us,” Newcombe said. She hopes her work will challenge researchers convinced that “babies are smarter than we think” to embrace a different point of view. Newcombe believes that babies are “really not so smart, and that there’s a lot of change that occurs during development, and that our task is indeed to embrace this change and to think about what creates this change.” Right now, Newcombe sees child cognitive development between infancy and 4 years of age as terra incognita. Infants, she argues, must learn many of the skills scientists previously assumed they were born with; mapping that learning process should be a new priority for developmental researchers.<\|endoftext\|>	3.65625
561	- Praise the student in a strong positive way for a correct or positive response. Use such terms as "excellent answer," "absolutely correct," and "bull's eye." These terms are quite different from the common mild phrases instructors often use such as "O,K.," "hm-hm," and "all right." Especially when the response is long, the instructor should try to find at least some part that deserves praise and then comment on it. - Make comments pertinent to the specific student response. For example, suppose that a student has offered an excellent response to the question, "What function did the invasion of the Falklands serve for Argentina?" The instructor might say, "That was excellent, Pat. You included national political reasons as well as mentioning the Argentine drive to become the South American leader." This response gives an excellent rating to the student in an explicit and strong form. It also demonstrates that the instructor has listened carefully to the student's ideas. - Build on the student's response. If the instructor continues to discuss a point after a student response, he or she should try to incorporate the key elements of the response into the discussion. By using the student's response, the instructor shows that he or she values the points made. By referring to the student explicitly by name (e.g., "As Pat pointed out, the Falklands' national political status… ") the instructor gives credit where credit is due. - Avoid the "Yes, but… "reaction. Instructors use "Yes, but…" or its equivalent when a response is wrong or at least partly wrong. The overall impact of these phrases is negative and deceptive even though the instructor's intent is probably positive. The "Yes, but…" fielding move says that the response is correct or appropriate with one breath and then takes away the praise with the next. Some straight-forward alternatives can be recommended: - Wait to a count of five with the expectation that another student will volunteer a correct or better response. - Ask, "How did you arrive at that response? (Be careful though, not to ask this question only when you receive inadequate responses, ask it also at times when you receive a perfectly good response). - Say, "You're right regarding X and that's great; wrong regarding Y. Now we need to correct Y so we can get everything correct." - Say, "Thanks. Is there someone who wants to respond to the question or comment on the response we've already heard?" These four alternatives are obviously not adequate to fit all cases. Indeed, it is generally difficult to field wrong or partially wrong responses because students are sensitive to instructor criticism. However, with these alternatives as examples, you will probably be able to generate others as needed.<\|endoftext\|>	3.875
807	A+, Satisfactory, 58%, Partial understanding, Needs more work, below average – yes there are many ways for us teachers to express how well we think a student has done in an assessment activity. This is very important as sometimes we want to give a very precise grade and other times we just want the student to know that they are doing well. To do this we use grading scales. Grading scales are a way of capturing, and communicating, student performance in a learning or assessment activity. A very typical grading scale is to use a percentage to represent the amount of correct responses from a given student. Another popular grading scale is to use letters to indicate performance e.g. from an ‘A’ grade for the best possible performance on the assessment to an ‘F’ grade, to represent a failing grade in the assessment. Moodle allows for the use of both numeric and non-numeric grading scales for Forums, Glossaries and Assignments. Numeric grading scales are defined from 1-100, where the instructor indicates the maximum grade for the activity, e.g. a max of 100 is a percentage grade or a max of 10 can be used where students are marked out of a maximum mark of 10. Moodle has one non-numeric scale defined out of the box called “Separate and Connected ways of knowing”. This scale allows an instructor to define a learner’s knowledge of an area in terms of connected knowing or separate knowing as defined by Belenky et. al (1986). Moodle also allows instructors and administrators to define new non-numeric course-wide and site-wide grading scales respectively. The screen-shot below shows the Moodle screen used to define a new grading scale. The “name” field is used to define the name of the new scale, “scale” is used to define the separate grading scales (each scale is separated by a comma) and the “description” field is used to describe the rationale behind the new grading scale. Each non-numeric grade is assigned a numeric value behind the scenes. When entering grades they should be entered in increasing order of value, therefore the grades A,B,C,D should be entered as D,C,B,A. The value of each grade is based on the number of grades in the scale. To illustrate how the value of grades are calculated, below, we have taken the grades generally used in university degree programmes: - University degree grading system – Fail, Pass, 2nd Class Honours (Grade 2), 2nd Class Honours (Grade 1), First Class Honours - (Valued as 0/4pts, 1/4pt, 2/4pts, 3/4pts and 4/4pts respectively in any normalized aggregation method) - (Valued as 1, 2, 3, 4 and 5 respectively in the sum aggregation method) Okay so you have your grading scale set up, now how do you use them in your assessments? Easy all you do is specify the appropriate grading scale when you are creating your assessments. To do this all you need to do is set the “grade” in the assessment settings. In the screen shot below you can see the custom scales at the top of the list. There are two non-numeric grading scales: - MyScale – Excellent, Very good, Good, Average, Poor, Very poor - Separate and Connected Ways of Knowing – Separate knowing, Connected knowing, Separate and connected knowing So what does the teacher see? The screen shot below displays what a teacher will see when correcting an offline assignment that is using the MyScale as a grading scale. For more information on grading scales have a look at the grading scales page on the moodle.org web site – http://docs.moodle.org/en/Scales<\|endoftext\|>	3.90625
339	# LCM Formula The Least Common Multiple of two integers a and b, usually denoted by LCM (a, b), is the smallest positive integer that is divisible by both a and b. In simple words, the smallest positive number that is a multiple of two or more numbers. L.C.M formula for any two numbers is, $\large L.C.M=\frac{a\times b}{gcd\left(a,b\right)}$ LCM formula for \fraction is given by, $\large L.C.M=\frac{L.C.M\;of\;Numerator}{H.C.F\;of\;Denominator}$ The GCD or HCF is the greatest divisor which is divisible by both the numbers. ### Solved Examples Question 1: Find the LCM of (50, 65). Solution: Given number is (50, 65). The numbers can be written in the form of their prime factors- 50 = 1$\times$2$\times$5$\times$5 65 = 1$\times$5$\times$13 The greatest common factors (gcf) 0f (50,65) is 5. Thus Least Common Multiple = $\frac{50 \times 65}{5} = \frac{10}{65} = 65$ Or, The primes common to both are 2, 5, 5, 13 . Hence, the LCM of (50, 65) = 2 $\times$ 5 $\times$ 5 $\times$ 13 = 650 LCM (50, 65) = 650<\|endoftext\|>	4.4375
166	By talking through their thinking at each step of a process, teachers can model what learning looks like. Observing thought processes and expected behaviors also helps students feel supported in their learning. Watch a kindergarten teacher model his thinking while reading a story aloud. He shows how he arrived at his questions and how asking questions is important when reading. Students then try the strategy out with a partner, allowing them to also practice their Social Awareness and Relationship Skills. Videos are chosen as examples of strategies in action. These choices are not endorsements of the products or evidence of use of research to develop the feature. Learn how an assistive reading app, the Sounding Out Machine, uses modeling to support learners' Decoding skills. By hearing how to break down unfamiliar words, learners develop Phonological Awareness and Auditory Processing skills.<\|endoftext\|>	4.0625
696	#### Transcript Factors, Fractions, and Exponents ```Midsegments of Triangles 5.1 Today’s goals By the end of class today, YOU should be able to… 1. Define and use the properties of midsegments to solve problems for unknowns. 2. Use the properties of midsegments to make statements about parallel segments in a given triangle. 3. Understand and write coordinate proofs. Midsegments  A midsegment of a triangle is a segment connecting the midpoints of two sides of a triangle. Triangle Midsegment Theorem  If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length. Proving the Triangle Midsegment Theorem…  To prove the Triangle Midsegement Theorem, use coordinate geometry and algebra. This style of proof is called a coordinate proof. To begin the proof:  Place the triangle in a convenient spot on the coordinate plane.  Choose variables for the coordinates of the vertices. Coordinate proofs cont… State what you are given.  State what you want to prove.  Use the midpoint formula to find the midpoints of two sides of the triangle.  To prove that the midsegment and third side are parallel, show that their slopes are equal.  To prove that the midsegment is half the length of the third side, use the distance formula to calculate the length of both segments.  *See section 5.1 for a complete version of a coordinate proof for the Triangle Midsegment Theorem Ex.1: Find the lengths Using the following illustration (where H, J, and K are the midpoints), find the lengths of HJ, JK, and FG: Ex.1: Solution HJ = ½ EG = ½ (100) = 50 JK = ½ EF = ½ (60) = 30 HK = ½ FG FG = 2 HK = 2 (40) = 80 Use the Triangle Midsegment Theorem to show that the midsegment = ½ the length of the third side You Try… Use the following triangle to solve the questions below (A, B, & C are midpoints): 1. DF=120, EF=90, BC=40. Find AB, AC, & DE. 2. EF=10x, AB=4x, AC=10, DE=12x. Find BC & AC. Ex.2: Identify the parallel segments If A, B, & C are midpoints, which segment is parallel to AC? Ex.2: Solution By the Triangle Midsegment Theorem, AC ll EF You Try… If A, B, & C are midpoints: 1. Which segment is parallel to DE? 1. Which segment is parallel to AB? Practice Solve with a partner The triangular face of the Rock and Roll Hall of Fame in Cleveland, Ohio is isosceles. The length of the base is 229 ft 6 in. What is the length of a segment located half way up the face of the Rock and Roll Hall of Fame<\|endoftext\|>	4.78125
1,283	APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS (Population Word Problems) To solve an exponential or logarithmic word problem, convert the narrative to an equation and solve the equation. In this section, we will review population problems. We will also discuss why the base of e is used so often with population problems. Example 1: Suppose that you are observing the behavior of cell duplication in a lab. In one experiment, you started with one cell and the cells doubled every minute. Write an equation with base 2 to determine the number (population) of cells after one hour. Solution and Explanations: First record your observations by making a table with two columns: one column for the time and one column for the number of cells. The number of cells (size of population) depends on the time. If you were to graph your findings, the points would be formed by (specific time, number of cells at the specific time). For example, at t = 0, there is 1 cell, and the corresponding point is (0, 1). At t = 1, there are 2 cells, and the corresponding point is (1, 2). At t = 2, there are 4 cells, and the corresponding point is (2, 4). At t = 3, there are 8 cells, and the corresponding point is (3, 8). It appears that the relationship between the two parts of the point is exponential. At time 0, the number of cells is 1 or 20 = 1. After 1 minute, when t = 1, there are two cells or 21 = 2. After 2 minutes, when t = 2, there are 4 cells or 22 = 4. Therefore, one formula to estimate the number of cells (size of population) after t minutes is the equation (model) f (t) = 2t. Determine the number of cells after one hour: • Convert one hour to minutes. . = 60 min • Substitute 60 for t in the equation. f (t) = 2t: f (60) = 260 = 1.15×1018 Example 2: Determine how long it would take the population (number of cells) to reach 100,000 cells. Solution and explanation: • In this example, you know the number of cells at the beginning of the experiment (1) and at the end of the experiment (100,000), but you do not know the time. Substitute 100,000 for f(t) in the equation f (t) = 2t: 100, 000 = 2t • Take the natural logarithm of both sides: ln(100, 000) = ln(2t) • Simplify the right side of the equation using the third rule of logarithms: ln(100, 000) = t ln(2) • Divide both sides by ln(2): t = = 16.60964 min It would take 16.6 minutes, rounded, for the population (number of cells) to reach 100,000. Example 3: Write an equation with base 5 that is equivalent to the equation f (t) = 2t. Solution and Explanation: f (t) = a . 5bt. • The f(t) represents the size of the population at time t, the t represents the time, and the a and b represent adjusters when we change the base. The value of a is the number of cells (size of population) at the beginning of the study, and the value of b is the relative growth rate based on a base of 5. We need to find the values of a and b. • We know that the population is 1 at time 0, so insert these numbers in the equation f (t) = a . 5bt. We have 1 = a . 5b . 0 = a . 50 = a . 1 = a. We now know that the value of a in the adjusted equation is 1. • Rewrite the equation f (t) = a . 5bt with a = 1. f (t) = 1 . 5bt which in turn can be rewritten as f (t) = 5b . t. • We know that the population after 1 minute is 2 cells, so insert these numbers in the equation f (t) = 5b . t to obtain 2 = 5b . 1. • Solve for b by taking the natural logarithm of both sides of the equation 2 = 5b. ln(2) = ln(5b). • Simplify the right side of the equation using the third rule of logarithms: ln(2) = b ln(5). • Divide both sides of the equation by ln(5) and simplify: b = = 0.43067658075, rounded to 0.4307. • Insert this value of b in the equation f (t) = 5b . t, and the equation is simplified to f (t) = 50.4307t • We know that the population is 8 after 3 seconds, so use these values to check the validity of the above equation. Substitute 3 for t in the right side of the above equation. If the answer is 8, or close to 8 because we rounded, then the model (equation) is correct. 50.4307(3) = 8.00906, rounded to 8. • We know from the original equation that after 4 seconds, the population is 16. Let's do a second check. 50.4307(4) = 16.02415, rounded to 16 cells. The check would be closer had we rounded b to more decimals. If you would like to work another example, click on Example [Menu Back to Solving Word Problems] [Exponential Rules] [Logarithms] [Algebra] [Trigonometry ] [Complex Variables]<\|endoftext\|>	4.90625
847	Not just heat: Climate change signs can be seen all around WASHINGTON (AP) — You don't just feel the heat of global warming, you can see it in action all around. Some examples of where climate change's effects have been measured: —Glaciers across the globe are melting and retreating, with 279 billion tons of ice lost since 2002, according to NASA's GRACE satellite. Jakobshavn Glacier in Greenland is flowing faster than any other glacier on Earth. In 2012, it hit a record pace of about 75 inches per hour (1.9 meters). In 2017, it slowed down to 40 inches per hour (1 meter). The Portage Glacier in Alaska has retreated so much it cannot be seen from the visitor center that opened in 1986. —In the Rocky Mountains, the first robins of spring are arriving 10.5 days earlier than 30 years ago. The first larkspur wildflower is showing up eight days earlier and the marmots are coming out of hibernation five days earlier, according to data gathered by the Rocky Mountain Biological Lab. —On average, during the past 30 years there have been more major hurricanes (those with winds of more than 110 mph), they have lasted longer and they produced more energy than the previous 30 years, according to an Associated Press analysis of storm data. Other studies have shown that the first named storm in the Atlantic forms nearly a month earlier than 30 years ago and storms are moving slower, allowing more rain to fall. —Across the globe, seas have risen about 3 inches since 1993. That doesn't sound like much, but it is enough to cover the entire United States in water about 9 feet deep. Places like Miami Beach, Florida, and Norfolk, Virginia, flood frequently with high tides. —The number of acres burned in the U.S. by wildfire has doubled compared with 30 years ago. Last year, more than 10 million acres burned. Over the last five years, an average of 6.7 million acres burned a year. From 1984 to 1988, about 2.8 million years burned, on average. —Allergies have gotten worse with longer growing seasons and more potent pollen. High ragweed pollen days have increased by between 15 and 29 days since 1990 in a swath of the country from Oklahoma City north to Winnipeg, Canada, according to a U.S. Department of Agriculture study. —In the western United States the cute rodent called a pika needs weather around freezing for most of the year. But those habitats are shrinking, forcing them to higher altitudes. University of Colorado's Chris Ray, a pika expert, said she hasn't definitively linked climate change to dramatic reductions in pika populations, but she found that they have disappeared more from places that are warming and drying. —Extreme one-day rainfall across the nation has increased 80 percent over the past 30 years. Ellicott City, Maryland, had so-called thousand-year floods in 2016 and this year. Flooding in Louisiana, West Virginia and Houston in 2016, South Carolina, Texas and Oklahoma in 2015, Michigan and parts of the Northeast in 2014 all caused more than $1 billion in damage, according to the National Oceanic and Atmospheric Administration. —The number of polar bears in parts of Alaska dropped 40 percent since the late 1990s. When scientists have weighed polar bears recently in certain locations they were losing 2.9 to 5.5 pounds per day at a time of year when they were supposed to be putting on weight. —Warmer water is repeatedly causing mass global bleaching events to Earth's fragile coral reefs. Before 1998 there had been no global mass bleaching events — which turn the living coral white and often lead to death. But there have been three in the last two decades. U.S. government coral reef specialist Mark Eakin said for multiple reasons, including global warming, "most of the reefs that were in great shape in the 1980s in Florida are just barely hanging on now." The Associated Press Health & Science Department receives support from the Howard Hughes Medical Institute's Department of Science Education. The AP is solely responsible for all content.<\|endoftext\|>	3.796875
569	Money Worksheets Use this 48 page resource with your 1st, 2nd, or 3rd grade classroom or home school students.Our free counting games, videos, and worksheets give children plenty of opportunities to count. Counting - Free, Fun Counting Games, Videos & Worksheets Enjoy this cool money counting game for kids and have fun learning online. Free Math Worksheets, Problems and Practice \| AdaptedMind Kindergarten Math Help for Standardized Tests - Beginning Printables. Counting Coins Worksheets. Lemonlilyfestival Money Worksheets - Common Core Money Coin Counting Money Worksheet for Teachers - InstructorWeb Data and Probability, Money and Time \| Third Grade Math Free Kindergarten Addition Worksheets - Learning to Add Number Patterns - Printable Math Worksheets at Children learn counting money while playing fun, free online math games. Counting Money Worksheets - Free Printable Worksheets forThey can practice adding, subtracting, calculating change, counting, and dividing all with the.Kindergarten, 1st Grade, 2nd Grade, 3rd Grade, 4th Grade, 5th.This Word Problems 1 (K-1) Morning Math is perfect to practice beginning math skills.Kindergarten Addition Worksheets Learning to Add Through Images and Numbers Addition at a beginning stage can be taught by combining objects and simply counting them.Money. Students will work on their money skills with these worksheets. With these Smartboard lessons, students practice counting coins and writing money amounts.Money Worksheet, Adding Money Worksheet, Counting Coins Worksheet.Free Printable Money Worksheets For Kids:count on to find the total amount. Free Printable Money Worksheets For Kids - Softschools.com Fourth grade and fourth Math Worksheets and Printable PDF Handouts, Math printables for 4th grade. First Grade Money Worksheets « Math Worksheet WizardNumber patterns, sequencing patterns, and more are all featured on the math worksheets on this page. Money Worksheets page 1 \| abcteachBEGINNING MONEY WORKSHEET. Practice counting coins and writing the respective amounts. Pennies. these are RANDOM worksheets. This page contains links to free math worksheets for Money Word Problems problems. Beginner Math Worksheets - Printable Worksheets for Money BINGO is a fun and educational game for kids to practice counting money. Help students improve their math skills with these ready-made percent, decimal, and money worksheets for grades 1-6.These problems are popular on standardized math tests, and these...<\|endoftext\|>	3.671875
2,015	## What is topological sort? Topological sort is a sorting technique that sets up a hierarchy in a process. In this technique, when a directed network has nodes connected by arrows, the root node is placed before the successor node. A root node is the one from which arrows arise. The nodes to which the arrows point are called successor nodes. Based on this root-first principle, topological sorting is done. The algorithms written for the topological sort are known as topological sort algorithms. The topological sort algorithm returns an array of nodes for a directed graph. The node that has the minimum number of arrows comes first. The order in which nodes appear in topological sort is called topological ordering. ## What is topological sort for Directed Acyclic Graph? Topological sort produces an array of nodes or vertices in descending order of the number of arrows directed towards nodes. Hence, topological sorting is possible only for acyclic graphs as the first node of the array should have an in-degree zero. A graph having at least one vertex with in-degree zero is called a Directed Acyclic Graph (DAG). Consider the directed acyclic graph given below. The graph “5 4 2 3 1 0” is a topological sort. As both 4 and 5 are in in-degree zero, another possible topological sort is “4 5 2 3 1 0”. Note that an in-degree zero vertex has no incoming arrows or edges. Let’s understand it better through the following example: Vertex 1 is in in-degree zero, so it appears first in the array. Vertex 1 is followed by vertices 2, 3, 4, and then 5. Thus, the topological order of the given graph is as shown below: There can be two topological orders for the system—[1, 2, 3, 4, 5] and [1, 3, 2, 4, 5]. Note: • You need a DAG for topological sorting. • For a DAG, more than one topological sort is possible. • A vertex that has in-degree zero comes first in the topological order. ## Cyclic graphs A directed graph in which a cycle appears, or there is only one node with both incoming and outgoing edges, is called a cyclic graph. • A graph that does not contain a cycle is called an acyclic graph. • A cyclic graph with exactly one unidirectional cycle is called a unicyclic graph. • A cyclic graph does not have a valid topological order. ## Directed Acyclic Graph (DAG): Topological Ordering The nodes or vertices of a directed graph are connected through arrows called edges. These edges run from one vertex to another. A directed graph that does not have a cycle is called a directed acyclic graph. No node in a directed acyclic path contains both incoming and outgoing edges. The following is the representation of directed graphs with and without cycles. • Directed graph with cycles • Directed graph without cycles Note: Directed graphs without cycles are called DAGs. ## Depth-First Search (DFS) Depth-First Search (DFS) is an uninformed search technique that allows backtracking. It uses a recursive algorithm. DFS starts with the first node of the tree and moves ahead. And if the node is not required according to the current path, it can go back, which is called backtracking. While moving backward to find nodes for the next path, it touches all the nodes of the current path till it traverses all the untouched ones. This process helps to select the next path. Using stacks, you can implement the recursive nature of DFS. The basic idea of the stack is as follows: • Select the in-degree zero node, and keep all the remaining nodes in a stack. • Pick the next node in the array of order while keeping the rest in the stack. • Repeat the process until all the nodes are picked from the stack. ## Topological Sorting vs Depth-First Search (DFS): Let’s discuss the basic differences between topological sort and depth-first search. Let us consider the below graph: The topological ordering of the graph is [5 4 2 3 1 0] whereas the DFS will return [5 2 3 1 0 4]. ## Algorithm to find Topological Sorting The steps to produce the topological order for any directed graph are listed below: • The first node in topological ordering is always an in-degree zero node. So, you need to find a node that has no edges coming into it. • Remove the node chosen in the topological order and its outgoing edges from the graph. • Add the next in-degree zero node in the topological order. • Remove the node and the outgoing edges, and repeat the process until you reach the last node. Consider the following graph: • Look for an in-degree zero node and add it to the order. Here, B is the in-degree zero vertex. It will come first in the topological order array. • Remove the graph of B and its outgoing edges to get the next in-degree zero node. • Similarly, E and C are in-degree zeros. As there can be more than one topological order for a graph, choose any of E and C. • Repeat the process to get the next in-degree zero vertex. • Pick the next node and repeat the process. • Repeat the process until you reach the last node. Hence, the topological order of the graph is [B E A C D]. ## Time and space complexity For obtaining an optimal algorithm, you should consider time and space complexity. There are various ways to design an algorithm. However, the most efficient ones are those that take less time to execute and occupy minimum space. In this section, you’ll understand time and space complexity one by one. ### Time complexity The time complexity of an algorithm is its execution time as a function. For huge input sizes and worst-case scenarios, the time complexity is examined. The time taken by an algorithm can be controlled by following the given points while writing the algorithm. • For each node in the graph, the in-degree zero is determined. The time taken in the process is referred to as the O(E) time. Here, E represents the number of edges. • The process of determining the in-degree zero requires going through each node. The time taken for this is referred to as the V(N) time. Here, N represents the number of nodes. • The time complexity is the sum of the time taken in passing through each node and path. It is represented as O(E) + V(N). ### Space complexity Space complexity is the space or memory occupied by the process in the system when it is executed as a function. The space occupied depends on the number of vertices in the graph. If there are N nodes and each takes one unit space, then O(N) is the space complexity of the graph system. ## Applications of topological sort Topological sort is used in • designing hierarchy for systems, • determining the priority list of files, and • serializing data. ## Common Mistakes Remember the following points while doing topological sort of a graph: • Make sure that the graph provided is not cyclic. • In the case of more than one topological sort, do not get confused; it is possible. ## Formulas • Time complexity is O(M+N), where M is the number of nodes and N is the number of edges in the graph. • The complexity of space is O(N), where N denotes the number of nodes. ## Context and Applications This topic is significant in the professional exams for both graduate and postgraduate courses, especially for Bachelor in Computer Science Bachelor in Computer Application Bachelor of Technology in Computer Science and Engineering Master in Computer Application Master of Technology in Computer Science and Engineering Selection Sort Bubble Sort Insertion Sort Merge Sort ## Practice Problems Q. 1 Topological sort is applicable on: (A) Direct Cyclic Graph (B) Directed Acyclic Graph (C) Both A and B (D) None of the above Correct Option: (B) Q. 2 Topological sort finds: (A) In-degree zero vertex (B) In-degree one vertex (C) Aligned vertex (D) None of the above Correct Option: (A) Q. 3 Which of the following is not a sorting type in the database management system? (A) Topological Sort (B) Quick Sort (C) Bubble Sort (D) First Sort Correct Option: (D) Q. 4 A cyclic graph having exactly one unidirectional cycle is called a/an: (A) Acyclic graph (B) Unicyclic graph (C) First cycle graph (D) None of the above Correct Option: (B) Q. 5 Topological sort returns: (A) An array (B) A word (C) A string (D) A file Correct Option: (A) ### Want more help with your computer science homework? We've got you covered with step-by-step solutions to millions of textbook problems, subject matter experts on standby 24/7 when you're stumped, and more. Check out a sample computer science Q&A solution here! *Response times may vary by subject and question complexity. Median response time is 34 minutes for paid subscribers and may be longer for promotional offers. ### Search. Solve. Succeed! Study smarter access to millions of step-by step textbook solutions, our Q&A library, and AI powered Math Solver. Plus, you get 30 questions to ask an expert each month. Tagged in EngineeringComputer Science<\|endoftext\|>	4.5
897	Thalassaemia and Sickle Cell Disorder Research has recently been carried out by York, Loughborough and De Montfort Universities with the objective of allowing the uninterrupted education of children who suffer from Thalassaemia and Sickle Cell Disorder. The full guidance is available from www.sicklecelleducation.com and www.sicklecellanaemia.org. What is Sickle Cell Disorder (SCD)? Sickle cell disorder (SCD) is a collective name for a series of serious inherited chronic conditions that can affect all systems of the body. These sickle cell disorders are associated with episodes of severe pain called sickle cell painful crises. Many systems of the body can be affected, meaning that different key organs can be damaged and many different symptoms can occur in many different parts of the body. What is Beta-Thalassaemia Major? Beta-thalassaemia major is a serious inherited blood condition in which the red blood cells are nearly empty of haemoglobin, the key part of the blood that carries oxygen around the body. The first life-saving step of treatment involves children having blood transfusions every 3-4 weeks for the rest of their lives. This extra blood introduces extra iron into the body that the body cannot get rid of easily. The second step of treatment involves drugs that get rid of the excess iron. Both these disorders are inherited and cannot, like a cold, be caught from another person. A person may be a carrier of either disease, in which case they have one normal and one affected gene. Such a person will normally be perfectly healthy and may not know they have the trait unless they have a blood test. If two carriers have children together there is a one in four chance that each child could have sickle-cell anaemia and a one in two chance that a child could be a carrier. Certain factors have been identified as more likely to precipitate a painful sickle cell crisis. These include infections, colds and/or damp conditions, pollution, dehydration, strenuous exertion, stress, sudden changes in temperature, alcohol, caffeine, and smoking. Young people with SCD need to be well hydrated to reduce the likelihood of becoming ill. This means that they need a ready supply of fresh drinking water available at all times and need to be able to access drinking water in class. People with SCD produce large quantities of dilute urine and need to go to the toilet more often so will need frequent toilet breaks. A child with SCD may experience severe anaemia. This may mean they feel tired, lethargic and unable to concentrate. It is possible that a sufferer will find everyday activities such as climbing stairs extremely tiring. Young people with betathalassaemia major are likely to be tired towards the end of their 4 week cycle of transfusions. It is important that teachers do not mistake serious medical symptoms of SCD or betathalassaemia major for laziness. A child with SCD will need to avoid hard, physical exercise that could precipitate a sickle cell crisis. For children with SCD, cold or wet weather, or exposure of the skin to cooling wind, may all be a trigger to episodes of illness. Obligatory sports and gym sessions out of doors in cold and wet weather is a potent stimulant to crisis for some children. It is important to listen to the child and parent, and follow advice from their specialist medical teams about this. Young people with SCD and thalassaemia may catch infections more easily. Safe storage and dispensing of any antibiotic drugs prescribed for a student with sickle cell disorder or thalassaemia is essential. Young people with SCD need to avoid activities that require outdoor work in cold or damp conditions as well as under-heated classrooms, especially mobile classrooms. They may need to wear a coat in class and should be allowed to stay inside at break times in cold or wet and windy weather. A student with SCD or betathalassaemia major will have an individual health care plan, which should be reviewed yearly. As both SCD and beta-thalassaemia major have numerous possible complications affecting many systems of the body, it is important, where possible, to include a specialist sickle cell or thalassaemia nurse in drawing up this plan.<\|endoftext\|>	3.84375
278	Check out our collection of printable First Grade Worksheets available in the following images! Loads of options of first grade exercises are available such as simple Math, English, and more. Get and print these free and printable first grade worksheets with fun topics to help improve your kids’ basic skill in counting and other important skills. Below are various worksheets and handouts for the greater study time. Check out and enjoy! Using these free and printable First Grade Worksheets will help your grade 1 children practice more on the knowledge that she/he has learned so far. The printable worksheets include interesting topics to improve your kids’ vocabulary, counting, and reading skills. With different kinds of problems for kids, watch these kids grasp and learn new skill. More worksheets are listed below. Worksheets are a great tool for practicing, and practice often helps children understand knowledge and practice it better. These worksheets include math and phonics worksheets specifically made for first grade students. There are various worksheets that you can choose for your kids. Click on the image to save it and print right from your browser. All pictures presented are printable resources of first grade worksheets for children. Make sure to print these worksheets and hand them to your children! Have a fun studying activity!<\|endoftext\|>	4.09375
453	Etymology is the study of the origin of words and the way their meanings change over time. The sources which are available to us suggest two theories relating to the where Strathardle got its name. The first theory is explained in Notes on Strathardle (1888): ‘The Gaelic name of the strath is Strath-ardshil, pronounced Sra-ardil, and is supposed to be derived from two Gaelic words meaning the strath of high stream or river – either having reference to the high sources of the river, or to its being more elevated in comparison with another stream, such as Athole – in Gaelic Athshil.’ Therefore, it would seem, the name would have evolved from Strath-Athshil to Strathardle. The second theory is more plausible and is the one most commonly held. The Rev. Allan Stewart, the parish minister of Kirkmichael in 1790s, wrote about Glenshee and Strathardle’s history in the First Statistical Account of Scotland. His entry for the Parish of Kirkmichael says that: ‘According to tradition, Strathardle was anciently called in Gaelic, Srath na muice brice; the strath of the spotted wild sow; which name it is said to have retained till the time of the Danish invasions, when, in a battle fought between the Danes and the Caledonians, at the head of the country, a chief, named Ard-fhuil, high, or noble blood, was killed, whose grave is shown to this day. From him the country got the name of Srath Ard-fhuil, Strathardle.’ Notes on Strathardle also admits that this is a more likely explanation, stating: ‘It appears that the strath’s original name was, in Gaelic, Strath na muic breac…[i.e.] the strath of the bridled boar; so it seems very probable that the changing of it was in some way connected with the above-named hero [Ardil].’<\|endoftext\|>	3.71875
541	A denizen of lofty, fog-shrouded mountain rain forests, the Hawai‘i Creeper, formerly called Olive-green Creeper, is a small green bark-picker typically seen creeping along trunks and large branches, peering back and forth, and probing under bark as it searches out its insect prey. The Hawai‘i Creeper was historically widespread and relatively common, but despite its former abundance, no Hawaiian name was ever recorded for this species. Lacking the spectacular bill, brilliant colors, complex songs, and conspicuous behavior of many Hawaiian birds, the drab Hawai‘i Creeper was frequently overlooked and perhaps underappreciated owing to its superficial similarity to the Hawai‘i ‘Amakihi. In fact, Scott Wilson, who collected some of the first specimens in the mid-1880s and described the species shortly thereafter, was unaware that he had encountered a new species until he examined his specimens in England, and consequently had no information about the Hawai‘i Creeper's habits. The first nest was found in 1975, yet the species has only in the past few years become the object of long-term field studies. Despite this neglect, the Hawai‘i Creeper can be quite conspicuous in its limited range when it forms large mixed-species postbreeding flocks with other Hawaiian honeycreepers—the Hawai‘i Creeper's incessant, stuttering dee-dee, dee-dee-dee juvenile calls are among the most characteristic and memorable sounds of summer in its high-mountain refugia. When the Hawai‘i Creeper was listed as a Federally Endangered Species in 1975, it was not known whether it was merely uncommon or onthe verge of extinction. The epic Hawaiian Forest Bird Survey of the late 1970s and early 1980s found this species to be relatively widespread in higher-elevation ‘öhi‘a and koa forests on the windward side of Hawai‘i Island, however, and estimated a total population of around 12,500 in-dividuals in 4 disjunct populations. No comprehensive surveys have been conducted since that time, but local census data indicate stable populations in at least some protected higher-eleva-tion forests and declines or extirpations in the few lower-elevation sites where the Hawai‘i Creeper had been found. Most data cited here are from studies at Hakalau Forest National Wildlife Refuge (NWR), located in high-elevation wet and mesic forest on eastern Mauna Kea volcano (Figure 1).<\|endoftext\|>	3.734375
567	Viruses in phosphorus-starved regions of the ocean are able to take advantage of bacteria which needs phosphorus as a nutrient. The bacteria uses viruses only to have the virus inject its own DNA into the bacteria, which results in the host bacterium exploding after about 10 hours. Researches have found out how the viruses uses bacteria to support the replication of their own DNA. They have discovered that photosynthetic ocean bacteria should beware of viruses as these viruses are carrying genetic material which are taken from their previous bacterial hosts that tricks the new host into using its own machinery to activate the genes. These bacteria are stressed by the lack of phosphorus uses a nutrient have their phosphorus-gathering machinery in high gear. The virus senses the host’s stress and offers what seems like a helping hand,bacterial genes nearly identical to the host’s own that enable the host to gather more phosphorus. The host uses these genes but the additional phosphorus goes towards supporting the virus’s replication of its own DNA. Once that process is completed, about 10 hours after infection, the virus explodes its host, releasing progeny viruses back into the ocean where they can invade other bacteria and repeat the same process. The additional phosphorus-gathering genes provided by the virus keep its reproduction cycle on schedule. The virus is choosing a very sophisticated component of the host’s regulatory machinery for enhancing its own reproduction.The phages have evolved the capability to sense the degree of phosphorus stress in the host they’re infecting and have captured, over evolutionary time, some components of the bacteria’s machinery to overcome the limitation. This research was performed using the bacterium Prochlorococcus and Synechococcus, which together produces about one-sixth of the oxygen in Earth’s atmosphere. Prochlorococcus is about one micron in diameter and can reach densities of up to 100 million per liter of seawater,Synechococcus is only slightly larger and a less abundant. The bacterial mechanism is a two-component regulatory system, that refers to the microbe’s ability to respond to external environmental conditions. This system prompts the bacteria to produce extra proteins that bind to phosphorus and bring it into the cell. The gene carried by the virus encodes this same protein. The research indicates that the phage which infects these bacteria have evolved right along with their hosts. The whole system is a bit of evidence for the incredible intimacy of the relationship of phage and host.Most of what we understand about phage and bacteria has come from model microorganisms used in biomedical research. The environment of the human body is dramatically different from that of the open oceans, and these oceanic phage have much to teach us about fundamental biological processes.<\|endoftext\|>	4.125
716	# Lesson 21 Rational Equations (Part 2) ### Problem 1 Solve $$x-1 = \dfrac{x^2 - 4x + 3}{x+2}$$ for $$x$$. ### Solution For access, consult one of our IM Certified Partners. ### Problem 2 Solve $$\frac{4}{4-x} = \frac{5}{4+x}$$ for $$x$$. ### Solution For access, consult one of our IM Certified Partners. ### Problem 3 Show that the equation $$\frac{1}{60} = \frac{2x+50}{x(x+50)}$$ is equivalent to $$x^2 - 70x - 3,\!000 = 0$$ for all values of $$x$$ not equal to 0 or -50. Explain each step as you rewrite the original equation. ### Solution For access, consult one of our IM Certified Partners. ### Problem 4 Kiran jogs at a speed of 6 miles per hour when there are no hills. He plans to jog up a mountain road, which will cause his speed to decrease by $$r$$ miles per hour. Which expression represents the time, $$t$$, in hours it will take him to jog 8 miles up the mountain road? A: $$t=\frac{8-r}{6}$$ B: $$t=\frac{8}{6+r}$$ C: $$t=\frac{6+r}{8}$$ D: $$t=\frac{8}{6-r}$$ ### Solution For access, consult one of our IM Certified Partners. ### Problem 5 The rational function $$g(x) = \frac{x+10}{x}$$ can be rewritten in the form $$g(x) = c + \frac{r}{x}$$, where $$c$$ and $$r$$ are constants. Which expression is the result? A: $$g(x)=x+\frac{10}{x}$$ B: $$g(x)=1+\frac{10}{x}$$ C: $$g(x)=x -\frac{10}{x+10}$$ D: $$g(x)=1-\frac{1}{x+10}$$ ### Solution For access, consult one of our IM Certified Partners. (From Unit 2, Lesson 18.) ### Problem 6 For each equation below, find the value(s) of $$x$$ that make it true. 1. $$10 = \frac{1+7x}{7+x}$$ 2. $$0.2=\frac{6+2x}{12+x}$$ 3. $$0.8= \frac{x}{0.5+x}$$ 4. $$3.5=\frac{4+2x}{0.5-x}$$ ### Solution For access, consult one of our IM Certified Partners. (From Unit 2, Lesson 20.) ### Problem 7 A softball player has had 8 base hits out of 25 at bats for a current batting average of $$\frac{8}{25}=.320$$. How many consecutive base hits does she need if she wants to raise her batting average to .400? Explain or show your reasoning. ### Solution For access, consult one of our IM Certified Partners. (From Unit 2, Lesson 20.)<\|endoftext\|>	4.5625
1,491	In Part 1 of this article, I outlined some of the variables which can affect Earth’s climate, and gave a brief overview of plate tectonics, and how changes in continental positioning can lead to climate change through albedo feedback and via the alteration of ocean circulation and heat distribution patterns. In doing so, I used the example of the Rodinia Supercontinent and the Snowball Earth hypothesis of the Neoproterozoic era in order to relate the concepts to events in Earth’s prehistory. For the sake of completeness, I want to finish up that example by briefly going over a few proposed triggering mechanisms that could have made a runaway albedo feedback loop possible in the Cryogenian period. After that, I want to go over the ways in which the presence of mountain ranges can affect local and regional climate. Although the exact causal sequence (and even the SnowBall Earth hypothesis itself) is still an unresolved area of active scientific debate, there are multiple candidates for such a cooling trigger mechanism: decreased solar luminosity, global cooling from a Super Volcano eruption, or perhaps a reduction in atmospheric methane – a much stronger greenhouse gas than CO2 – due to reactions with atmospheric oxygen could plausibly have contributed. There is also evidence suggesting increased sequestration of CO2 by rocks due to weathering effects from high precipitation levels. The idea behind the latter hypothesis is the following: Carbonate and Silicate rock weathering reactions are important carbon sinks in Earth’s carbon cycle: Carbonate rock weathering reaction: CaCO3 + CO2 + H2O → 2HCO3– + Ca2+ Silicate rock weathering reaction: CaSiO3 + 2CO2 + H2O → 2HCO3– + Ca2+ + SiO2 Precipitation levels tend to be high around the equator, which therefore increases these weathering reactions, and thus increases CO2 sequestration, thereby decreasing the greenhouse effect, and leading to global cooling. Once sufficient equatorial land ice could accumulate, the aforementioned ice albedo feedback effect could ensue. The end of the extreme Neoproterozoic glaciation is an interesting story in its own right, but in addition to the breakup of Rodinia via tectonic plate movement, it involves a greenhouse effect facilitated by a very long period of extreme volcanism, which I’ll be covering in a subsequent post. It also directly preceded the Cambrian explosion. Rodinia was not the last time Earth had a tropical continental arrangement, but a critical change occurred: the evolution of forests. The Devonian Period (419 – 372 mya) featured the Greening of the Continents, and the evolution of forests, which serve as carbon sinks. So, by the time the Pangea Supercontinent formed (about 250 mya), an equatorial continental arrangement did not necessarily mean excessive rock weathering reactions, let alone runaway ice albedo feedback. Plate tectonics can also influence climate through the formation of mountains (orogeny). Mountains can have profound effects on climate, particularly in their effect on precipitation patterns on the surrounding lands. Higher points on mountains tend to correspond to lower temperatures, so as rising moist warm air makes its way up the windward side of a mountain, it cools down, thus causing its ability to hold water to decline, which in turn leads to precipitation (rain or snow). Consequently, the leeward side of the mountain will often receive less precipitation, and by the time that air reaches the adjacent flat lands, it may not have any moisture left. This is one reason deserts are sometimes located directly next to mountain ranges. Mountains can also affect air circulation patterns great distances away. You can read more about how mountains affect climate and weather patterns at the Mountain Professor. It’s perfectly natural to wonder what (if any) role changes in Continental positioning have played in the global warming and climate change we’ve been experiencing currently. The answer is very little (if any). The issue is the RATE at which the current change has been occurring. These processes I’ve described take place over tens or even hundreds of millions of years. The continents tend to move at roughly 2 cm a year, and even faster moving plates rarely exceed 10 cm even on a fast year. The continents haven’t moved all that much in the last few hundred thousand years, and less than meter or so within the last 150 years during which the recent changes have been occurring. So continental movement is ruled out as a principle causal determinant of the current warming. It simply doesn’t induce climatic changes fast enough to explain the rapidity with which the recent warming has been occurring. However, its effects on the planet over the long term can be quite profound indeed. Brady, P. V. (1991). The effect of silicate weathering on global temperature and atmospheric CO2. Journal of Geophysical Research: Solid Earth,96(B11), 18101-18106. Davies, N. S., Gibling, M. R., & Rygel, M. C. (2011). Alluvial facies evolution during the Palaeozoic greening of the continents: case studies, conceptual models and modern analogues. Sedimentology, 58(1), 220-258. Eyles, N., & Januszczak, N. (2004). ‘Zipper-rift’: a tectonic model for Neoproterozoic glaciations during the breakup of Rodinia after 750 Ma. Earth-Science Reviews, 65(1), 1-73. Goodale, C. L., Apps, M. J., Birdsey, R. A., Field, C. B., Heath, L. S., Houghton, R. A., … & Nabuurs, G. J. (2002). Forest carbon sinks in the Northern Hemisphere. Ecological Applications, 12(3), 891-899. Harris, B. (2008). The potential impact of super-volcanic eruptions on the Earth’s atmosphere. Weather, 63(8), 221. Kerr, R. A. (1999). Early life thrived despite earthly travails. Science,284(5423), 2111-2113. Lackner, K. S. (2002). Carbonate chemistry for sequestering fossil carbon. Annual review of energy and the environment, 27(1), 193-232. Murphy, J. B., Nance, R. D., & Cawood, P. A. (2009). Contrasting modes of supercontinent formation and the conundrum of Pangea. Gondwana Research, 15(3), 408-420. Schrag, D. P., Berner, R. A., Hoffman, P. F., & Halverson, G. P. (2002). On the initiation of 814 a snowball Earth. Geochemistry Geophysics Geosystems, 3. Torsvik, T. H. (2003). The Rodinia jigsaw puzzle. Science, 300(5624), 1379-1381.<\|endoftext\|>	3.96875
783	The Lugbara plateau is extremely fertile, supporting, in its center, a population density of more than 80 to the square kilometer during the 1950s. The Lugbara are highly efficient peasant farmers, their staples being grains (traditionally millets and sorghums, now with some maize), root crops (traditionally sweet potatoes, now also cassava), and legumes of many kinds. With increasing dependence on cassava, the formerly highly nutritious diet of the Lugbara has been drastically worsened. Cash crops were encouraged during the colonial period, but, owing to edaphic and climatic factors and the long distance to the nearest markets for cash crops (some 800 kilometers to the south), few have been profitable. Groundnuts, sunflower, cotton, and tobacco have all been tried, only the latter two with success. The main export has been that of male labor to the Indian-owned sugar plantations and the African-owned farms of southern Uganda; about one-quarter of the men are absent at any one time. Until the Obote atrocities of the 1970s, the Lugbara peasant society could maintain its members on a level of nutrition and health that was at least equal to those of most Third World societies. The Lugbara keep some livestock: cattle, goats, sheep, fowl, dogs, and cats; before the cattle epidemics of the 1980s, they had far greater herds. Cattle, goats, and sheep are not killed for consumption, but rather for ancestral sacrifices (although the meat is actually consumed by those attending); the sale of hides and skins earns valuable income. Traditionally, local exchange of surplus foodstuffs was in the form of gifts between kin and barter with others. Small local weekly markets came into being during the 1920s, with the introduction of cash, maize (used for beer brewing), and consumer goods such as kerosene, cigarettes, and cloth. (As late as the 1950s, women wore only pubic leaves and beads, and elder men, animal skins.) The division of labor is sharply defined. Men and women share agricultural tasks, the men opening the fields and the women doing most of the remaining work. Men hunt and herd cattle; women do the arduous and the time-consuming everyday domestic tasks. Formerly, men were responsible for the physical protection of their families and for waging feuds and war. Men hold formal authority over their kin, but older women informally exercise considerable domestic and lineage authority. Land is held by lineages, as land is traditionally not sold or rented. Women are allocated rights of use by their husbands' lineage elders. The country is open, composed of countless small ridges with streams between them; the compounds and fields are set on the ridges. Houses, round and made of mud and wattle and thatch, are dispersed throughout the almost continuous fields. Few settlements today have more than three or four houses. A century ago compounds were large for defense, but colonial administration removed the threat of war and feud. A house is the dwelling place of one wife and her children. If she is the only wife, her husband also sleeps there. If a man has more than one wife, he moves from one house to another in turn. The house, and especially its hearth, is very much a female domain. A compound, of one or more women's houses, is typically surrounded by a euphorbia fence, often with a nearby cattle kraal; beyond lie fields. The fields vary in type: small gardens typically on the sites of earlier houses, home fields under permanent cultivation and often irrigated, and farther fields under shifting cultivation, the fallow used for grazing. Stretching out beyond the fields is untilled grazing land, and near the edges of the country are wide extents of bush and forest, used for hunting.<\|endoftext\|>	3.765625
1,464	\|Optimization of a mechanical heart The heart has four chambers. The upper two are the right and left atria. The lower two are the right and left ventricles. Blood is pumped through the chambers, aided by four heart valves. The valves open and close to let the blood flow in only one direction. What are the four heart valves? - The tricuspid valve is between the right atrium and right - The pulmonary or pulmonic valve is between the right ventricle and the pulmonary artery. - The mitral valve is between the left atrium and left ventricle. - The aortic valve is between the left ventricle and the Each valve has a set of flaps (also called leaflets or cusps). When working properly, the heart valves open and close fully. Heart valves don't always work as they should. A person can be born with an abnormal heart valve, a type of congenital heart defect. Also, a valve can become damaged by - infections (e.g. infective endocarditis) - changes in valve structure in the elderly - rheumatic fever A mechanical (or artificial) heart valve is a man-made device that is used to replace one of a patient’s own damaged or diseased heart valve that cannot be repaired. A biological valve, from either an animal (xenograft) or a deceased human donor (allograft) may also be used to replace the patient’s original valve. In most cases, the use of a mechanical heart valve can lengthen or even save a patient’s life. The valves are durable and can last 30 years or longer. However, there is a risk of complications, and most patients will need to take anticoagulants for the rest of their lives to reduce the risk of blood clot Here we perform a numerical study of a valve construction based on a curved central guide strut and a flat disc. This has two advantages: (i) It allows assembly of the valve and disc without imparting stress on the valve housing and (ii) it allows the disc to move out of the annular plane (which is the tightest constricture of the resultant outflow tract). This type of valve is produced by Medtronic. Turbulence in the cardiovascular system leads to higher flow resistance, resulting in increased pressure gradients. Furthermore, elevated levels of turbulent shear stresses may create hemolysis or platelet activation , . This may in turn lead to thrombosis and embolism. It was also shown that turbulent shear stresses can be associated with the development of aortic aneurysms , . The damage done to red blood cells can be described as a function of spatial distribution, exposure time and magnitude of turbulent shear stresses. In particular, in and the critical parameters of turbulent shear stress were identified, which lead to lethal or sublethal damage of blood cells. The goal of an optimal heart valve is to retain a near physiologic turbulence profile. The benefits are minimal pressure gradients and very low levels of thrombosis and thromboembolism. This was experimentally prooved for the Medtronic Hall valve in To achieve this design goal, computational fluid dynamics simulations are of great advantage. The valve and the adjacent arteries were modeled using NURBS to facilitate (i) adaptive mesh refinement and (ii) a parametric geometrical model which is suitable for design optimization. The fluid zone was meshed automatically by a hexahedral mesh using approx. 400.000 elements. The fluid problem was modeled using a non-Newtonian model which was solved using a modified SIMPLE algorithm. The moving bodies were considered using fluid structure interaction considered. Thereby, the arterial wall is simulated as a hyperelastic medium and the valve disc as a rigid, but movable body. The results of the simulation are shown in Fig. 1 and 2. Fig. 1: CFD simulation of the mechanical heart valve. The image depicts the flow situation at the systolic phase (the valve is fully opened under 75°). Streamlines are colored by the local pressure. Section planes show the Despite a common belief in a symmetrical (i.e., bullet-shaped) flow pattern in the ascending aorta, research has confirmed that natural aortic flows are eccentric—with the region of highest velocity occurring in the non-coronary sinus [Paulsen et al.]. \|Fig. 2: Velocity magnitude at fully opened valve (three representative sections). MH, Hellums JD, McIntyre LV, et al. Platelets and Shear Stress. Blood 1996;88:1525-41. P, Perthal M, Nygaard H, et al. Medtronic Hall versus St. Jude Mechanical Aortic Valve: Downstream Turbulences with Respect to Rotation in Pigs. J Heart Valve Dis 1998;7:548-55. WW, O'Rourke MF. McDonald's Blood Flow in Arteries. Theoretic, Experimental, and Clinical Properties. 3rd ed. Philadelphia: Lea & Febiger, 1990;54-71. PK, Nygaard H, Hasenkam JM, et al. Analysis of Velocity in the Ascending Aorta in Humans. A Comparative Study Among Normal Aortic Valves, St. Jude Medical and Starr-Edwards Silastic Ball Valves Int. J Artif Org 1988; 11:293-302. ZM. Mechanisms of Shear-Induced Platelet Adhesion and Aggregation. Thromb Haemost 1993:70:119-123. PD, Sabbah HN. Hemorheology of Turbulence. Bioheol 1980;17:301-19. W, Reul H, Herold M, et al. In Vitro Wall Shear Measurements in Aortic Valve Prostheses. J Biomech 1984;17:263-79. AP, Wick TM, Reul H., The Influence of Flow Characteristics of Prosthetic Heart Valves on Thrombus Formation. I: Butchart EG, Bodner E (eds.) Current Issues in Heart Valve Disease: Thrombosis, Embolism and Bleeding. London: ICR, 1992;123-48. AP, Woo Y-R, Sung H-W. Turbulent Shear Stress Measurements in Aortic Valve Prostheses. J Biomech 1986;19:433-42.<\|endoftext\|>	3.703125
3,378	# Solving percent problems \| Decimals \| Pre-Algebra \| Khan Academy percent, amount, and base in this problem. And they ask us, 150 is 25% of what number? They don’t ask us to solve it, but it’s too tempting. So what I want to do is first us to solve. But first, I want to answer this question. And then we can think about what the percent, the amount, and the base is, because those are just words. Those are just definitions. The important thing is to be able to solve a problem like this. So they’re saying 150 is 25% of what number? Or another way to view this, 150 is 25% of some number. So let’s let x, x is equal to the number that 150 is 25% of, right? That’s what we need to figure out. 150 is 25% of what number? That number right here we’re seeing is x. So that tells us that if we start with x, and if we were to take 25% of x, you could imagine, that’s the same thing as multiplying it by 25%, which is the same thing as multiplying it, if you view it as a decimal, times 0.25 times x. These two statements number, you take 25% of it, or you multiply it by 0.25, that is going to be equal to 150. 150 is 25% of this number. And then you can solve for x. So let’s just start with this one over here. Let me just write it separately, so you understand what I’m doing. 0.25 times some number is equal to 150. Now there’s two ways we can do this. We can divide both sides of this equation by 0.25, or if you recognize that four quarters make a dollar, you could say, let’s multiply both sides of this equation by 4. You could do either one. I’ll do the first, because that’s how we normally do algebra problems like this. So let’s just multiply both by 0.25. That will just be an x. And then the right-hand side will be 150 divided by 0.25. And the reason why I wanted to is really it’s just good practice dividing by a decimal. So let’s do that. So we want to figure out what 150 divided by 0.25 is. And we’ve done this before. When you divide by a decimal, what you can do is you can make the number that you’re dividing into the other number, you can turn this into a whole number by essentially shifting the decimal two to the right. But if you do that for the number in the denominator, you also have to do that to the numerator. So right now you can view this as 150.00. If you multiply 0.25 times 100, you’re shifting the decimal two to the right. Then you’d also have to do that with 150, so then it becomes 15,000. Shift it two to the right. So our decimal place becomes like this. So 150 divided by 0.25 is the same thing as 15,000 divided by 25. And let’s just work it out really fast. So 25 doesn’t go into 1, doesn’t go into 15, it goes into 150, what is that? Six times, right? If it goes into 100 four times, then it goes into 150 six times. 6 times 0.25 is– or actually, this is now a 25. We’ve shifted the decimal. This decimal is sitting right over there. So 6 times 25 is 150. You subtract. You get no remainder. Bring down this 0 right here. 25 goes into 0 zero times. 0 times 25 is 0. Subtract. No remainder. Bring down this last 0. 25 goes into 0 zero times. 0 times 25 is 0. Subtract. No remainder. So 150 divided by 0.25 is equal to 600. And you might have been able to do that in your head, because when we were at this point in our equation, 0.25x is equal to 150, you could have just multiplied both sides of this equation times 4. 4 times 0.25 is the same thing as 4 times 1/4, which is a whole. And 4 times 150 is 600. So you would have gotten it either way. And this makes total sense. If 150 is 25% of some number, that means 150 should be 1/4 of that number. It should be a lot smaller than that number, and it is. 150 is 1/4 of 600. Now let’s answer their actual question. Identify the percent. Well, that looks like 25%, that’s the percent. The amount and the base in this problem. And based on how they’re wording it, I assume amount means when you take the 25% of the base, so they’re saying that the amount– as my best sense of it– is that the amount is equal to the percent times the base. Let me do the base in green. So the base is the number you’re taking the percent of. The amount is the quantity that that percentage represents. So here we already saw the percent is 25%. That’s the percent. The number that we’re taking 25% of, or the base, is x. The value of it is 600. We figured it out. And the amount is 150. This right here is the amount. The amount is 150. 150 is 25% of the base, of 600. The important thing is how you solve this problem. The words themselves, you know, those are all really just definitions. ### 93 Replies to “Solving percent problems \| Decimals \| Pre-Algebra \| Khan Academy” 1. norwayte says: Good one – doing it in slow motion and color coded. 2. jsr9422 says: thanks a lot 🙂 3. nando5561 says: good video 4. Junerey Piojo says: Is/Of = %/100 5. Saif AlZubaidi says: cool, what program r u usin? 6. Vianney Rodriguez says: People are always asking me where i go to school. I tell em khanacademy 7. Glory7studios says: just do this 150 * 100/25 ۞ 8. Dimensions100 says: My dad's whole family is full of teachers….after I learn from this, I'm gonna pass this gift called KhanAcademy on to them. 9. SaKomunikasyong Glooball says: dota players are so lucky, these percent problems are a bit easy for them 10. Mathew Lin says: Anyone notice at 3:12 he draws a cat face 11. Matt Mazur says: Anyone know what software/inputs he is using to create this video? 12. danielwild794 says: just multiply the % by the number what program is that ?? 🙂 14. minhaj khan says: what confused………………. 8( 15. Yousra Isaac says: In Khan we trust!!!!! <3 16. Megan Elizabeth says: This is still very hard . 17. tyler burg says: Nadia took two practice exams to prepare for her college entrance exam. On the first practice exam, she scored one thousand one hundred. After studying, her score increased by thirty percent. What was her score on the second practice exam? one thousand one hundred thirty one thousand four hundred one thousand four hundred thirty one thousand five hundred seventy-one This was on my flvs class 😀 awesome 18. Ama O says: u r a life saver!! 19. Trekole says: I am going to use your work to study for the GRE, but I don't know which book you're using? Please let me know, thanks! 20. Aimee Rudd says: Dude! You so rock! Thanks for your help! 21. Maximilian Cole says: it all makes sense now 22. Its. Ric0 says: THAnks man you helped i wondering if i should subscribe 23. Dominick Hall says: hi 24. Destroyer_Of_All says: subscribe to Destroyer_Of_All 25. BabelaSweet says: hi i hope u can help me. I am looking for a formula to fin the following: I need to know 3 things. subtotal, 7% of subtotal, total (sum of subtotal and 7%) I have the total which is 4100 and i know the percent used to calculate that is 7%…how can I find out which the SUBTOTAL number should be? 26. Chili Pepper Gaming says: THANK YOU 27. Solid Director says: Daurde Sandstorm 28. Quincy Smith says: At 3:03 when he was writing 150 as 150.00 and then he was carrying the decimal, the two zeroes at the end of it wight he carried decimal line under it made it look like a :3 face. lol 29. jovan singh says: khan acedemy 30. Christina Gabriel says: I have been watching your videos for YEARS and I cannot thank you enough for all the help you've given me!!!!!!! 31. Everett Brown says: these vidieos are geting me through school. i went in to school unconfodent but now iam cofodent and ready for a test 32. Ameil Tirant says: 33. MrCowman says: You go way to fast in all of your videos, still love you though. 34. Kasie Hernandez says: thank you sooo much helps me alot 35. Kitzya says: I just did 150×4 and I got 600 ._. 36. Tiernan McDougall says: u know war u r awesome 37. Tiernan McDougall says: THX THX THX 38. Sanay .S. says: You could have just done 4*150 39. Michaela says: So what Is 150 25% of? Oh lemme see. Since 4 quarters make a dollar. (25×4=100) so that means 150×4=600 since 150 is only a quarter of the number. Yall make it overly complicated for no reason. 40. Lucas Langer says: thx very helpful u r awsome 😉 41. Mijaz Mljaz says: It is easy 42. Raj Thomas says: Can u do a video with percents with proportion 43. miami boy187 says: iiiy papi 44. david bean says: Thanks for doing the process instead of using a calculator 45. Seattle206723 says: In the First 1:53 minutes I was saying the mans a Genius. 46. maxwolf 22 says: it doent understand wath a shit =((((( 47. Videogamer 2014 says: You do all of you act like jerks who try teach you important academic skills needed in your life. Micheala im talking to you 48. Junie Joseph says: cool 49. Evelyn Reyes says: Thx really helpful. But still not sure 50. maxwolf 22 says: you now well fock you 51. Grace Higley says: there is no sound 52. Wellem Evangelista says: To solve percentage I think it's much easier to use what we call " Regra de três" ( rule of tree, literally). In this problem it was obvious that we could multiply for 4, but in other cases this way is better. (IN MY OPINION). 150= 25 \| 25x = 150.100 X = 100\| X = 15000/25=600 If 150 is 25% of X, so X is 100%. 150 times 100 is equal 15000; 25 times X is equal 25X ; — you can take out the last two zeros if you want to make it easier, and at the end you add them back.— So 150 by 25 is 6 (plus those two zeros you took off before) it is 600. I hope you got it, I'm not a native English speaker, I'm just trying to help. 53. hiten chaudhary says: you could also do 150+150+150+150 =600 54. Lj Wilkerson says: Or you can just do 150×4 55. Lj Wilkerson says: So this is the slower way 56. lala#77 rules says: 150 is 25% of 600 57. Christian Anthony Maninang says: I really don't understand to solve by using the formula triangle can you make a video about the formula triangle 58. KKGGT says: look at the bottom right corner of 3:26 you see my face 59. jeanab100 says: you help me 60. jeanab100 says: i still do not understand 61. Yalda Noori says: This is confusing. 62. slope. says: thanks 63. Jasmin Panumpang says: Green screen? 64. andrejustin bayarong says: tnx you help me @1:32 — what is the "it" ?? 66. IQIS_ MYGIRL says: Thank u 67. Nicholas Dillon says: I have to watch this for test corrections 68. comedy and documentary says: You cn alw 69. comedy and documentary says: You can always just do 150×25-. 70. Tsering Dorjee says: WOW this really helped out with my test 71. Ivan Leung says: this is useful 72. lavinth says: By the 1st minute I was able to understand very clearly. Thank you. 73. AnthoCal12 says: THANK YOU 74. zeezo bluoshi says: We can solve it much easier by multiplying 150 by 25 then divide it by 100 that will answer 600 75. The Llama says: 76. YNS KNAVE says: I do not get it 77. Tomislav Pavlovic says: 78. Rania Rezwana says: Admit it. You're here because you have a test😅 79. Eva Li says: Who else is watching this for a last-minute test? 80. Tim Yin says: wrong video 4 me lol i found de wrong video 81. The Khans says: wow, this guy is a GENIOUS… MY MIND IS BLOWN 82. santi Bergonzi says: 60-0 equals 60 not 0 83. William Clark says: 84. Silver Unknown says: :[] …………………………… 85. Tahjai Patrick says: this is a grate video 86. Josie Berast says: Still makes no sense You could have timed it by 4 88. Llawliet Otero says: is not 37.5? 89. zoe mr. clean says: This video is for nine years of people cramming for a test. 90. Czar Tuzon says: you could have saved time by just multiplying 150 into 4 91. bu says: SAVED ME FROM THIS HOMEWORK 🤢💘 92. Mr Jaws says: Ohhhhhhhhhhh after I watched I finally understand clearly. 93. Maria Sajini Durairaj says: We could have done this method 25 _ × 100 150<\|endoftext\|>	4.6875
651	Rational Exponents and Radicals Related Topics: Intermediate Algebra Lessons More Algebra Lessons A series of free, online Intermediate Algebra Lessons or Algebra II lessons. Videos, worksheets, and activities to help Algebra students. In this lesson, we will learn • about rational exponents • how to evaluate rational exponents • how to evaluate rational exponents with negative coefficients • how to use the law of exponents for rational exponents Rational Exponents Any radical can also be expressed as a rational exponent. For example, a cube root is equivalent to an exponent of 1/3; a fourth root is an exponent of 1/4. When using this method to simplify roots, we need to remember that raising a power to a power multiplies the exponents. This topic is important when finding derivatives and in integral calculus. How to understand the correlation between radicals and fractional exponents? Rational Exponents Part 1 covers the meaning of a fractional exponent when the numerator is 1, and provides examples. Part 2 covers the meaning of a fractional exponent, and provides examples. Evaluating Rational Exponents Rational exponents indicate two properties: the numerator is the base's power and the denominator is the power of the root. When evaluating rational exponents it is often helpful to break apart the exponent into its two parts: the power and the root. To decide if it is easier to perform the root first or the exponent first, see if there exists a whole number root of the base; if not, we perform the exponent operation first. How to evaluate a fractional exponent? Evaluating Numbers with Rational Exponents by using Radical Notation Evaluating Numbers Raised to Fractional Exponents Rational Exponents with Negative Coefficients A negative coefficient of a term with a rational exponent can mean that we either (1) apply the rational exponent and then take the opposite of the result, or (2) the rational exponent applies to a negative term. In case 2 of rational exponents with negative coefficients, the answer will be not real if the denominator of the exponent is even. If the root is odd, the answer will be a negative number. Rules for Rational Exponents The rules for multiplying and dividing exponents apply to rational exponents as well - however the operations will be slightly more complicated because of the fractions. Some basic rational exponent rules apply for standard operations. When multiplying exponents, we add them. When dividing exponents, we subtract them. When raising an exponent to an exponent, we multiply them. If the problem has root symbols, we change them into rational exponents first. How to simplify an expression with rational exponents. Using exponent rules with rational exponents Simplify Fractional Exponents Using the Laws of Exponents Part 1 Simplify Fractional Exponents Using the Laws of Exponents Part 2 Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.<\|endoftext\|>	4.71875
310	The National Center for PTSD has designated June National PTSD awareness month. PTSD has become a household acronym in the US but do we have an understanding of what it really is? Let’s talk about some history. Post-Traumatic Stress Disorder (PTSD) gained notoriety during the Vietnam war in the 1970s when the phrase was coined, but symptoms were being detected in military service men back during the second World War where people classified it as “shell shock.” It was not until the 1980’s when the third edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM) classified it as a diagnosis. Since that time research has taken off and in the recent years it has gained more attention. Here are some basics: PTSD is a stress reaction to a traumatic event lasting longer than 3 months causing impairment in home or work life. Many people report reliving the event and the feelings of the event, hypervigilance, anxiety, nightmares, poor sleep, avoidant behavior, difficulty concentrating, and dramatic changes in beliefs or feelings. Although PTSD is more frequently seen in military population, many other people who have suffered a trauma can be living with it. Many people seek counseling and medication to help overcome the symptoms. Learn more about PTSD at the National Center for PTSD. If you or someone you know is struggling with PTSD and are ready to start counseling, please contact us today. We have therapists who work with individuals with PTSD. submitted by: Steven Plummer, LCPC<\|endoftext\|>	3.71875
456	Next: Second derivatives, third derivatives, Up: Derivatives: How to Find Previous: The Quotient Rule # Derivatives of Trig Functions When you were a kid, there were two key facts to remember, your name and your address. You confused the two, and you might have been lost forever. You're not a kid anymore, but in this section, there are still just two key facts to remember that you don't want to mix up, namely: and The derivatives of all of the other trig functions follow from these. It is easy to get confused about which of these two derivatives has the negative sign in front. The easiest way to keep it straight is to remember, Sine keeps its sign, when you differentiate". That is to say, when you differentiate the sine function, you do not change the sign for the result. I guess you could remember, Cosine changes sign," but it's not as catchy. Showing that the derivative of the sine and cosine functions are what they are by using the limit definition of the derivative is a little tricky. It uses the fact that we already made a big deal over, namely You should determine whether or not the instructor expects you to be able to derive trigonometric derivatives using this limit. But anyway, once you know these derivatives, the rest are easy. Say you want to know the derivative of the tangent function. Well: By the quotient rule : Since , and since , This derivative occurs enough that it is probably worth memorizing: But the derivatives of and are most likely not worth memorizing, as they are easily derived. Again, this depends a lot on the professor. Make sure that you can find the derivatives of these functions using the derivatives of sine and cosine. It makes a typical test question. And by the way, just as cosine picks up a minus sign upon differentiation, so do the other two trig functions that begin with c', namely cosec and cot. So you can just remember, `To avoid a grade of C-, C gets a negative". Next: Second derivatives, third derivatives, Up: Derivatives: How to Find Previous: The Quotient Rule Joel Hass 1999-05-26<\|endoftext\|>	4.5
542	Although the body only requires a small amount of micronutrients, these important vitamins and minerals play a crucial role in overall health and wellness. Adequate micronutrient consumption is particularly important for young children, the elderly and pregnant women. Consult your doctor to determine whether you might benefit from a micronutrient supplement. The term micronutrient actually refers to a broad list of vitamins and minerals. They are called micronutrients because the body only needs small amounts to function properly. Because micronutrients are found in a large variety of food sources, most people who live in developed countries obtain adequate amounts. However, some micronutrients, like vitamin A, are only found in a few food sources, resulting in more frequent deficiencies. According to the World Food Programme, approximately 2 billion people around the world suffer from micronutrient deficiencies, which include anemia, iodine deficiency disorder and vitamin A deficiency. Micronutrients play a crucial role in healthy growth and development. Calcium, for example, contributes to proper development of bones and teeth, and iodine is important for proper thyroid development. Other micronutrients, such as iron, contribute to metabolism and energy balance. Magnesium helps prevent heart disease by regulating the rhythm of heartbeats and muscular activity in the heart. Some micronutrients, like zinc, selenium and phosphorus play an important role in the regulation and activation of other micronutrients. For example, B-complex vitamins are better absorbed and assimilated by the body when combined with adequate levels of zinc. When micronutrients are not consumed in adequate quantities, a variety of undesirable symptoms may develop. Iron deficiency, for example, may cause iron deficient anemia, which leads to fatigue and breathlessness. Iron deficient anemia is common in both developed and undeveloped countries. In some extreme cases, micronutrient deficiencies may lead to the development of chronic disease or disability. Vitamin A deficiency, for example, causes 250,000 to 500,000 children to become blind each year, according to the United Nations World Food Programme. Ideally, a healthy and balanced diet that includes all of the food groups and meets daily calorie recommendations should provide adequate micronutrients for body functions. Micronutrients are present in a wide variety of fruits, vegetables, whole grains and dairy products. If you find that you do not obtain enough of one or several micronutrients, a dietary supplement can help fill the gaps. However, bear in mind that whole food sources are always best, as emphasized in a 2005 study published in the "Proceedings of the Nutrition Society."<\|endoftext\|>	3.9375
515	## D A T A W O K Creation: January 01 1970 Modified: October 12 2019 # The Monty Hall Problem ## Introduction The Monty Hall problem is a counter-intuitive statistics puzzle loosely based on the American television game show Let's Make a Deal and named after its original host, Manty Hall. The Game: • There are three doors, behind which are two goats and a car; • You randomly pick a door; • Monty, examines the other two doors and always opens one of them with a goat. Do you stick with the original guess or switch to the other unopened door? Surprisingly, the odds to win the car ar 1/3 if you stick with the original one but goes to 2/3 if you switch the door. The game is about re-evaluating your decisions as new information emerges. ## Evident Proof Initially you can pick up one of three doors. Only one with the car. Assume that the door with the car is W and the other two are B and C. Let's enumerate the possible cases: 1. You choose W. Monty removes B or C. If you stick with the original choice you win, else you loose. 2. You choose B. Monty removes C. If you stick with the original choice you loose, else you win. 3. You choose C. Monty removes B. If you stick with the original choice you loose, else you win. As you can see from the evidence: If you don't change your choice you loose 2/3 of the times. If you change your choice you win 2/3 of the times. ## Explanation The problem can be generalized to N doors. Where only one has a car behind. If you stick with the original choice the odds to win are just 1/N, but if you change your mind, after that Monty has removed N-2 goats, then the odds to win magically becomes (N-1)/N. Pretty impressive. To better understand why this works, another, alternative, approach to the problem is needed: If you change your initial choice is like if, initially, you are not trying to pick the door with the car but a door with a goat, which has probability (N-1)/N). Then Monty will always make you the favour of removing the other N-2 bad cases and what is left is, eventually, the fortunate case.<\|endoftext\|>	4.5625
532	The Permanent Council was the highest administrative authority in the Polish-Lithuanian Commonwealth between 1775 and 1789 and the first modern government in Europe. In Polish it was renamed as Zdrada Nieustająca - Permanent Betrayal. The Permanent Council was created on the insistence of Catherine II of Russia, who saw it as a way to secure her influence over the internal and external politics of Poland. Contrary to the Sejm, which previously had the same prerogatives, the Council could not be vetoed nor disbanded. Also, it was much less prone to influence of the minor gentry. Finally, both Catherine and her ambassador to Poland, Otto Magnus von Stackelberg, believed that the Council would be dominated by anti-royal magnates and that it would put an end to his push towards the reforms. The Council was composed of the King Stanisław August Poniatowski (who acted as a modern prime minister and had two votes instead of one), 18 members of the Polish Senate and 18 members of the Sejm. The meetings were supervised by marshal Roman Ignacy Potocki. In reality all of the Council's staff was nominated in accordance with Russian ambassador Otto Magnus von Stackelberg, who acted as a representative of Empress Catherine II, protectress of Polish-Lithuanian Commonwealth since 1768. Soon after its creation, the Council became an instrument of Russian surveillance over Poland. The council was divided into 5 separate ministries called Departie: Among the prerogatives of the Council was supervising the state administration, preparation of projects of laws and Sejm acts, which were later accepted by the parliament, control over law obedience and interpretation of the law. Although heavily-criticized, most notably by the so-called Patriotic Party and the Familia, the Council managed to start a period of economical prosperity in Poland and significantly strengthened the power of the monarch in Poland. It was liquidated in 1789 by the Four-Year Sejm and briefly reinstituted in 1793 by the Sejm of Grodno. However, this time it was directly headed by the Russian ambassador. The Majority of it's members were bribed by the Russian embassy in Warsaw. King Stanisław August Poniatowski marshal Roman Ignacy Potocki Tomasz Adam Ostrowski Ludwik Szymon Gutakowski Stanisław Poniatowski (kings' relative) Michał Jerzy Poniatowski (primate of Poland)<\|endoftext\|>	3.890625
715	What is Play Therapy? Click here to read more great articles in our news room. Play therapy is a structured, theoretically based approach to therapy that builds on the normal communicative and learning processes of children. The curative powers inherent in play are used in many ways. Therapists strategically utilize play therapy to help children express what is troubling them when they do not have the verbal language to express their thoughts and feelings. In play therapy, toys are like the child’s words and play is the child’s language. Play therapy is a well established discipline based upon a number of psychological theories. Research, both qualitative and quantitative shows that it is highly effective in many cases. Recent research by PTUK, an organization affiliated to Play Therapy International, suggests that 71% of the children referred to play therapy will show a positive change. Who Comes to Play Therapy? Children are referred for play therapy to resolve their problems which can include behavioral problems such as acting out at school or in the home, emotional difficulties such as depression, anxiety, or OCD, or poor social skills. Children also are referred to come in to work through family issues such as divorce, death, loss and abandonment issues. Why Play in Therapy? Play is a fun, enjoyable activity that is the natural way in which children learn about and explore the world around them. It elevates our spirits and brightens our outlook on life. Play Therapy utilizes this natural ability of children and uses developmentally appropriate techniques to expand self-expression, self-knowledge, self-actualization and self-efficacy. Play relieves feelings of stress and boredom, connects us to people in a positive way, stimulates creative thinking and exploration, regulates our emotions, and boosts our ego. In addition, play allows us to practice skills and roles needed for survival. Researchers now have evidence that learning and development are best fostered through play. Play Therapy Can: - Help children learn more adaptive behaviors. - Provides a corrective emotional experience necessary for healing - Promote cognitive development - Provide insight about and resolution of inner conflicts or dysfunctional thinking in the child - Heal from trauma, loss, and/or grief experiences How does play therapy work? A safe, confidential and caring environment is created between the therapist and child, which allows the child to play with as few limits as possible but as many as necessary (for physical and emotional safety). This allows healing to occur on many levels. Play and creativity operate on impulses from outside our awareness – the unconscious. Play Therapy provides a safe psychological distance from their problems and allows expression of thoughts and feelings appropriate to the child’s development. The therapist may reflect back to the child observations of what has happened during the session if this is felt to be appropriate. Above all the child is given “Special Time” where they are the focus of attention and where they can begin to learn how to feel and express feelings in a healthy manner. The child is given strategies to cope with difficulties they face in life and which they themselves cannot change. It provides a more positive view of their future life. Play therapy differs from regular play in that the therapist helps children to address and resolve their own problems. Play therapy builds on the natural way that children learn about themselves and their relationships in the world around them. Through play therapy, children learn to: - communicate with others - express feelings - modify behavior - develop problem-solving skills - learn a variety of ways of relating to others.<\|endoftext\|>	3.765625
1,093	In English, many things are named after a particular country – but have you ever wondered what those things are called in those countries? 1(class)nobleza femininethe nobility — la nobleza - The prerogative of nobles was to command, and nobilities everywhere dominated the machineries of state. - Instead, they were answerable to a complex of hereditary or franchise jurisdictions in the hands of the feudal nobility. - To this extent, the novel could be seen as a celebration of the values of the English nobility. - At all levels of government, the nobility dominated decision making. - Around this castle were the smaller houses of lesser nobility and the members of court. - That there were fewer revolts in the second half of the century was due in no small part to a growing mutual understanding between rulers and nobilities, the history of which has attracted less attention than the revolts themselves. - With its glittering population of titled courtiers, it also symbolized a whole social system dominated by a privileged nobility. - All this made her popular with the French nobility, eventually including the King and Queen of France. - By insinuating himself into the French nobility, he systematically destroys the men who manipulated and enslaved him. - In the Czech Republic, the old nobility is enjoying a new lease of life. - Even he could be persuaded that a man's conduct was so markedly honourable as to justify elevation to the nobility. - He spent most of his life in the service of the English nobility, partly as a music tutor. - The night of 4 August also transformed the character of the French nobility. - The nobilities of the Italian states (except Piedmont) were broken by the process of unification, and the new state was run by a bourgeois political class of lawyers, civil servants, and landowners. - For the landowning nobility, the portents were not good. - But the English nobility keep themselves to themselves and only dine with the pick of the bunch. - They enjoyed abundant mineral wealth, stunning yields of maize and cacao, as well as strong and enduring ties to the Mesoamerican nobilities of Oaxaca and central Mexico. - Meanwhile, the older nobility was losing income due to declining rents. - Yet sceptics argued that a large modern republic was not possible in Europe, with its overpowerful feudal nobilities and its hordes of miserable poor. - Though the civil service was dominated by the nobility, it became progressively more open to commoners. 2(of appearance, action)nobleza feminine - Sport is used as a tool for defining so-called Australian nobility of spirit. - Yes, let's take the classics and teach about nobility, honor, character, courage, commitment. - At such moments nobility and strength of character propel us way beyond our means to be kind and helpful. - I, of course, have remained above all this, not out of any nobility of character, but out of sheer laziness. - But if Othello dies a deluded and confused figure, would that not rob him of all dignity and nobility, turning him into the pitiful victim of a vicious, hostile society? - His face was reasonably happy and his standard expression seemed to be one of aloof nobility, even though he knew he wasn't noble. - The basic premise of the story is that noble birth doesn't guarantee a noble person and nobility can be present in the most humble peasant. - With all the nobility of her character, she kept Margaret's secret. - She was admired for that nobility of spirit, it seems. - Audrey Hepburn is luminous, waif-like, but with nobility that itself transcended her character's station. - I don't think Erica is programmed to understand nobility of character. - The yardstick for gauging the inherent nobility of a character in major films these days is the slowness of the slow-motion in which their death is captured. - Her long black hair was tied back in a thick braid, and her blue-gray eyes gazed into mine with a mixture of wisdom, kindness, and nobility. - Such a limitation requires a strong breed of man, however, with a quality of character and nobility of soul. - She was reputedly of great beauty, and aside from that also possessed much grace, kindness, nobility, and, among other things, charm. - But they wanted to be recognized for their nobility of character. - Most of the characters reveal sorry weaknesses but also unsuspected bits of nobility. - He spent the entire film buried under a ton of make-up as Frankenstein's Monster but captured the essential nobility of his put-upon character really well. - There is a nobility to his character that the other villagers find almost impossible to understand. English has borrowed many of the following foreign expressions of parting, so you’ve probably encountered some of these ways to say goodbye in other languages. Many words formed by the addition of the suffix –ster are now obsolete - which ones are due a resurgence? As their breed names often attest, dogs are a truly international bunch. Let’s take a look at 12 different dog breed names and their backstories.<\|endoftext\|>	3.703125
8,182	Tamil Nadu Board of Secondary EducationHSC Science Class 12th # Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 7 - Applications of Differential Calculus [Latest edition] ## Chapter 7: Applications of Differential Calculus Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5Exercise 7.6Exercise 7.7Exercise 7.8Exercise 7.9Exercise 7.10 Exercise 7.1 [Pages 8 - 9] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.1 [Pages 8 - 9] Exercise 7.1 \| Q 1. (i) \| Page 8 A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres. Find the average velocity between t = 3 and t = 6 seconds Exercise 7.1 \| Q 1. (ii) \| Page 8 A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres. Find the instantaneous velocities at t = 3 and t = 6 seconds Exercise 7.1 \| Q 2. (i) \| Page 8 A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t2 in t seconds. How long does the camera fall before it hits the ground? Exercise 7.1 \| Q 2. (ii) \| Page 8 A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t2 in t seconds. What is the average velocity with which the camera falls during the last 2 seconds? Exercise 7.1 \| Q 2. (iii) \| Page 8 A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t2 in t seconds. What is the instantaneous velocity of the camera when it hits the ground? Exercise 7.1 \| Q 3. (i) \| Page 8 A particle moves along a line according to the law s(t) = 2t3 – 9t2 + 12t – 4, where t ≥ 0. At what times the particle changes direction? Exercise 7.1 \| Q 3. (ii) \| Page 8 A particle moves along a line according to the law s(t) = 2t3 – 9t2 + 12t – 4, where t ≥ 0. Find the total distance travelled by the particle in the first 4 seconds Exercise 7.1 \| Q 3. (iii) \| Page 8 A particle moves along a line according to the law s(t) = 2t3 – 9t2 + 12t – 4, where t ≥ 0. Find the particle’s acceleration each time the velocity is zero Exercise 7.1 \| Q 4 \| Page 8 If the volume of a cube of side length x is v = x3. Find the rate of change of the volume with respect to x when x = 5 units Exercise 7.1 \| Q 5 \| Page 8 If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = sqrt(3x) then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres Exercise 7.1 \| Q 6 \| Page 8 A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water? Exercise 7.1 \| Q 7 \| Page 8 A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shoreline. How fast is the beam moving along the shoreline when it makes an angle of 45° with the shore? Exercise 7.1 \| Q 8 \| Page 8 A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep? Exercise 7.1 \| Q 9. (i) \| Page 8 A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall. How fast is the top of the ladder moving down the wall? Exercise 7.1 \| Q 9. (ii) \| Page 8 A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall, at what rate, the area of the triangle formed by the ladder, wall, and the floor, is changing? Exercise 7.1 \| Q 10 \| Page 9 A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car? Exercise 7.2 [Pages 14 - 15] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.2 [Pages 14 - 15] Exercise 7.2 \| Q 1. (i) \| Page 14 Find the slope of the tangent to the following curves at the respective given points. y = x4 + 2x2 – x at x = 1 Exercise 7.2 \| Q 1. (ii) \| Page 14 Find the slope of the tangent to the following curves at the respective given points. x = a cos3t, y = b sin3t at t = pi/2 Exercise 7.2 \| Q 2 \| Page 14 Find the point on the curve y = x2 – 5x + 4 at which the tangent is parallel to the line 3x + y = 7 Exercise 7.2 \| Q 3 \| Page 14 Find the points on curve y = x3 – 6x2 + x + 3 where the normal is parallel to the line x + y = 1729 Exercise 7.2 \| Q 4 \| Page 14 Find the points on the curve y2 – 4xy = x2 + 5 for which the tangent is horizontal Exercise 7.2 \| Q 5. (i) \| Page 15 Find the tangent and normal to the following curves at the given points on the curve y = x2 – x4 at (1, 0) Exercise 7.2 \| Q 5. (ii) \| Page 15 Find the tangent and normal to the following curves at the given points on the curve y = x4 + 2ex at (0, 2) Exercise 7.2 \| Q 5. (iii) \| Page 15 Find the tangent and normal to the following curves at the given points on the curve y = x sin x at (pi/2, pi/2) Exercise 7.2 \| Q 5. (iv) \| Page 15 Find the tangent and normal to the following curves at the given points on the curve x = cos t, y = 2 sin2t at t = pi/2 Exercise 7.2 \| Q 6 \| Page 15 Find the equations of the tangents to the curve y = 1 + x3 for which the tangent is orthogonal with the line x + 12y = 12 Exercise 7.2 \| Q 7 \| Page 15 Find the equations of the tangents to the curve y = - (x + 1)/(x - 1) which are parallel to the line x + 2y = 6 Exercise 7.2 \| Q 8 \| Page 15 Find the equation of tangent and normal to the curve given by x – 7 cos t andy = 2 sin t, t ∈ R at any point on the curve Exercise 7.2 \| Q 9 \| Page 15 Find the angle between the rectangular hyperbola xy = 2 and the parabola x2 + 4y = 0 Exercise 7.2 \| Q 10 \| Page 15 Show that the two curves x2 – y2 = r2 and xy = c2 where c, r are constants, cut orthogonally Exercise 7.3 [Pages 21 - 22] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.3 [Pages 21 - 22] Exercise 7.3 \| Q 1. (i) \| Page 21 Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals f(x) = \|1/x\|, x ∈ [- 1, 1] Exercise 7.3 \| Q 1. (ii) \| Page 21 Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals f(x) = tan x, x ∈ [0, π] Exercise 7.3 \| Q 1. (iii) \| Page 21 Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals f(x) = x – 2 log x, x ∈ [2, 7] Exercise 7.3 \| Q 2. (i) \| Page 21 Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions: f(x) = x2 – x, x ∈ [0, 1] Exercise 7.3 \| Q 2. (ii) \| Page 21 Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions: f(x) = (x^2 - 2x)/(x + 2), x ∈ [-1, 6] Exercise 7.3 \| Q 2. (iii) \| Page 21 Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions: f(x) = sqrt(x) - x/3, x ∈ [0, 9] Exercise 7.3 \| Q 3. (i) \| Page 21 Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals: f(x) = (x + 1)/x, x ∈ [-1, 2] Exercise 7.3 \| Q 3. (ii) \| Page 21 Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals: f(x) = \|3x + 1\|, x ∈ [-1, 3] Exercise 7.3 \| Q 4. (i) \| Page 21 Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval: f(x) = x^3 - 3x + 2, x ∈ [-2, 2] Exercise 7.3 \| Q 4. (ii) \| Page 21 Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval: f(x) = (x - 2)(x - 7), x ∈ [3, 11] Exercise 7.3 \| Q 5. (i) \| Page 21 Show that the value in the conclusion of the mean value theorem for f(x) = 1/x on a closed interval of positive numbers [a, b] is sqrt("ab") Exercise 7.3 \| Q 5. (ii) \| Page 21 Show that the value in the conclusion of the mean value theorem for f(x) = "A"x^2 + "B"x + "C" on any interval [a, b] is ("a" + "b")/2 Exercise 7.3 \| Q 6 \| Page 21 A race car driver is kilometer stone 20. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours Exercise 7.3 \| Q 7 \| Page 21 Suppose that for a function f(x), f'(x) ≤ 1 for all 1 ≤ x ≤ 4. Show that f(4) – f(1) ≤ 3 Exercise 7.3 \| Q 8 \| Page 22 Does there exist a differentiable function f(x) such that f(0) = – 1, f(2) = 4 and f(x) ≤ 2 for all x. Justify your answer Exercise 7.3 \| Q 9 \| Page 22 Show that there lies a point on the curve f(x) = x(x + 3)e^(pi/2), -3 ≤ x ≤ 0 where tangent drawn is parallel to the x-axis Exercise 7.3 \| Q 10 \| Page 22 Using Mean Value Theorem prove that for, a > 0, b > 0, \|e–a – eb\| < \|a – b\| Exercise 7.4 [Page 25] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.4 [Page 25] Exercise 7.4 \| Q 1. (i) \| Page 25 Write the Maclaurin series expansion of the following functions: ex Exercise 7.4 \| Q 1. (ii) \| Page 25 Write the Maclaurin series expansion of the following functions: sin x Exercise 7.4 \| Q 1. (iii) \| Page 25 Write the Maclaurin series expansion of the following functions: cos x Exercise 7.4 \| Q 1. (iii) \| Page 25 Write the Maclaurin series expansion of the following functions: log(1 – x); – 1 ≤ x ≤ 1 Exercise 7.4 \| Q 1. (v) \| Page 25 Write the Maclaurin series expansion of the following functions: tan–1 (x); – 1 ≤ x ≤ 1 Exercise 7.4 \| Q 1. (vi) \| Page 25 Write the Maclaurin series expansion of the following functions: cos2x Exercise 7.4 \| Q 2 \| Page 25 Write down the Taylor series expansion, of the function log x about x = 1 upto three non-zero terms for x > 0 Exercise 7.4 \| Q 3 \| Page 25 Expand sin x in ascending powers x - pi/4 upto three non-zero terms Exercise 7.4 \| Q 4 \| Page 25 Expand the polynomial f(x) = x2 – 3x + 2 in power of x – 1 Exercise 7.5 [Pages 31 - 32] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.5 [Pages 31 - 32] Exercise 7.5 \| Q 1 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> 0) (1 - cosx)/x^2 Exercise 7.5 \| Q 2 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x - oo) (2x^2 - 3)/(x^2 -5x + 3) Exercise 7.5 \| Q 3 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> oo) x/logx Exercise 7.5 \| Q 4 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> x/2) secx/tanx Exercise 7.5 \| Q 5 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> oo) "e"^-x sqrt(x) Exercise 7.5 \| Q 6 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> 0) (1/sinx - 1/x) Exercise 7.5 \| Q 7 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> 1^+) (2/(x^2 - 1) - x/(x - 1)) Exercise 7.5 \| Q 8 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> 0^+) x^x Exercise 7.5 \| Q 9 \| Page 31 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> oo) (1 + 1/x)^x Exercise 7.5 \| Q 10 \| Page 32 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> pi/2) (sin x)^tanx Exercise 7.5 \| Q 11 \| Page 32 Evaluate the following limits, if necessary use l’Hôpital Rule: lim_(x -> 0^+) (cos x)^(1/x^2) Exercise 7.5 \| Q 12 \| Page 32 Evaluate the following limits, if necessary use l’Hôpital Rule: If an initial amount A0 of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is A = "A"_0 (1 + "r"/"n")^"nt". If the interest is compounded continuously, (that is as n → ∞), show that the amount after t years is A = A0ert Exercise 7.6 [Page 40] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.6 [Page 40] Exercise 7.6 \| Q 1. (i) \| Page 40 Find the absolute extrema of the following functions on the given closed interval. f(x) = x2 – 12x + 10; [1, 2] Exercise 7.6 \| Q 1. (ii) \| Page 40 Find the absolute extrema of the following functions on the given closed interval. f(x) = 3x4 – 4x3 ; [– 1, 2] Exercise 7.6 \| Q 1. (iii) \| Page 40 Find the absolute extrema of the following functions on the given closed interval. f(x) = 6x^(4/3) - 3x^(1/3) ; [-1, 1] Exercise 7.6 \| Q 1. (iv) \| Page 40 Find the absolute extrema of the following functions on the given closed interval. f(x) = 2 cos x + sin 2x; [0, pi/2] Exercise 7.6 \| Q 2. (i) \| Page 40 Find the intervals of monotonicities and hence find the local extremum for the following functions: f(x) = 2x3 + 3x2 – 12x Exercise 7.6 \| Q 2. (ii) \| Page 40 Find the intervals of monotonicities and hence find the local extremum for the following functions: f(x) = x/(x - 5) Exercise 7.6 \| Q 2. (iii) \| Page 40 Find the intervals of monotonicities and hence find the local extremum for the following functions: f(x) = "e"^x/(1 - "e"^x) Exercise 7.6 \| Q 2. (iv) \| Page 40 Find the intervals of monotonicities and hence find the local extremum for the following functions: f(x) = x^3/3 - log x Exercise 7.6 \| Q 2. (v) \| Page 40 Find the intervals of monotonicities and hence find the local extremum for the following functions: f(x) = sin x cos x + 5, x ∈ (0, 2π) Exercise 7.7 [Page 44] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.7 [Page 44] Exercise 7.7 \| Q 1. (i) \| Page 44 Find intervals of concavity and points of inflexion for the following functions: f(x) = x(x – 4)3 Exercise 7.7 \| Q 1. (ii) \| Page 44 Find intervals of concavity and points of inflection for the following functions: f(x) = sin x + cos x, 0 < x < 2π Exercise 7.7 \| Q 1. (iii) \| Page 44 Find intervals of concavity and points of inflection for the following functions: f(x) = 1/2 ("e"^x - "e"^-x) Exercise 7.7 \| Q 2. (i) \| Page 44 Find the local extrema for the following functions using second derivative test: f(x) = – 3x5 + 5x3 Exercise 7.7 \| Q 2. (ii) \| Page 44 Find the local extrema for the following functions using second derivative test: f(x) = x log x Exercise 7.7 \| Q 2. (iii) \| Page 44 Find the local extrema for the following functions using second derivative test: f(x) = x2 e–2x Exercise 7.7 \| Q 3 \| Page 44 For the function f(x) = 4x3 + 3x2 – 6x + 1 find the intervals of monotonicity, local extrema, intervals of concavity and points of inflection Exercise 7.8 [Page 47] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.8 [Page 47] Exercise 7.8 \| Q 1 \| Page 47 Find two positive numbers whose sum is 12 and their product is maximum Exercise 7.8 \| Q 2 \| Page 47 Find two positive numbers whose product is 20 and their sum is minimum Exercise 7.8 \| Q 3 \| Page 47 Find the smallest possible value of x2 + y2 given that x + y = 10 Exercise 7.8 \| Q 4 \| Page 47 A garden is to be laid out in a rectangular area and protected by a wire fence. What is the largest possible area of the fenced garden with 40 meters of wire? Exercise 7.8 \| Q 5 \| Page 47 A rectangular page is to contain 24 cm2 of print. The margins at the top and bottom of the page are 1.5 cm and the margins at the other sides of the page are 1 cm. What should be the dimensions’ of the page so that the area of the paper used is minimum? Exercise 7.8 \| Q 6 \| Page 47 A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 1,80,000 sq. mtrs in order to provide enough grass for herds. No fencing is needed along the river. What is the length of the minimum needed fencing material? Exercise 7.8 \| Q 7 \| Page 47 Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10 cm Exercise 7.8 \| Q 8 \| Page 47 Prove that among all the rectangles of the given perimeter, the square has the maximum area Exercise 7.8 \| Q 9 \| Page 47 Find the dimensions of the largest rectangle that can be inscribed in a semi-circle of radius r cm Exercise 7.8 \| Q 10 \| Page 47 A manufacturer wants to design an open box having a square base and a surface area of 108 sq.cm. Determine the dimensions of the box for the maximum volume Exercise 7.8 \| Q 11 \| Page 47 The volume of a cylinder is given by the formula V = pi"r"^2"h". Find the greatest and least values of V if r + h = 6 Exercise 7.8 \| Q 12 \| Page 47 A hollow cone with a base radius of a cm and’ height of b cm is placed on a table. Show that) the volume of the largest cylinder that can be hidden underneath is 4/9 times the volume of the cone Exercise 7.9 [Page 53] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.9 [Page 53] Exercise 7.9 \| Q 1. (i) \| Page 53 Find the asymptotes of the following curves: f(x) = x^2/(x^2 - 1) Exercise 7.9 \| Q 1. (ii) \| Page 53 Find the asymptotes of the following curves: f(x) = x^2/(x + 1) Exercise 7.9 \| Q 1. (iii) \| Page 53 Find the asymptotes of the following curves: f(x) = (x^2 - 6x - 1)/(x + 3) Exercise 7.9 \| Q 1. (iv) \| Page 53 Find the asymptotes of the following curves: f(x) = (x^2 - 6x - 1)/(x + 3) Exercise 7.9 \| Q 1. (v) \| Page 53 Find the asymptotes of the following curves: f(x) = (x^2 + 6x - 4)/(3x - 6) Exercise 7.9 \| Q 2. (i) \| Page 53 Sketch the graphs of the following functions y = - 1/3 (x^3 - 3x + 2) Exercise 7.9 \| Q 2. (ii) \| Page 53 Sketch the graphs of the following functions: y = xsqrt(4 - x) Exercise 7.9 \| Q 2. (iii) \| Page 53 Sketch the graphs of the following functions: y = (x^2 + 1)/(x^2 - 4) Exercise 7.9 \| Q 2. (iv) \| Page 53 Sketch the graphs of the following functions: y = 1/(1 + "e"^-x) Exercise 7.10 [Pages 54 - 55] ### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.10 [Pages 54 - 55] Exercise 7.10 \| Q 1 \| Page 54 Choose the correct alternative: The volume of a sphere is increasing in volume at the rate of 3π cm3/ sec. The rate of change of its radius when radius is 1/2 cm • 3 cm/s • 2 cm/s • 1 cm/s • 1/2 cm/s Exercise 7.10 \| Q 2 \| Page 54 Choose the correct alternative: A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon’s angle of elevation in radian per second when the balloon is 30 metres above the ground • 3/25 radians/sec • 4/25 radians/sec • 1/5 radians/sec • 1/3 radians/sec Exercise 7.10 \| Q 3 \| Page 54 Choose the correct alternative: The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t2 – 2t – 8. The time at which the particle is at rest is • t = 0 • t = 1/3 • t = 1 • t = 3 Exercise 7.10 \| Q 4 \| Page 54 Choose the correct alternative: A stone is thrown, up vertically. The height reaches at time t seconds is given by x = 80t – 16t2. The stone reaches the maximum! height in time t seconds is given by • 2 • 2.5 • 3 • 3.5 Exercise 7.10 \| Q 5 \| Page 54 Choose the correct alternative: Find the point on the curve 6y = x3 + 2 at which y-coordinate changes 8 times as fast as x-coordinate is • (4, 11) • (4, – 11) • (– 4, 11) • (– 4, – 11) Exercise 7.10 \| Q 6 \| Page 54 Choose the correct alternative: The abscissa of the point on the curve f(x) = sqrt(8 - 2x) at which the slope of the tangent is – 0.25? • – 8 • – 4 • – 2 • 0 Exercise 7.10 \| Q 7 \| Page 54 Choose the correct alternative: The slope of the line normal to the curve f(x) = 2 cos 4x at x = pi/12 is • - 4sqrt(3) • – 4 • - sqrt(3)/12 • 4sqrt(3) Exercise 7.10 \| Q 8 \| Page 54 Choose the correct alternative: The tangent to the curve y2 – xy + 9 = 0 is vertical when • y = 0 • y = +- sqrt(3) • y = 1/2 • y = +- 3 Exercise 7.10 \| Q 9 \| Page 54 Choose the correct alternative: Angle between y2 = x and x2 = y at the origin is • tan^-1 3/4 • tan^-1 (4/3) • pi/2 • pi/4 Exercise 7.10 \| Q 10 \| Page 54 Choose the correct alternative: The value of the limit lim_(x -> 0) (cot x - 1/x) is • 0 • 1 • 2 • oo Exercise 7.10 \| Q 11 \| Page 55 Choose the correct alternative: The function sin4x + cos4x is increasing in the interval • [(5pi)/8, (3pi)/4] • [pi/2, (5pi)/8] • [pi/4, pi/2] • [0, pi/4] Exercise 7.10 \| Q 12 \| Page 55 Choose the correct alternative: The number given by the Rolle’s theorem for the function x3 – 3x2, x ∈ [0, 3] is • 1 • sqrt(2) • 3/2 • 2 Exercise 7.10 \| Q 13 \| Page 55 Choose the correct alternative: The number given by the Mean value theorem for the function 1/x, x ∈ [1, 9] is • 2 • 2.5 • 3 • 3.5 Exercise 7.10 \| Q 14 \| Page 55 Choose the correct alternative: The minimum value of the function \|3 - x\| + 9 is • 0 • 3 • 6 • 9 Exercise 7.10 \| Q 15 \| Page 55 Choose the correct alternative: The maximum slope of the tangent to the curve y = ex sin x, x ∈ [0, 2π] is at • x = pi/4 • x = pi/2 • x = pi • x = (3pi)/2 Exercise 7.10 \| Q 16 \| Page 55 Choose the correct alternative: The maximum value of the function x2 e-2x, x > 0 is • 1/"e" • 1/(2"e") • 1/"e"^2 • 4/"e"^4 Exercise 7.10 \| Q 17 \| Page 55 Choose the correct alternative: One of the closest points on the curve x2 – y2 = 4 to the point (6, 0) is • (2, 0) • (sqrt(5), 1) • (3, sqrt(5)) • (sqrt(13), - sqrt(13)) Exercise 7.10 \| Q 18 \| Page 55 Choose the correct alternative: The maximum value of the product of two positive numbers, when their sum of the squares is 200, is • 100 • 25sqrt(7) • 28 • 24sqrt(14)` Exercise 7.10 \| Q 19 \| Page 55 Choose the correct alternative: The curve y = ax4 + bx2 with ab > 0 • has no horizontal tangent • is concave up • is concave down • has no points of inflection Exercise 7.10 \| Q 20 \| Page 55 Choose the correct alternative: The point of inflection of the curve y = (x – 1)3 is • (0, 0) • (0, 1) • (0, 1) • (1, 1) ## Chapter 7: Applications of Differential Calculus Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5Exercise 7.6Exercise 7.7Exercise 7.8Exercise 7.9Exercise 7.10 ## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 7 - Applications of Differential Calculus Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 7 (Applications of Differential Calculus) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Class 12th Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster. Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Tamil Nadu Board Samacheer Kalvi textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students. Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 7 Applications of Differential Calculus are Applications of Differential Calculus, Meaning of Derivatives, Mean Value Theorem, Series Expansions, Indeterminate Forms, Applications of First Derivative, Applications of Second Derivative, Applications in Optimization, Symmetry and Asymptotes, Sketching of Curves. Using Tamil Nadu Board Samacheer Kalvi Class 12th solutions Applications of Differential Calculus exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Tamil Nadu Board Samacheer Kalvi Solutions are important questions that can be asked in the final exam. Maximum students of Tamil Nadu Board of Secondary Education Class 12th prefer Tamil Nadu Board Samacheer Kalvi Textbook Solutions to score more in exam. Get the free view of chapter 7 Applications of Differential Calculus Class 12th extra questions for Class 12th Mathematics Volume 1 and 2 Answers Guide and can use Shaalaa.com to keep it handy for your exam preparation<\|endoftext\|>	4.625
1,808	Multiplication Worksheets Grade 3 [with PDF] Multiplication is one of the basic operations in mathematics that represents repeated addition. So in multiplication one number is added a number of times to get the results. Multiplication is represented by the signs cross ‘×’, asterisk ‘*’, or dot ‘·’. The numbers that are multiplied are called the factors and the result after multiplication is called the product. So, multiplication is an arithmetic operation to find the product of two or more numbers. Properties of Multiplication 1. Multiplication by 1: the multiplication of a number with 1 is always the number itself, for example- 15X1=15. 2. Multiplication by 0: the multiplication of a number with 0 is always zero, for example-95X0=0 3. The order does not matter in multiplication, the result remains the same. For example, 8X5=5X8=40. 4. While multiplying three or more numbers, grouping can be done in any order to get the result. For example: 2x4x6x3=(2×4)x(6X3)=(2×6)x(4X3)=(2×3)x(4×6)=(2x4x3)x6. In all the above cases the result is the same as 144. So numbers can be grouped in any order to get the product. 5. Multiplying by 10: While multiplying by 10 simply insert one 0 to the right of the number. For example, 4×10=40, here simply a 0 is inserted on the right of the number 4. 6. Multiplying by 100: while multiplying by 100 simply insert two “0” s to the right of the number. For example, 6×100=600. 7. Multiplying by 20, 30, 40, etc.: While multiplying a number with 20, 30, 40, etc. simply write a “0” at one’s place and then multiply the remaining numbers. For example, to multiply 30 with 7 put zero at one’s place, and then multiply 3 with 7 and write before 0. The answer is 30×7=210. 8. Solving Multiplication Worksheet or multiplication problems will be easier if you learn the multiplication chart or multiplication table by heart. The multiplication table is provided in Fig. 1 for your reference. See also What is Unitary Method? Worksheets and Problems of Unitary Method (PDF) Multiplication Chart Multiplication chart or Multiplication table is a very useful mathematical tool to find the product of two numbers in the chart. Refer to Fig. 1 to find an example of a multiplication chart. Using this multiplication chart is very easy. To multiply two numbers in this table simply find the numbers in the column and rows and find the overlapping cell to find the product of multiplication. For example in the above figure multiplication of 6 X 4 is 24. Find 4 in column and 6 in rows and find the overlapping cell 24 to find the result. Let’s practice the following Multiplication Worksheets along with https://practiceworksheet.com/. Note that option to print and making pdf is provided at the end of each article. Simply click on the button to convert the multiplication worksheet grade 3 into pdf or print. Q1) Fill in the blanks to solve the following multiplication worksheets 2. 6 + 6 + 6 + 6 + 6=__________ X _______________ =30 3. 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 =________________ X 9= __________ 4. 3 X 7= _______+__________+__________=___________ 5. Write 5 X 10 in addition format: ________________________________ 6. 8 X________ = 7 X _________________=56 7. 7 X 100=____________=70 X_____ 8. 23 X _________= 2 X 3 X__________=15 X__________= 0 X 3 X 5= __________ 9. 2 X 3 X 4 X 5 = (2 X 3 X 4) X______=(2 X_____) X (3 X_____) 10. The result that we get in a multiplication problem is called ___________. 11. 15 X 23= 15 X (8 + __________) 12. 67 X 8=_______________. 13. 12 X 29 =______________. 14. 567 X 48 =_____________. 15. 234 X 23 X 4 X 0= ________________. 16. 7 weeks= _______________ days. 17. 8 years= ________________months. 18. 3 years=________________days 19. 6 years= ______________ weeks 20. 1 day=______________minutes 21. 4 hours= _____________seconds. 22. The product of 14 and 5 is ________________ tens. See also Decimal to Fraction and Fraction to Decimal Worksheets with PDF Q2) Write True or False for the following multiplication worksheets statements based on your understanding of multiplication. 1. The numbers that are multiplied are called Products 2. To find the double of a number we have to multiply the number with 2. 3. One can jump 6 times higher on the moon than on the earth. If Swarnali jumps 50 cm in the earth she will jump 300 cm on the moon. 4. 40 X 75 =4 X 25 X 30 5. 5 X (3+4)= 5X3+5X4 6. 32 X 60= 32 X 6 X 10 7. A man has 32 teeth, So 4 men will have 108 teeth. 8. An adult has 206 bones in his body. If a group consists of 5 such members, there will be a total of 1,030 bones. 9. 8 sevens= 4 X 7 + 2 X 7. 10. The symbol “X” indicates multiplication. 11. Multiplication problems can be checked using addition. 12. 6 X 1000 >60 X 100 13. A multiplication problem can have a remainder. 14. Multiplication table helps in easily solving multiplication worksheet problems. 15. The term subtraction is associated with multiplication. Q3) Multiplication Worksheets Grade 3 Word Problems: Solve the following Mental Math Multiplication word problems 1. A toy costs Rs. 678 per piece. A team is planning to distribute 78 such tops to the poor children in a NGO. What will be the total expense of the team? 2. Aharsi thought of an even number between 50 and 60 which is a multiple of 7. What is the number? 3. What will be product of the largest 3-digit number and the largest 2-digit number? 4. Firan wants to buy a nice gift for his wife Mohor on her birthday. So he started saving 26 OMR per month. So, in a year what will be amount that Firan will be saving? 5. Sumita sold 8 necklaces to Dipusree. Each necklace set costs Rs 136/-. If Dipusree gave a Rs 2000/- note to Sumita, how much she will get as return? 6. Uttam purchased 500 stamps for Swarnali’s craftwork. Swarnali has to prepare 9 toys using those stamps. She used 48 stamps in each toy. So how many stamps will remain once she finishes all toys? 7. What is the missing number in the problem: 3___5 X 3 =1095? 8. Ismat have a collection of few pens. The number of pens is between 50 and 60. So, how many pens does Ismat have if the number comes in the multiplication chart of 7 and 8 both? 9. Apurba went to Nesto market and purchased a dozen egg trays. Each tray consists of 30 eggs. After returning to home he found that in each tray two of the eggs are rotten which needs to be thrown away. So how many good eggs did Apurba purchase? 10. In a competitive exam 36 question to be answered. All questions are compulsory for submission. For each correct answer 6 marks will be rewarded and for each wrong answer 2 marks will be deducted. Aharsi answered 29 questions correctly. So what will be his score?<\|endoftext\|>	4.71875
303	# Prove that the difference between any two sides of a triangle is less than its third side Prove that the difference between any two sides of a triangle is less than its third side. Let us consider a ∆ABC, whose sides are AB, BC and CA. To prove AC - AB < BC, BC - AC < AB and BC - AB < AC Construction Let AC > AB. Then, along AC, cut-off AD = AB and join BD. Proof In ∆ ABD, AB=AD⇒ ∠2=∠1 [∵ angles opposite to equal sides are equal] Side CD of ∆ BCD has been produced to A. ∠2 >∠4 [∵ exterior angle is greater than each interior opposite angle] Again, side AD of ∆ABD has been produced to C. ∴∠3>∠1 …(iii) [∵ exterior angle is greater than each interior opposite angle] From Eqs. (i) and (iii), we get∠3 > ∠2 Now, ∠3 >∠2 and ∠2 > ∠4 => ∠3 > ∠4 ∴BC >CD [∵ sides opposite to greater angle is longer] ⇒ CD < BC ⇒ AC - AD < BC Hence, AC - AB < BC [∵ AD = AB] Similarly, BC - AC < AB and BC - AB< AC Hence proved.<\|endoftext\|>	4.4375
662	Homework is a central characteristic of Benjamin Franklin Schools. It has three major purposes: - To teach personal responsibility and time management skills. - To keep parents informed about what their children are studying. - To provide additional academic growth and development. Parent responsibilities concerning homework are as follows: - To provide a time and place free from distraction for the work to occur. - To sign the daily homework sheets or papers (whether completed or not) and review the child's work and provide words of encouragement, not correction. - To work closely with the teacher if a problem occurs. Below are the minimum and the maximum time guidelines for homework for which parents should plan. The times by grade level are listed below: Grade Minimum Maximum Kindergarten 10 min. 15 min. 1st Grade 15 min. 30 min. 2nd Grade 20 min. 45 min. 3rd Grade 30 min. 60 min. 4th Grade 30 min. 75 min. 5th Grade 30 min. 90 min. 6th Grade 30 min. 90 min. All student's are expected to read aloud or be read to for a minimum of 10 minutes each night as a part of their homework. Parents should insist that the students dedicate at least the minimum amount of time to some academic study at home. Reading and vocabulary study is the recommended activity during that time if the homework from school is not enough to take the minimum amount of time. Remember, the first purpose of homework is to help children learn personal responsibility and time management skills. Parents should also teach their children that if they do not finish their daily work in class, that work is in addition to the actual homework, and it is not part of the actual homework time. Therefore, reports, themes, and unfinished daily work could account for some students more than the maximum amount of time stated. Student punctuality and personal responsibility will ensure that this does not happen. All students receive homework four nights per week, Monday through Thursday. Homework will be directly related to daily instruction. Daily work not completed is not considered homework, but must be completed by the next day. Homework will be evaluated by the teacher or as a class assignment. Standards of neatness and accuracy are to be maintained regardless of the subject matter area. The homework grade is determined by the percent of homework completed or worked on for the maximum time. Grades 2nd - 6th use daily homework assignment sheets to assist in communication with the home. First grade uses homework assignment sheets at the end of the year. Students should be taught regularly that they are responsible for their homework and that no one other than the classroom teacher should help them complete the assignments. This will help the students learn to concentrate during instructional time, as well as teach personal responsibility. Problems with incomplete homework may be referred for disciplinary action. Signing the daily homework log is a parent responsibility that should be a part of the daily home routine. This log lets you know exactly what the child is supposed to do and serves as a communication tool between home and school. Please support your child's education by doing your part. If students are absent, they are given two school days for every school day of absence to make-up missed work.<\|endoftext\|>	3.6875
6,562	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 16 Sep 2019, 14:15 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # 40% of the dogs at a certain animal shelter have been microchipped. Ho Author Message TAGS: ### Hide Tags Manager Joined: 30 May 2019 Posts: 108 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 18 Jul 2019, 21:03 1 We know that 40% are microchipped, other 60% must have not been microchipped. We need to find exact number of either microchipped, not chipped, or total number of dogs, then we will be able to answer to the questions. (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. - Not sufficient. We just know that 40% of 100 must be integer and that 37.5% of 40% of 100% also must be integer. But there are many such numbers. A is eliminated (2) There are less than 50 dogs at the animal shelter - at least 45 dogs must be present at shelter or else 40% will not be integer. But 40, 35, etc all multiples of 5 work as well. We have many options, thus not sufficient again. B is gone too. Combined, we still do not know the total number of dogs because two numbers satisfy condition above 37.5%40%100%. Those numbers are 40 dogs total, out of which 16 ($$\frac{2}{5}$$40) are microchipped, 24($$\frac{3}{5}$$40) not chipped. And of those chipped(16$$\frac{3}{8}$$), 6 are also sprayed. Another number that matches is 20 dogs total, out of which 8 ($$\frac{2}{5}$$20) are chipped and 12 ($$\frac{3}{5}$$20) not chipped. Out of those chipped ($$\frac{3}{8}$$8), 3 are also sprayed. We arrived at two different values, thus not sufficient again. Manager Joined: 12 Mar 2018 Posts: 83 Location: United States Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 18 Jul 2019, 21:25 1 40% of the dogs at a certain animal shelter have been microchipped. How many of the dogs have not been microchipped? Could be that 60% of the dogs are not microchipped, but to find a value for this, we need the total number of dogs. (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. --> Does not give us any info about the total number of dogs or any way to find the # of dogs not microchipped. Hence not sufficient. (2) There are less than 50 dogs at the animal shelter Clearly insufficient, because there could be 20 dogs 40 dogs, any number of possibilities for the total number of dogs. Stmts 1 + 2 together also doesn't provide any additional information. Hence E. Senior Manager Joined: 31 May 2018 Posts: 272 Location: United States Concentration: Finance, Marketing 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 18 Jul 2019, 21:47 1 let the total number of dogs be x 40% of x microchipped = $$\frac{2x}{5}$$ (from here we know that total number of dogs are multiple of 5) we have to find 60% of x = $$\frac{3x}{5}$$ STATEMENT (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. Dogs that have been microchipped =$$\frac{2x}{5}$$ $$\frac{37.5}{100}$$$$\frac{2x}{5}$$ are either spayed or neutered = $$\frac{3x}{20}$$ from here we know that total number of dogs are multiple of 20 x can be = 20,40,60...... INSUFFICIENT STATEMENT (2) There are less than 50 dogs at the animal shelter We know that the total number of dogs are multiple of 5 so from here, the total number of dogs can be = 5,10,15........,45 INSUFFICIENT STATEMENT (1)&STATEMENT (2) combined we know from statement (1) that the total number of dogs is a multiple of 20 and from statement (2) that the number of dogs is less that 50 we get total number of dogs = 20 and 40 we cant find the total number of dogs not been microchipped so INSUFFICIENT Manager Joined: 06 Jun 2019 Posts: 116 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 18 Jul 2019, 22:10 1 What we know from the stem itself is that the number of dogs must be an integer. Additionally, $$40$$% of this number must also be an integer. $$x$$ - is the number of dogs. $$\frac{2}{5}x$$ - must be an integer. ST1 says that $$37.5$$% of the $$40$$% of $$x$$ also must be an integer. That means that $$\frac{3}{8} * \frac{2}{5} * x$$ must be an integer. More importantly, $$x$$ must be a fixed number so that we are able to find the number of not-microchipped dogs. However, if simplified, $$\frac{3}{20}x$$ can be different integers depending on $$x$$. For example: If $$x=20$$, then $$\frac{3}{20}20= 3$$ If $$x=40$$, then $$\frac{3}{20}40= 6$$ Insufficient ST2 says that $$x<50$$. From the stem we know that $$\frac{2}{5}x$$ - must be an integer. There are many numbers less than $$50$$ that can meet this requirement. For example: If $$x=20$$, then $$\frac{2}{5}20=8$$ If $$x=40$$, then $$\frac{2}{5}40=16$$ Insufficient ST1+ST2. Now $$\frac{3}{20}x$$ must be an integer provided that $$x<50$$. Once again, there are two numbers that can meet both of these requirements: If $$x=20$$, then $$\frac{3}{20}20= 3$$ If $$x=40$$, then $$\frac{3}{20}40= 6$$ Insufficient Hence E _________________ Bruce Lee: “I fear not the man who has practiced 10,000 kicks once, but I fear the man who has practiced one kick 10,000 times.” GMAC: “I fear not the aspirant who has practiced 10,000 questions, but I fear the aspirant who has learnt the most out of every single question.” Manager Joined: 17 Jan 2017 Posts: 86 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 18 Jul 2019, 22:26 1 40% of the dogs at a certain animal shelter have been microchipped. How many of the dogs have not been microchipped? (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. (2) There are less than 50 dogs at the animal shelter For knowing the number of dogs has not been microchipped, statement should provide some numeric value. Stmt 1: it doesn't provide any numeric value. so knowing percentage information is not enough. so insufficient. Stmt 2: it shows condition like dogs<50. but, many answer can be possible . so insufficient. Combining 1 and 2, still many answer can be possible. So, the correct anwer choice is (E) Director Joined: 20 Jul 2017 Posts: 661 Location: India Concentration: Entrepreneurship, Marketing WE: Education (Education) Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 18 Jul 2019, 23:00 1 (1) 37.5% of the dogs that have been micro chipped are also either spayed or neutered. Does not talk about total number Insufficient (2) There are less than 50 dogs at the animal shelter Does not talk about further division of micro chipped Insufficient Combining (1) & (2) X can take values that are multiples of 20 --> Possible values of x = 20 or 40 Insufficient IMO Option E Pls Hit Kudos if you like the solution Attachments 1.png [ 8.03 KiB \| Viewed 150 times ] Manager Joined: 27 May 2010 Posts: 200 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 18 Jul 2019, 23:03 1 40% of the dogs at a certain animal shelter have been microchipped. How many of the dogs have not been microchipped? (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. 37.5% of 40% of microchipped = 15%. Not enough to determine the number not microchipped. (2) There are less than 50 dogs at the animal shelter Total < 50 dogs Not enough Combine both together, 15% of (<50) should be a whole number. The total number of dogs could be 20 or 40. Total number of dogs not microchipped could be 12 or 24. Both together are not enough. Option E. _________________ Please give Kudos if you like the post Manager Joined: 04 Dec 2017 Posts: 93 Location: India Concentration: Other, Entrepreneurship Schools: ISB '20 (D) GMAT 1: 570 Q36 V33 GMAT 2: 620 Q44 V32 GMAT 3: 720 Q49 V39 GPA: 3 WE: Engineering (Other) Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 18 Jul 2019, 23:21 1 Let the total dogs be D. 40% of D = micro-chipped = X Hence, 60% D = Not micro-chipped = Y. We need to find numerical value Y. For that we need to find D. D can be found if X is given. St: 1 37.5% of 40% of D = 37.5% of X = spayed or neutered. We are not given a definite numerical value, and hence, we cannot get D and hence Y. St 1 not sufficient. St: 2 D<50. Multiple values possible. St 2 not sufficient. St1 + St2: Even both statements together don't give definite value of D and hence, that of Y. Manager Joined: 08 Jan 2018 Posts: 98 Location: India GPA: 4 WE: Information Technology (Computer Software) Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 00:23 1 How many of the dogs have not been microchipped? 1. 37.5% of the dogs that have been microchipped are also either spayed or neutered. But it does not tell us how many dogs are not microchipped. Insufficient. 2. There are less than 50 dogs at the animal shelter The number of dogs < 50 It again does not tell us about not microchipped dogs. Insufficient. Combining 1 and 2: Let dogs are 40 =>16 dogs are microchipped and then 6 dogs are also either spayed or neutered. We cannot say anything about not microchipped dogs. Let dogs are 20 => 8 dogs are microchipped and then 3 dogs are also either spayed or neutered. We cannot say anything about not microchipped dogs. We don`t have the exact number of dogs. So, IMO the answer is E. Please hit kudos if you like the solution. Manager Joined: 12 Jan 2018 Posts: 112 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 00:54 1 40% of the dogs at a certain animal shelter have been microchipped. How many of the dogs have not been microchipped? (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. Since the number of dogs must be integral value, least number of dogs is= $$\frac{40}{100} \frac{375}{1000} = \frac{3}{20}$$ Therefore, minimum number of total dogs N= 20. But no. of dogs not microchipped will vary based on the N. -Not sufficient. (2) There are less than 50 dogs at the animal shelter, _Clearly this alone not sufficient. (1)+(2). If N=20, No.of dogs not micro chipped is 12. But if N=4, which also satifies condition in (2), no. of dogs not micro chipped will be 24. Clearly not sufficient Ans: E _________________ "Remember that guy that gave up? Neither does anybody else" Senior Manager Joined: 18 May 2019 Posts: 250 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 01:02 1 We are informed in the question that 40% of the dogs in an animal shelter have been microchipped. Based on the given information, we expected to determine the number of dogs that are not microchipped. Clearly, we know that 60% of the dogs are not microchipped. We therefore need information that leads us to find the total number of dogs in the shelter to enable us find the number of dogs that are not microchipped. Statement 1 is clearly insufficient since it does not give us any useful information or data about the total number of dogs within the shelter. From statement 2, we know that there are less than 50 dogs in the shelter. We only know that the number of dogs is less 50, meaning if we argue for the number of dogs microchipped to be an integer, we will still have many possibilities total dogs in the shelter such as: 45, 16, 35, 30, 25, 20, 15, 10 and 5. Hence statement 2 is also insufficient on its own. Combining 1 and 2. The additional information about the proportion of dogs that were spayed or neutered does not help to narrow down thecpossilities in statement 2 to a definite figure considering taking 37.5% of the dogs that have been microchipped results in fractions. Hence both statements are insufficient. Posted from my mobile device Senior Manager Joined: 18 Jan 2018 Posts: 308 Location: India Concentration: General Management, Healthcare Schools: Booth '22, ISB '21, IIMB GPA: 3.87 WE: Design (Manufacturing) Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 01:28 1 40% of the dogs at a certain animal shelter have been microchipped. How many of the dogs have not been microchipped? Given 40% of dogs at a shelter are microchipped . ==> 60% not microchipped . But we don't know the total no. of dogs in shelter? (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. We need total of dogs , which we cannot infer from the given data --- So Insufficient (2) There are less than 50 dogs at the animal shelter Given n<50 , lets say n = 49 then 60%of 49 = 29.4 = app 30 n =40 , then 60%of 40 =24 If total dogs change , dogs not microchipped change ---Insufficient. Even if take data from A and B combined , we cannot find ---So Answer to this question is E Intern Joined: 09 Jul 2019 Posts: 38 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 02:25 1 Given 40 percent of dogs being microchipped, and 60 percent not, we need to find what is 60 percent in numerical terms. Statement 1 does not provide us with such information neither does statement 2. Combined we are still not able to find number of dogs not microchipped as total number of dogs can be either 40 or 20. Answer is E Manager Joined: 22 Oct 2018 Posts: 73 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 02:27 1 Statement 2: there are less than 50 dogs at the anial shelter. Again no substantial information i being povided.Just that we can consider the various value for number of dogs being less than 50 i.e 49,48,..........1. Not sufficient. Now combining both ,if we take any value less than 50 , we have to calculate 40% of it which should be a whole number and also 37.5% of 40% of dogs also has to be a whole number. Now if we take vlaue as 49 and calculate number of dogs which are microchipped then we get 19.6 which is not a whole number also 37.5% of it also wont be a whole number (we get 7.35) Now if we take number of dogs a 40 and calculate 40% of it we get 16 and 37.5% of it would be 6.It seems that we have arrived at the solution .But there is another value of dogs which gives a whole number as the answers and that number is 20.We get microchipped dogs as 8 and 37.5% of it would be 3.Hence we get two values and that cant hold to be true. Hence E IMO Intern Joined: 24 Mar 2018 Posts: 48 Location: India Concentration: Operations, Strategy Schools: ISB '21 WE: Project Management (Energy and Utilities) Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 03:12 1 Let the total no. of dogs be x. No. of dogs microchipped = 0.4x No. of dogs not microchipped = 0.6x We need to find the value of 0.6x (or simply the value of x) Statement 1: No. of dogs that have been microchipped that are also either spayed or neutered = 37.5% of 0.4x = 0.15x No. of dogs that have been microchipped that are neither spayed or neutered = 0.4x-0.15x = 0.25x This info doesn't help in finding the value of x. Not sufficient. Statement 2: x<50 If x=40, no. of dogs not microchipped = 0.6x = 24 If x=30, no. of dogs not microchipped = 0.6x = 18 Not sufficient. Statement 1 & 2: No. of dogs that have been microchipped that are also either spayed or neutered = 37.5% of 0.4x = 0.15x x<50 x should be an integer (and so should 0.15x) But we cannot get a unique value for 0.15 x (consider the following 2 cases) If x=40, 0.15x=6 If x=20, 0.15x=3 Not sufficient. Option (E) Senior Manager Joined: 25 Sep 2018 Posts: 404 Location: United States (CA) Concentration: Finance, Strategy GMAT 1: 640 Q47 V30 GPA: 3.97 WE: Investment Banking (Investment Banking) Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 03:49 1 40% of the dogs at a certain animal shelter have been microchipped. How many of the dogs have not been microchipped? (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. (2) There are less than 50 dogs at the animal shelter Solution: Question Stem analysis: If there are 40 % of dogs who have been micro chipped in a set, the rest 60 % aren't. we need a specified value for the number of dogs Statement One Alone: 37.5% of the dogs that have been micro chipped are also either spayed or neutered. This statement does not tell us any information about the dogs who have not been micro-chipped or the total number of dogs. Hence statement one alone is insufficient. We can eliminate A & D Statement two alone: Total number of dogs is less than 50. this does not gives us a specified number but this gives us a range. Statement one & two together We know that , the number of dogs is less than 50 and that 40 % of them are micro chipped , considering the numbers less than 50, we can test some values , for eg, 49 now 40 % of 49 is surely not an integer, we can try 40, we see that the total number of micro-chipped dogs is 16. so we have a number, but can it satisfy the first condition and be an integer? Yes. it does. 37.5 % of 16 is 6 and hence it can be the total number of dogs, but testing some further values, we notice that 40 % of 20 is 8 & 37.5 % of 8 is 3. So we are getting two different values satisfying both the conditions and hence there isn't a specific answer or value. _________________ Why do we fall?...So we can learn to pick ourselves up again Manager Joined: 01 Oct 2018 Posts: 112 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 04:43 1 40% of the dogs at a certain animal shelter have been microchipped. How many of the dogs have not been microchipped? (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. Give nothing. INSUFFICIENT (2) There are less than 50 dogs at the animal shelter n - number of dogs n<50 If 40% of the dogs at a certain animal shelter have been microchipped, 0,4n = integer (must be) Possible variants: 5,10,15...45 INSUFFICIENT (1) and (2) If 37.5% of the dogs that have been microchipped are also either spayed or neutered, than 0,40,375n = integer (must be) 0,40,375 = 0,15 0,15n = integer n<50 Possible variants: 20 and 40 INSUFFICIENT Manager Joined: 31 Dec 2017 Posts: 83 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 04:44 1 40% of the dogs have been microchipped---->60% of the dogs haven't been microchipped., how many of the dogs account for 60%????? So, we need to know the total of the dogs in a real number in that shelter ST1: We know the percentage of the dogs regarding to its types.---->Obviously not helpful to answer the question----->NS ST2: total of the dogs < 50 --->the number of the dogs with microchip + the number of the dogs without microchip < 50------>This piece of info looks like a condition for an equation; however, we couldn't get anything from this statement to answer the question---->NS ST1 + ST2: Still couldn't answer the question, as the we need at least 1 real number to form an equation; meanwhile, we don't have any after combining all the given info.------>My answer is E Manager Joined: 25 Jul 2018 Posts: 207 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 05:02 1 The number of total dogs is 'a' --> a40%=a40/100=2/5a(microchipped dogs--it should be integer) Statement1: "37.5% of microchipped dogs": --> 37.5%(2/5a)= (37.5/100) (2/5a)=3/20a That means 3/20a must be integer. Still no info about how many dogs there are. Any multiples of 20 could help to find the solution. --> (if a=20, then 202/5=8 microchipped ones. 20-8 =12(not been microchipped) --> (if a=60, then 602/5=24 microchipped ones. 60-24 =36(not been microchipped) .... Insufficient. Statement2: a<50 Also, the solution depends on 'a' if a=45, then 45-(4540/100)=27 (not been microchipped) if a=35, then 35-(3540/100)=19 (not been microchipped) .... Insufficient Taken together 1 and 2: ---> '3/20a' and a<50 There are 2 multiples of 20 between 1 and 50,(not inclusive): (20 and 40) --> that means two different solutions Insufficient Manager Joined: 03 Aug 2009 Posts: 58 Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] ### Show Tags 19 Jul 2019, 05:20 1 40% of the dogs at a certain animal shelter have been microchipped. How many of the dogs have not been microchipped? As we are talking about alive beings, their number always has to be an integer (1) 37.5% of the dogs that have been microchipped are also either spayed or neutered. Not sufficient, as 37.5% could of 100 or 200 or ...1000 of dogs (2) There are less than 50 dogs at the animal shelter Not sufficient, as 40% could of 10, 20,30,40 (multiple of 10<50) dogs Combined, we get that 37,5% spayed of 40% chipped of X dogs needs to be an integer less than 50. Is there is such unique integer? $$0.375 0.40X = 0.15X$$, then to be an integer X must be multiple of 20 less than 50: 20, 40 Back-test : $$40.4.375 = 6$$ $$20.4.375 = 3$$ So, since there are 2 possibilities of X, then insufficient, answer is E. Re: 40% of the dogs at a certain animal shelter have been microchipped. Ho [#permalink] 19 Jul 2019, 05:20 Go to page Previous 1 2 3 4 Next [ 75 posts ] Display posts from previous: Sort by<\|endoftext\|>	4.4375
578	# Electric power definition Electric power definition: Electric power is the product of current and voltage. Electric power = current × voltage The electric power is expressed in watts. 1 watt = 1 ampere × 1 volt Say for instance an iron is rated at 1200 watts. If you home outlet is 120 volts, the iron will draw a current of 10 amperes. Why? It is because 1200 = 10 × 120 At this point, you may not quite see why we are multiplying the current and the voltage to get the power. A little math will clarify this! Recall that in the lesson about electric current, we saw that 1 ampere = 1 coulomb of charge per second. The word per means divided by. Therefore, we can rewrite 1 coulomb of charge per second as shown below. 1 ampere = 1 coulomb / 1 second Recall also that in the lesson about electric potential, we saw that 1 volt = 1 joule of energy per 1 coulomb of charge Therefore, we can rewrite 1 joule of energy per 1 coulomb of charge as shown below. 1 volt = 1 joule / 1 Coulomb We can multiply a current of 1 ampere by a voltage of 1 volt to see what we end up with. current × voltage = 1 coulomb / 1 second × 1 joule / 1 coulomb 1 coulomb gets cancelled since it is on top and at the bottom. current × voltage = 1 joule / 1 second The above also means 1 joule per second. In the lesson about power, we saw that 1 joule of energy per second = 1 watt and watt is a unit of power. Therefore, power = current × voltage = 1 joule / 1 second We can generalize the formula. Keep in mind that joule represents energy and 1 second represents time ### Electric power definition #2 Electric power = energy / time From the formula, you can see that the electric power is the rate at which energy is transferred. This second definition of electric power can also help us to calculate the cost of electric energy and we will show you how in a different lesson. If 10 / 2 = 5, then 10 = 5 × 2 If electric power = energy / time then, energy = electric power × time energy = electric power × time is the formula to use to calculate the cost of electric energy. Enter Your E-mail Address Enter Your First Name (optional) Then Don't worry — your e-mail address is totally secure. I promise to use it only to send you Physics lessons.<\|endoftext\|>	4.53125
250	Purpose of study English has a pre-eminent place in education and in society. A high-quality education in English will teach pupils to write and speak fluently so that they can communicate their ideas and emotions to others and through their reading and listening, others can communicate with them. Through reading in particular, pupils have a chance to develop culturally, emotionally, intellectually, socially and spiritually. Literature, especially, plays a key role in such development. Reading also enables pupils both to acquire knowledge and to build on what they already know. All the skills of language are essential to participating fully as a member of society; pupils, therefore, who do not learn to speak, read and write fluently and confidently are effectively disenfranchised. The overarching aim for English in the national curriculum is to promote high standards of literacy by equipping pupils with a strong command of the written and spoken word, and to develop their love of literature through widespread reading for enjoyment. The national curriculum for English aims to ensure that all pupils: The National Curriculum in England 2014 For more information on your child's learning in English, please look at the Curriculum Leaflet on your child's class page, found in our 'Children' section.<\|endoftext\|>	4.1875
1,661	# HW7_soln - Math 33B HW#7 Solutions Problems Graded Section... • Notes • 6 This preview shows page 1. Sign up to view the full content. This is the end of the preview. Sign up to access the rest of the document. Unformatted text preview: Math 33B HW #7 Solutions Problems Graded Section 8.4: 28 Section 8.5: 8, 26 Section 9.1: 10, 20 Solutions Section 8.4: Problem 28 We follow the analysis in Example 4.8 (starting on page 356). We see that the first mass has the force from both springs acting on it while the second mass has the force from the second spring and the external force acting upon it. From Newton's second law, the equations are: Equation (1) cos Equation (2) Now, we let , , , , where we have: , , , We can now rewrite equations (1) and (2), respectively, as follows: Additionally, we have (from the definition of ): We can write these equations in matrix form: 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 Since both masses start from rest and at their equilibrium positions, we have the following initial conditions: 0 0 0, 0 0 0 In terms of our , we have: 0 0, 0, 0, 0 T and not simply , (i.e. we have an external force), the system Since we have is inhomogeneous. Section 8.5: Problem 8 The system in Problem 2 can be written as: 6 4 8 6 e and t We want to show that t e We first show x : e e 6 4 8 6 Similarly, we show y e 2e e e : 2e 4e 2e 2e 6e 8e 4e 6e e 2e are solutions to the system above. 2e 2e 6 4 4e e 6e 2e 8 6 2e 8e 12e 4e We now verify that any linear combination is also a solution. In other words, we want to show : 2 e 2 e 2e 2e 2 e 4 e 2e 4e 6 4 e e 8 6 e 2e e e 6 4 8 6 e 2 e e e 6 4 8 6 e 2 e 6 e e 4 e 2 e e e 2 e 8 6 e 6e 4e 6e 8e 6e 8e 12e 8e 2 e 2 e 2 e 4 e Section 8.5: Problem 26 We are given the following solutions: cos sin , cos The system and the initial conditions are: 1 1 1 , 0 0 2 1 We first verify that cos cos 1 2 cos cos cos 1 1 cos cos sin sin sin sin t cos cos sin and sin sin sin cos are solutions to the homogeneous solution: sin sin cos sin sin 1 2 sin sin sin 1 1 sin cos cos cos cos sin sin cos cos cos sin sin cos t We can show that the solutions are linearly independent by assuring that the Wronskian is not identically zero: , det cos sin sin cos cos cos cos sin sin cos sin sin t sin t sin sin sin cos cos t cos cos We see that the Wronskian is not identically zero and thus the solutions are linearly independent. Another, perhaps simpler, way to see if two solutions is by looking at a particular value of . Take an arbitrary , say 0, and see if one solution is a multiple of the other: cos 0 sin 0 0 cos 0 1 sin 0 cos 0 0 sin 0 0 Since the two solutions are not multiples of one another at one time instant, they are linearly independent at this one time instant. From Definition 5.13, if a set of solutions is linearly independent at one value of , then they are linearly independent for any value of . Therefore, the two solutions are linearly independent. Finally, we can get a general solution using the two given solutions: cos sin sin cos sin we use the initial condition to solve for the undermined coefficients: 1 0 0 cos 0 sin 0 sin 0 cos 0 0 cos 0 sin 0 1 cos 0 We have the following set of equations: 0 1 2 The solution is thus: sin 0 2 2 cos sin sin 2sin cos Section 9.1: Problem 10 We have the following: 1 0 0 4 3 2 8 4 3 Therefore, we have: 1 0 0 4 3 2 8 4 3 We want to find out when the determinant of the above matrix is zero in order to get the eigenvalues. It is simplest to take the determinant along the first row since it has two zeros: det 1 3 3 2 4 1 9 8 1 1 1 1 1 0 The eigenvalues are: 1 (multiplicity 2) 1 Section 9.1: Problem 20 We are given: 3 2 4 3 To get the eigenvalues: 3 2 4 3 det 3 3 2 4 9 8 1 1 The eigenvalues are: 1, 1 Now, we can get the eigenvectors for each eigenvalue. For 1: 3 2 1 4 3 The two equations are: 3 2 2 2 0 3 4 4 0 4 We see that the two equations lead to the same result of . Each equation resulting from the eigenvalue problem will always be multiples of the other equations. This shows that you can always just take one equation resulting from and know that the other equations do not 1 and thus our eigenvector is: give any new information. We arbitrarily set 1 1 For 1: 3 2 1 4 3 The two equations are: 3 2 4 2 0 3 4 2 0 4 Again, the two equations lead to the same result of 2 consequently 2 2. The eigenvector is: 1 2 The solution can then be written as: 1 1 . We arbitrarily set 1 and 1 2 ... View Full Document {[ snackBarMessage ]} ### What students are saying • As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students. Kiran Temple University Fox School of Business ‘17, Course Hero Intern • I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero. Dana University of Pennsylvania ‘17, Course Hero Intern • The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time. Jill Tulane University ‘16, Course Hero Intern<\|endoftext\|>	4.4375
1,391	Why do this problem? offers an opportunity to explore and discuss two types of probability: experimental and theoretical. The simulation generates lots of experimental data quickly, freeing time to focus on predictions, analysis and justifications. Calculating the theoretical probabilities provides a motivation for using sample space diagrams or perhaps tree diagrams. The final question in the problem offers the opportunity for exploration of a rich context where collaborative working makes it possible to tackle an otherwise unmanageable task. This printable worksheet may be useful: Odds and Evens. You may wish to use the start of What Numbers Can We Make? as a preliminary activity to get students thinking about the effect of combining odd and even numbers. The notes that follow are based on a lesson with a group of 14 year old students. They are in two parts: the first part for teachers who wish to use the activity for a single lesson on probability and sample space diagrams or tree diagrams, and the second part for teachers who wish to follow this up with a collaborative task that leads to interesting and Start by showing how the game is played using Set A with the interactivity (or using numbered counters in a bag). Play the game no more than ten times, so that students have a feel for the game but don't have sufficient results to draw conclusions about the probabilities. Then ask them to decide whether they think the game is fair, and to do some maths to support While students are working, circulate and observe the methods being used: Bring the class together and choose individuals who used different methods to explain what they did to the class, recording what they did on the board. Perhaps choose those who used less sophisticated methods first. Emphasise the merits of a sample space method rather than a listing method, to prepare students for tackling examples with a large number of balls. Those who are confident with tree diagrams may prefer to continue using them. Use the interactivity to confirm that the experimental probability matches closely to the theoretical probability that students have calculated. There are opportunities here for rich discussion about how closely we expect an experimental probability to match the theory. Now show sets B, C and D, and ask them to think on their own, without writing, about which of the four sets they would choose to play with, to maximise their chances of winning. Once they have had a short time to reflect, ask them to discuss in pairs their choice, and to justify their decisions (again, without writing). There is often disagreement about which set offers the best chance of winning, so bring the class together to compare ideas before setting them the task of calculating the probabilities - discourage them from using inefficient listing methods. Once the probabilities have been calculated, use the interactivity again to confirm that the experimental probability is close to the calculated one. Now write up on the board a set E, which contains four large even numbers and two large odd numbers. Make them large enough that calculations would be offputting! Ask the class to work in pairs to calculate the probability of winning with set E, and give them a short time frame in which to do this. The intention is to alert students that the numbers themselves don't matter, but the numbers of odds and evens is the important point. Set E has the same structure as Set C, so we already know the chance of winning. Then the class can be introduced to this sort of sample space diagram where odds and evens are collected together: Point out that none of the sets looked at so far yields a fair game. "How could we go about finding out whether there are any sets that would give a fair game?" One way of organising the search is to draw up a table on the board showing different combinations of odds and evens: Those already identified as not being fair games (sets A, B, C and D) can be crossed off. Then divide the class into groups working on different combinations and ask them to report back. Students could record combinations that have been checked on the board with a tick or a cross to show whether they are fair or not. If something has two ticks or two crosses, it could be accepted as being confirmed. When disagreements arise, ask other groups to resolve them. There will be opportunities while the class are working to stop everyone and share students' insights that will make the job easier. For example: "None of the combinations with zero will work because..." "If 3 odds and 2 evens won't work, 2 odds and 3 evens won't either, because..." "You can't have the same number of evens and odds because..." Eventually, there will be a sea of crosses on the board and just a few combinations that work (four, if the class have gone up to 9 balls in total). Ask the class to stop and consider what the fair sets have in common. This may lead to some new conjectures about the total number of balls, so organise the class to test the conjecture out on the next obvious total. Once there is some confirmation about the total number of balls needed for fair games, conjectures can also be made about how these should be split into odds and evens. Students can be set to work to test examples with large numbers, using the simplified sample space method above. Draw attention to how valuable it is to work collaboratively as part of a mathematical community, and how difficult it would have been to have reached the same insights working alone. Although it is unlikely that many students will be able to prove their conjectures algebraically on their own, this proof may be sufficiently accessible to be worth sharing with the class. There are a number of ways of using this resource: - To be presented as an elegant way of proving the ideas the students have discovered - As a proof presented on the board for students to recreate for themselves after it's been rubbed out - To be printed out and distributed to students for them to make sense of, and for them to annotate so that they can talk through the proof, line by line, for someone who hadn't met it yet. - As a 'proof sorting' exercise where the proof is cut into sections and mixed up for students to reassemble into the correct order How can you decide if a game is fair? What are the most efficient methods for recording possible combinations? How can we make this difficult task (of finding a fair game) more manageable? The problem In a Box offers another context for exploring exactly the same underlying mathematical structure, and could be used as a follow-up problem a few weeks after working on this one. The first parts of this problem should be accessible to most students, and can be used for focussing on the benefits of using sample space diagrams instead of listing combinations.<\|endoftext\|>	4.65625
1,506	# NCERT Solutions For Class 6 Maths Decimals Exercise 8.2 ncert textbook ## NCERT Solutions For Class 6 Maths Decimals Exercise 8.2 NCERT Solutions for Class 6 Maths Chapter 8 Decimals Ex 8.2 Exercise 8.2 Question 1. Complete the table with the help of these boxes and use decimals to write the number. Solution: Ones Tenths Hundredths Number (a) 0 2 6 0.26 (b) 1 3 8 1.38 (c) 1 2 8 1.28 Explanation: (a) In this figure, If small block out of 100 are shaded ∴ Decimal representation = 0.26 (b ) 100 small blocks + 38 small blocks are shaded ∴ Decimal representation = 1.38 (c) 100 small blocks + 28 small blocks are shaded ∴ Decimal representation = 1.28 Question 2. Write the numbers given in the following place value table in decimal form: Solution: (a) 0 Hundreds + 0 Tens + 3 Ones + 2 Tenths + 5 Hundredths + 0 Thousandths = 3 + 0.2 + 0.05 + 0.000 = 3.250 = 3.25 (b) 1 Hundreds + 0 Tens + 2 Ones + 6 Tenths + 3 Hundredths + 0 Thousandths = 102.630 = 102.63 (c) 0 Hundreds + 3 Tens + 0 Ones + 0 Tenths + 2 Hundredths + 5 Thousandths = 0 x 100 + 3 x 10 + 0 x 1 + 0 x $\frac { 1 }{ 10 }$+ 2 x $\frac { 1 }{ 100 }$+ 5 x $\frac { 1 }{ 1000 }$ = 0 + 30 + 0 + 0.0 + 0.02 + 0.005 = 30.025 (d) 2 Hundreds + 1 Tens + 1 Ones + 9 Tenths + 0 Hundredths + 2 Thousandths = 2 x 100 + 1 x 10 + 2 x $\frac { 1 }{ 10 }$+ 0 x $\frac { 1 }{ 100 }$+ 2 x $\frac { 1 }{ 100 }$ = 200 + 10 + 1 + 0.9 + 0.00 + 0.002 = 211 + 0.902 = 211.902 (e) 0 Hundreds + 1 Tens + 2 Ones + 2 Tenths + 4 Hundredths + 1 Thousandths = 0 x 100 + 1 x 10 + 2 x 1 + 2 x $\frac { 1 }{ 10 }$+ 4 x $\frac { 1 }{ 100 }$+ 1 x $\frac { 1 }{ 1000 }$ = 0 + 10 + 2 + $\frac { 2 }{ 10 }$+ $\frac { 4 }{ 100 }$+ $\frac { 1 }{ 1000}$ = 12 + 0.2 + 0.04 + 0.001 = 12.241 Question 3. Write the following decimals in the place value table. (a) 0.29 (b) 2.08 (c) 19.60 (d) 148.32 (e) 200.812 Solution: (a) 0.29 = 0 + 0.2 + 0.09 = 0 Ones + 2 Tenths + 9 Hundredths (b) 2.08 = 2 + 0.0 + 0.08 = 2 Ones + 0 Tenths + 8 Hundredths (c) 19.60 = 10 + 9 + 0.6 + 0.00 = 1 Tens + 9 Ones + 6 Tenths + 0 Hundredths (d) 148.32 = 100 + 40 + 8 + 0.3 + 0.02 = 1 Hundred + 4 Tens + 8 Ones + 3 Tenths + 2 Hundredths (e) 200.812 = 200 + 0.8 + 0.01 + 0.002 = 2 Hundreds + 8 Tenths + 1 Hundredth + 2 Thousandths The above information, we can gives in place value Table: Hundreds (100) Tens (10) Ones (1) Tenths (1/10) Hundredths (1/100) Thousandths (1/1000) (a) 0 0 0 2 9 0 (b) 0 0 2 0 8 0 (c) 0 1 9 6 0 0 (d) 1 4 8 3 2 0 (e) 2 0 0 8 1 2 Question 4. Write each of the following as decimals. Solution: Question 5. Write each of the following decimals in words. (a) 0.03 (b) 1.20 (c) 108.56 (d) 10.07 (e) 0.032 (f) 5.008 Solution: (a) 0.03 = Zero point zero three (b) 1.20 = One point two zero (c) 108.56 = One hundred eight point fifty-six (d) 10.07 = Ten point zero seven (e) 0.032 = Zero point zero three two (f) 5.008 = Five point zero zero eight Question 6. Between which two numbers in tenths place on the number line does each of the given numbers lie? (a) 0.06 (b) 0.45 (c) 0.19 (d) 0.66 (e) 0.92 (f) 0.57 Solution: (a) 0.06 lies between 0 and 0.1 (b) 0.45 lies between 0.4 and 0.5 (c) 0.19 lies between 0.1 and 0.2 (d) 0.66 lies between 0.6 and 0.7 (e) 0.92 lies between 0.9 and 1.0 (f) 0.57 lies between 0.5 and 0.6 Question 7. Write as fraction in lowest terms. (a) 0.60 (b) 0.05 (c) 0.75 (e) 0.25 (f) 0 .125 (g) 0.066<\|endoftext\|>	4.8125
175	# How do you solve ln8-ln(x+4)=1? Aug 21, 2016 $x = \frac{8}{e} - 4 = - 1.057$ #### Explanation: $\ln 8 - \ln \left(x + 4\right) = 1$ can be written as $\ln 8 - \ln \left(x + 4\right) = \ln e$ or $\ln \frac{8}{x + 4} = \ln e$ or $\frac{8}{x + 4} = e$ or $x + 4 = \frac{8}{e}$ i.e. $x = \frac{8}{e} - 4$ or $x = 2.943 - 4 = - 1.057$<\|endoftext\|>	4.4375
520	# Additive identity Jump to navigation Jump to search In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings. ## Formal definition Let N be a set which is closed under the operation of addition, denoted +. An additive identity for N is any element e such that for any element n in N, e + n = n = n + e Example: The formula is n + 0 = n = 0 + n. ## Proofs ### The additive identity is unique in a group Let (G, +) be a group and let 0 and 0' in G both denote additive identities, so for any g in G, 0 + g = g = g + 0 and 0' + g = g = g + 0' It follows from the above that (0') = (0') + 0 = 0' + (0) = (0) ### The additive identity annihilates ring elements In a system with a multiplication operation that distributes over addition, the additive identity is a multiplicative absorbing element, meaning that for any s in S, s·0 = 0. This can be seen because: {\displaystyle {\begin{aligned}s\cdot 0&=s\cdot (0+0)=s\cdot 0+s\cdot 0\\\Rightarrow s\cdot 0&=s\cdot 0-s\cdot 0\\\Rightarrow s\cdot 0&=0\end{aligned}}} ### The additive and multiplicative identities are different in a non-trivial ring Let R be a ring and suppose that the additive identity 0 and the multiplicative identity 1 are equal, or 0 = 1. Let r be any element of R. Then r = r × 1 = r × 0 = 0 proving that R is trivial, that is, R = {0}. The contrapositive, that if R is non-trivial then 0 is not equal to 1, is therefore shown. ## References • David S. Dummit, Richard M. Foote, Abstract Algebra, Wiley (3d ed.): 2003, ISBN 0-471-43334-9.<\|endoftext\|>	4.71875
1,010	# Difference between revisions of "1959 IMO Problems/Problem 1" ## Problem Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$. ## Solutions ### First Solution For this fraction to be reducible there must be a number $x$ such that $x \cdot (14n+3) = (21n+4)$, and a $1/x$ such that $1/x \cdot (21n+4) = (14n+3)$. Since $x$ can only be one number ($x$ is a linear term) we only have to evaluate $x$ for one of these equations. Using the first one, $x$ would have to equal $3/2$. However, $3*3/2$ results in $9/2$, and is not equal to our desired $4$. Since there is no $x$ to make the numerator and denominator equal, we can conclude the fraction is irreducible. EDIT: It appears to me that this solution is incorrect because it assumes that if $ax + b = cx + d$, then $a = c$ and $b = d$ -- the problem says irreducible for ALL $n$. If you agree, please remove this solution. 12/17/2017 EDIT 2: Disregard the first edit, because when comparing two polynomials (linear functions in this case), the coefficients for each term must be equal to eachother. For example, if $x$ can be any number, then in $ax^2+4=6x^2+b$, we must have $(a,b)=(6,4)$. Thus, if $ax+b=cx+d$ for all values $x$, then $a$ must equal $c$ and $b$ must equal $d$. Clarification for edit 2: In this problem, we have $14nx+3x=21n+4$, with an unknown $n$, so we must have $14x=21$ and $3x=4$, which cannot be true for any $1$ value of $x$. ### Second Solution Denoting the greatest common divisor of $a, b$ as $(a,b)$, we use the Euclidean algorithm as follows: $( 21n+4, 14n+3 ) = ( 7n+1, 14n+3 ) = ( 7n+1, 1 ) = 1$ As in the first solution, it follows that $\frac{21n+4}{14n+3}$ is irreducible. Q.E.D. ### Third Solution Assume that $\dfrac{14n+3}{21n+4}$ is a reducible fraction where $p$ is a divisor of both the numerator and the denominator: $14n+3\equiv 0\pmod{p} \implies 42n+9\equiv 0\pmod{p}$ $21n+4\equiv 0\pmod{p} \implies 42n+8\equiv 0\pmod{p}$ Subtracting the second equation from the first equation we get $1\equiv 0\pmod{p}$ which is clearly absurd. Hence $\frac{21n+4}{14n+3}$ is irreducible. Q.E.D. ### Fourth Solution We notice that: $\frac{21n+4}{14n+3} = \frac{(14n+3)+(7n+1)}{14n+3} = 1+\frac{7n+1}{14n+3}$ So it follows that $7n+1$ and $14n+3$ must be coprime for every natural number $n$ for the fraction to be irreducible. Now the problem simplifies to proving $\frac{7n+1}{14n+3}$ irreducible. We re-write this fraction as: $\frac{7n+1}{(7n+1)+(7n+1) + 1} = \frac{7n+1}{2(7n+1)+1}$ Since the denominator $2(7n+1) + 1$ differs from a multiple of the numerator $7n+1$ by 1, the numerator and the denominator must be relatively prime natural numbers. Hence it follows that $\frac{21n+4}{14n+3}$ is irreducible. Q.E.D ### Fifth Solution By Bezout's Theorem, $3 \cdot (14n+3) - 2 \cdot (21n + 4) = 1$, so the gcd of the numerator and denominator is 1 and the fraction is irreducible.<\|endoftext\|>	4.8125
1,297	# 2013 AMC 10B Problems/Problem 16 ## Problem In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC$? $[asy] unitsize(0.2cm); pair A,B,C,D,E,P; A=(0,0); B=(80,0); C=(20,40); D=(50,20); E=(40,0); P=(33.3,13.3); draw(A--B); draw(B--C); draw(A--C); draw(C--E); draw(A--D); draw(D--E); dot(A); dot(B); dot(C); dot(D); dot(E); dot(P); label("A",A,SW); label("B",B,SE); label("C",C,N); label("D",D,NE); label("E",E,SSE); label("P",P,SSW); [/asy]$ $\textbf{(A) }13 \qquad \textbf{(B) }13.5 \qquad \textbf{(C) }14 \qquad \textbf{(D) }14.5 \qquad \textbf{(E) }15$ ## Solution 1 Let us use mass points: Assign $B$ mass $1$. Thus, because $E$ is the midpoint of $AB$, $A$ also has a mass of $1$. Similarly, $C$ has a mass of $1$. $D$ and $E$ each have a mass of $2$ because they are between $B$ and $C$ and $A$ and $B$ respectively. Note that the mass of $D$ is twice the mass of $A$, so $AP$ must be twice as long as $PD$. PD has length $2$, so $AP$ has length $4$ and $AD$ has length $6$. Similarly, $CP$ is twice $PE$ and $PE=1.5$, so $CP=3$ and $CE=4.5$. Now note that triangle $PED$ is a $3-4-5$ right triangle with the right angle $DPE$. Since the diagonals of quadrilaterals $AEDC$, $AD$ and $CE$, are perpendicular, the area of $AEDC$ is $\frac{6 \times 4.5}{2}=\boxed{\textbf{(B)} 13.5}$ ## Solution 2 Note that triangle $DPE$ is a right triangle, and that the four angles (angles $APC, CPD, DPE,$ and $EPA$) that have point $P$ are all right angles. Using the fact that the centroid ($P$) divides each median in a $2:1$ ratio, $AP=4$ and $CP=3$. Quadrilateral $AEDC$ is now just four right triangles. The area is $\frac{4\cdot 1.5+4\cdot 3+3\cdot 2+2\cdot 1.5}{2}=\boxed{\textbf{(B)} 13.5}$ ## Solution 3 From the solution above, we can find that the lengths of the diagonals are $6$ and $4.5$. Now, since the diagonals of AEDC are perpendicular, we use the area formula to find that the total area is $\frac{6\times4.5}{2} = \frac{27}{2} = 13.5 = \boxed{\textbf{(B)} 13.5}$ ## Solution 4 From the solutions above, we know that the sides CP and AP are 3 and 4 respectively because of the properties of medians that divide cevians into 1:2 ratios. We can then proceed to use the heron's formula on the middle triangle EPD and get the area of EPD as 3/2, (its simple computation really, nothing large). Then we can find the areas of the remaining triangles based on side and ratio length of the bases. ## Solution 5 We know that $[AEDC]=\frac{3}{4}[ABC]$, and $[ABC]=3[PAC]$ using median properties. So Now we try to find $[PAC]$. Since $\triangle PAC\sim \triangle PDE$, then the side lengths of $\triangle PAC$ are twice as long as $\triangle PDE$ since $D$ and $E$ are midpoints. Therefore, $\frac{[PAC]}{[PDE]}=2^2=4$. It suffices to compute $[PDE]$. Notice that $(1.5, 2, 2.5)$ is a Pythagorean Triple, so $[PDE]=\frac{1.5\times 2}{2}=1.5$. This implies $[PAC]=1.5\cdot 4=6$, and then $[ABC]=3\cdot 6=18$. Finally, $[AEDC]=\frac{3}{4}\times 18=\boxed{13.5}$. ~CoolJupiter ## Solution 6 As from Solution 4, we find the area of $\triangle DPE$ to be $\frac{3}{2}$. Because $DE = \frac{5}{2}$, the altitude perpendicular to $DE = \frac{6}{5}$. Also, because $DE \|\| AC$, $\triangle ABC$ is similar to $\triangle{DBE}$ with side length ratio $2:1$, so $AC=5$ and the altitude perpendicular to $AC = \frac{12}{5}$. The altitude of trapezoid $ACDE$ is then $\frac{18}{5}$ and the bases are $\frac{5}{2}$ and $5$. So, we use the formula for the area of a trapezoid to find the area of $ACDE = \boxed{13.5}$<\|endoftext\|>	4.78125
222	The SMU temperature-at-depth maps start from the actual temperature measured in the Earth at as many sites as possible. In addition, the thermal conductance of the rocks (changing as the rock minerals change with deeper depths), the area heat flow, and the rock density (sedimentary rocks are less dense than basement rocks) are used to calculate the deeper temperatures. SMU Geothermal Lab calculates temperatures at specific depth intervals using these variables to produce the temperature maps at different depth slices for the United States. Most of the measured temperatures used in the calculations are from sedimentary rocks which overlie the harder basement rock. The oil and gas industry has drilled into sedimentary rock as deep as 26,000 (ft) or 8 km in West Texas, yet more typical oil and gas drilling is 4,000 to 10,000 ft (1.2 to 3 km) depending on the depth to the resource. In areas with geothermal power production, drilling is usually in the 1 to 3 km depth range for western United States. Temperature-at-depth maps are available for the following depths:<\|endoftext\|>	3.859375
5,210	# 5.1: Graphical Solutions to Systems of Equations Difficulty Level: At Grade Created by: CK-12 When you graph two linear functions on the same Cartesian plane, the resulting lines may intersect. Do the following two lines intersect? If so, where? \begin{align}& \begin{Bmatrix} 2x+y = 5 \\ x-y = 1 \end{Bmatrix}\end{align} ### Guidance A \begin{align}2 \times 2\end{align} system of linear equations consists of two equations with two variables, such as athe one below: \begin{align}& \begin{Bmatrix} 2x+y = 5 \\ x-y = 1 \end{Bmatrix}\end{align} When graphed, a system of linear equations is two lines. To solve a system of linear equations, figure out if the two lines intersect and if so, at what point. One way to solve a system of equations is by graphing. Graph both lines and look for the point where they intersect. Keep in mind that even though most of the time when you graph two lines they will intersect in just one point, there are two other possibilities: 1. The lines might never intersect (they are parallel lines) 2. The lines might coincide (be exactly the same line) A system that results in one point of intersection is consistent and independent. A system that results in lines that coincide is consistent and dependent. A system that results in two parallel lines is inconsistent. #### Example A Solve the following system of linear equations graphically: \begin{align}\begin{Bmatrix} x-2y -2= 0 \\ 3x+4y = 16 \end{Bmatrix}\end{align} Solution: Both equations will be graphed on the same Cartesian plane using the slope-intercept method. Begin by writing each linear equation in slope-intercept form. \begin{align}& x-2y-2 = 0\\ & x {\color{red}-x}-2y-2 = 0 {\color{red}-x}\\ & -2y-2 = -x\\ & -2y-2 {\color{red}+2} = -x {\color{red}+2}\\ & -2y = -x+2\\ & \frac{-2y}{{\color{red}-2}} = \frac{-x}{{\color{red}-2}}+\frac{2}{{\color{red}-2}}\\ & \boxed{y = \frac{1}{2}x-1} \qquad \text{Equation One}\end{align} \begin{align}& 3x+4y = 16\\ & 3x {\color{red}-3x}+4y = 16 {\color{red}-3x}\\ & 4y = 16-3x\\ & \frac{4y}{{\color{red}4}} = \frac{16}{{\color{red}4}}-\frac{3x}{{\color{red}4}}\\ & y = 4-\frac{3}{4}x\\ & \boxed{y = -\frac{3}{4}x+4} \qquad \text{Equation Two}\end{align} Graph both equations on the same Cartesian plane. The lines have intersected at the point (4, 1). When two equations place two conditions on two same variables at the same time, a system of simultaneous equations is formed. The solution is an ordered pair which satisfies both of the equations in the system. If an ordered pair satisfies an equation, replacement of the variables with the values will result in both sides of the equation being equal. Test (4, 1) in equation one: \begin{align}x-2y-2 &= 0 && \text{Use the original equation}\\ ({\color{red}4})-2({\color{red}1})-2 &= 0 && \text{Replace} \ x \ \text{with} \ 4 \ \text{and replace} \ y \ \text{with} \ 1.\\ 4-2-2 &= 0 && \text{Perform the indicated operations and simplify the result.}\\ 4-{\color{red}4} &= 0\\ {\color{red}0} &= 0 && \text{Both sides of the equation are equal. The ordered pair} \ (4, 1) \ \text{satisfies the equation.}\end{align} Test (4, 1) in equation two: \begin{align}3x+4y &= 16 && \text{Use the original equation}\\ 3({\color{red}4})+4({\color{red}1}) &= 16 && \text{Replace} \ x \ \text{with} \ 4 \ \text{and replace} \ y \ \text{with} \ 1.\\ 12+4 &= 16 && \text{Perform the indicated operations and simplify the result.}\\ {\color{red}16} &= 16 && \text{Both sides of the equation are equal. The ordered pair} \ (4, 1) \ \text{satisfies the equation.}\end{align} This system of simultaneous equations has a solution and is therefore called a consistent system. Because it has only one ordered pair as a solution, it is called an independent system. #### Example B Solve the following system of linear equations graphically: \begin{align}\begin{Bmatrix} 2y-3x = 6 \\ 4y-6x = 12 \end{Bmatrix}\end{align} Solution: Both equations will be graphed on the same Cartesian plane using the intercept method. Let \begin{align}x = 0\end{align}. Solve for \begin{align}y\end{align} \begin{align}& 2y-3x = 6\\ & 2y-3 ({\color{red}0}) = 6 \quad \text{Replace} \ x \ \text{with zero.}\\ & 2y = 6 \qquad \qquad \text{Simplify}\\ & \frac{2y}{{\color{red}2}} = \frac{6}{{\color{red}2}} \qquad \quad \ \ \text{Solve for} \ y.\\ & \boxed{y = 3} \qquad \qquad \text{The} \ y \text{-intercept is} \ (0, 3)\end{align} Let \begin{align}y = 0\end{align}. Solve for \begin{align}x\end{align}. \begin{align}& 2y-3x = 6\\ & 2({\color{red}0})-3x = 6 \qquad \text{Replace} \ y \ \text{with zero.}\\ & -3x = 6 \qquad \quad \ \ \text{Simplify}\\ & \frac{-3x}{{\color{red}-3}} = \frac{6}{{\color{red}-3}} \qquad \ \ \ \text{Solve for} \ y.\\ & \boxed{x = -2} \qquad \qquad \text{The} \ x \text{-intercept is} \ (-2, 0)\end{align} \begin{align}& 4y-6x = 12\\ & 4y-6 ({\color{red}0}) = 12 \qquad \text{Replace} \ x \ \text{with zero.}\\ & 4y = 12 \qquad \qquad \quad \text{Simplify}\\ & \frac{4y}{{\color{red}4}} = \frac{12}{{\color{red}4}} \qquad \qquad \ \ \text{Solve for} \ y.\\ & \boxed{y = 3} \qquad \qquad \quad \ \text{The} \ y \text{-intercept is} \ (0, 3)\end{align} Let \begin{align}y = 0\end{align}. Solve for \begin{align}x\end{align}. \begin{align}& 4y-6x = 12\\ & 4 ({\color{red}0})-6x = 12 \qquad \text{Replace} \ y \ \text{with zero.}\\ & -6x = 12 \qquad \quad \ \ \text{Simplify}\\ & \frac{-6x}{{\color{red}-6}} = \frac{12}{{\color{red}-6}} \qquad \quad \ \text{Solve for} \ y.\\ & \boxed{x = -2} \qquad \qquad \ \text{The} \ x \text{-intercept is} \ (-2, 0)\end{align} When the \begin{align}x\end{align} and \begin{align}y\end{align}-intercepts were calculated for each equation, they were the same for both lines. The graph resulted in the same line being graphed twice. The blue line is longer to show that the same line is graphed directly on top of the red line. The system does have solutions so it is also known as a consistent system. However, the system does not have one solution; it has an infinite number of solutions. This type of consistent system is called a dependent system. All the ordered pairs found on the line will satisfy both equations. If you look at the two given equations \begin{align}\begin{Bmatrix} 2y-3x = 6 \\ 4y-6x = 12 \end{Bmatrix}\end{align}, equation two is simply a multiple of equation. #### Example C Solve the following system of linear equations graphically: \begin{align}\begin{Bmatrix} 3x+4y=12 \\ 6x+8y=-8 \end{Bmatrix}\end{align} Solution: Both equations will be graphed on the same Cartesian plane using the slope-intercept method. Begin by writing each linear equation in slope-intercept form. \begin{align}& 3x+4y = 12\\ & 3x {\color{red}-3x}+4y = 12 {\color{red}-3x}\\ & 4y = 12 {\color{red}-3x}\\ & \frac{4y}{{\color{red}4}} = \frac{12}{{\color{red}4}}-\frac{3x}{{\color{red}4}}\\ & y = 3-\frac{3}{4}x\\ & \boxed{y = -\frac{3}{4}x+3}\end{align} \begin{align}& 6x+8y = -8\\ & 6x {\color{red}-6x}+8y = -8 {\color{red}-6x}\\ & 8y = -8 {\color{red}-6x}\\ & \frac{8y}{{\color{red}8}} = \frac{-8}{{\color{red}8}}-\frac{6x}{{\color{red}8}}\\ & y = -1-\frac{6}{{\color{red}8}}x\\ & \boxed{y = -\frac{6}{8}x-1}\end{align} The lines do not intersect. This means that the system of equations has no solution. The lines are parallel and will never intersect. If you look at the equations that were written in slope-intercept form \begin{align}y=-\frac{3}{4}x+3\end{align} and \begin{align}y=-\frac{6}{8}x-1\end{align}, the slopes are the same \begin{align}\left(-\frac{6}{8}=-\frac{3}{4}\right)\end{align}. A system of linear equations that has no solution is called an inconsistent system. #### Example D Before graphing calculators, graphing was not considered the best way to determine the solution for a system of linear equations, especially if the solutions were not integers. However, technology has changed this outlook. In this example, a graphing calculator will be used to determine the solution for \begin{align}\begin{Bmatrix} x+4y=-14 \\ 2x-y=4 \end{Bmatrix}\end{align}. Solution: To use a graphing calculator, the equations must be written in slope-intercept form: \begin{align}& x+4y = -14\\ & x {\color{red}-x}+4y = {\color{red}-x}-14\\ & 4y = {\color{red}-x}-14\\ & \frac{4y}{{\color{red}4}} = -\frac{x}{{\color{red}4}}-\frac{14}{{\color{red}4}}\\ & y = -\frac{1}{4}x-\frac{14}{4}\\ & \boxed{y = -\frac{1}{4}x-\frac{7}{2}}\end{align} \begin{align}& 2x-y = 4\\ & 2x {\color{red}-2x}-y = {\color{red}-2x}+4\\ & -y = -2x+4\\ & \frac{-y}{{\color{red}-1}} = \frac{-2x}{{\color{red}-1}}+\frac{4}{{\color{red}-1}}\\ & \boxed{y = 2x-4}\end{align} The equations are both in slope-intercept form. Set the window on the calculator as shown below: The intersection point of the linear equations is (0.22, -3.56). The following represents the keys that were pressed on the calculator to obtain the above results: #### Concept Problem Revisited The following linear equations will be graphed by using the slope-intercept method. \begin{align}& \begin{Bmatrix} 2x+y = 5 \\ x-y = 1 \end{Bmatrix}\\ & 2x+y = 5\\ & 2x {\color{red}-2x}+y = 5 {\color{red}-2x}\\ & y = 5 {\color{red}-2x}\\ & \boxed{y = -2x+5}\\ & x-y = 1\\ & x {\color{red}-x}-y = 1 {\color{red}-x}\\ & -y = 1-x\\ & \frac{-y}{{\color{red}-1}} = \frac{1}{{\color{red}-1}}-\frac{x}{{\color{red}-1}}\\ & y = -1+x\\ & \boxed{y = x-1}\end{align} The two lines intersect at one point. The coordinates of the point of intersection are (2, 1). ### Vocabulary Consistent System of Linear Equations A consistent system of linear equations is a system of linear equations that has a solution. The solution may be one solution or an infinite number of solutions. Dependent System of Linear Equations An dependent system of linear equations is a system of linear equations that has an infinite number of solutions. The equations are multiples and the graphs of each equation are the same. Therefore, the infinite number of ordered pairs satisfies both equations. Equivalent Systems of Linear Equations Equivalent systems of linear equations are systems of linear equations that have the same solution set. Inconsistent System of Linear Equations An inconsistent system of linear equations is a system of linear equations that has no solution. The graphs of an inconsistent system of linear equations are parallel lines. The lines never intersect so there is no common point of intersection. Independent System of Linear Equations An independent system of linear equations is a system of linear equations that has one solution. There is only one ordered pair that satisfies both equations. System of Linear Equations A system of linear equations is two linear equations each having two variables. This type of system – two equations with two unknowns-is called a \begin{align}2 \times 2\end{align} system of linear equations. ### Guided Practice 1. Solve the following system of linear equations by graphing: \begin{align}\begin{Bmatrix} -3x+4y=20 \\ x-2y=-8 \end{Bmatrix}\end{align} Is the system consistent and dependent, consistent and independent, or inconsistent? Use technology to determine whether the system is consistent and independent, consistent and dependent, or inconsistent. 2. \begin{align}\begin{Bmatrix} 3x-2y=8 \\ 6x-4y=20 \end{Bmatrix}\end{align} 3. \begin{align}\begin{Bmatrix} x+3y=4 \\ 5x-y=4 \end{Bmatrix}\end{align} 1. \begin{align}\begin{Bmatrix} -3x+4y=20 \\ x-2y=-8 \end{Bmatrix}\end{align} Begin by writing the equations in slope-intercept form. \begin{align}& -3x+4y = 20\\ & -3x {\color{red}+3x}+4y = 20 {\color{red}+3x}\\ & 4y = 20+3x\\ & \frac{4y}{{\color{red}4}} = \frac{20}{{\color{red}4}}+\frac{3x}{{\color{red}4}}\\ & y = {\color{red}5}+\frac{3}{4}x\\ & \boxed{y = \frac{3}{4}x+5}\end{align} \begin{align}& x-2y = -8\\ & x {\color{red}-x}-2y = -8 {\color{red}-x}\\ & -2y = -8-x\\ & \frac{-2y}{{\color{red}-2}} = \frac{-8}{{\color{red}-2}}-\frac{x}{{\color{red}-2}}\\ & y = {\color{red}4}+\frac{1}{2}x\\ & \boxed{y = \frac{1}{2}x+4}\end{align} The lines intersect at the point (-4, 2). This ordered pair is the one solution for the system of linear equations. The system is consistent and independent. 2. \begin{align}\begin{Bmatrix} 3x-2y=8 \\ 6x-4y=20 \end{Bmatrix}\end{align} \begin{align}& 3x-2y = 8 && 6x-4y=20\\ & 3x-3x-2y = -3x+8 && 6x-6x-4y=-6x+20\\ & -2y=-3x+8 && -4y=-6x+20\\ & \frac{-2y}{-2} = \frac{-3x}{-2}+\frac{8}{-2} && \frac{-4y}{-4}=\frac{-6x}{-4}+\frac{20}{-4}\\ & \boxed{y = \frac{3}{2}x-4} \quad \text{Slope-intercept form} && \boxed{y = \frac{6}{4}x-5}\end{align} The lines are parallel. The lines will never intersect so there is no solution. The system is inconsistent. 3. \begin{align}& \begin{Bmatrix} x+3y=4 \\ 5x-y=4 \end{Bmatrix}\end{align} \begin{align}& x+3y = 4 && 5x-y=4\\ & x-x+3y = -x+4 && 5x-5x-y=-5x+4\\ & 3y=-x+4 && -y=-5x+4\\ & \frac{3y}{3} = \frac{-x}{3}+\frac{4}{3} && \frac{-y}{-1}=\frac{-5x}{-1}+\frac{4}{-1}\\ & \boxed{y = \frac{-1}{3}x+\frac{4}{3}} && \boxed{y=5x-4}\end{align} There is one point of intersection (1, 1). The system is consistent and independent. ### Practice Without graphing, determine whether the system is consistent and independent, consistent and dependent, or inconsistent. \begin{align}\begin{Bmatrix} 2x+3y=6 \\ 6x+9y=18 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 2x-y=-14 \\ 12x-6y=-11 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 3x+2y=14 \\ 5x-y=6 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 2x+3y=-12 \\ 3x-y=3 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 20x+15y=-30 \\ 4x+3y=18 \end{Bmatrix}\end{align} Solve the following systems of linear equations by graphing. \begin{align}\begin{Bmatrix} x+2y=8 \\ 3x+6y=24 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 4x+2y=-2 \\ 2x-3y=9 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 3x+5y=11 \\ 4x-2y=-20 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 2x+y=5 \\ 6x=15-3y \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 2x-y=2 \\ 4x-3y=-2 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 2x-3y=15 \\ 4x+y=2 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 2x+3y=-6 \\ 9y+6x+18=0 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} 6x+12y=-24 \\ 5x+10y=30 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} x-3y=7 \\ 2x+5y=-19 \end{Bmatrix}\end{align} \begin{align}\begin{Bmatrix} x+3y=9 \\ x-y=-3 \end{Bmatrix}\end{align} ### Notes/Highlights Having trouble? Report an issue. Color Highlighted Text Notes Show Hide Details Description Difficulty Level: Authors: Tags: Subjects: Date Created: Dec 19, 2012<\|endoftext\|>	4.9375
3,231	# corresponding angles are congruent ###### Hello world! noiembrie 26, 2016 We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. To learn more, visit our Earning Credit Page. In the simple case below, the two triangles PQR and LMN are congruent because every corresponding side has the same length, and every corresponding angle has the same measure. But if we substitute this answer back into the expressions for our angle measurements, we find that the angle measurements are: (4(100) - 50) / 10 = 35 and (3/10)(10) + 5 = 35. If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as: {\displaystyle \triangle \mathrm {ABC} \cong \triangle \mathrm {DEF}.} A transversal line is a line that crosses or passes through two other lines. Similarly, ∠2 and ∠6, ∠3 and 7, and ∠4 and ∠8 are also corresponding angles. TRIANGLE CONGRUENCE 2 Triangles are congruent if their vertices can be paired such that corresponding sides are congruent and corresponding angles are congruent. Notice that the lines are parallel. ∠3 ≅ ∠60° since ∠3 and ∠7 are corresponding angles, and m and n are parallel. triangle is congruent to: triangle (See Solving AAS Triangles to find out more) If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. Angles are congruent, m<2 = 101, angles that form a linear pair are supplementary 7. m<1 = 76 Alt int. In the diagram below transversal l intersects lines m and n. ∠1 and ∠5 are a pair of corresponding angles. Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. This one point where two rays meet is called a vertex. Likewise, side is across from and side is across from, so and corresponds to each other. Corresponding angles form are supplementary angles if the transversal perpendicularly intersects two parallel lines. These shapes must either be similar or congruent . How do you know? One angle is 3(3) + 7 = 9 + 7 = 16 degrees. lessons in math, English, science, history, and more. Since two angles of ABC are congruent to two angles of PQR, the third pair of angles must also be congruent, so ∠C≅∠R, and ABC≅ PQR by ASA. As a member, you'll also get unlimited access to over 83,000 These arrows indicate that lines m and n are parallel. ∠8 ≅ ∠120° since ∠4 and ∠8 are corresponding angles, and m and n are parallel. Angles: ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R. They are both above the parallel lines and to the right of the transversal. imaginable degree, area of 85 degrees is not equal to 87 degrees, so the lines cannot be parallel, by the Corresponding Angle Theorem. Find the measure of the missing angles in the following diagram. Imagine a transversal cutting across two lines. We had earlier said axiomatically, with no proof, that if two lines are parallel, the corresponding angles created by a transversal line are congruent. Earn Transferable Credit & Get your Degree, Alternate Exterior Angles: Definition & Theorem, Alternate Interior Angles: Definition, Theorem & Examples, Same-Side Interior Angles: Definition & Theorem, Consecutive Interior Angles: Definition & Theorem, Same-Side Exterior Angles: Definition & Theorem, The Perpendicular Transversal Theorem & Its Converse, Transversal in Geometry: Definition & Angles, Median of a Trapezoid: Definition & Theorem, The Parallel Postulate: Definition & Examples, Betweenness of Points: Definition & Problems, Supplementary Angle: Definition & Theorem, Complementary Angles: Definition, Theorem & Examples, Linear Pair: Definition, Theorem & Example, Vertical Angles in Geometry: Definition & Examples, Interior and Exterior Angles of Triangles: Definition & Examples, Perpendicular Bisector Theorem: Proof and Example, Constructing Triangles: Types of Geometric Construction, College Preparatory Mathematics: Help and Review, High School Precalculus: Tutoring Solution, McDougal Littell Algebra 2: Online Textbook Help, Holt McDougal Larson Geometry: Online Textbook Help, MTTC Mathematics (Secondary) (022): Practice & Study Guide, Ohio Graduation Test: Study Guide & Practice, Alberta Education Diploma - Mathematics 30-1: Exam Prep & Study Guide, Alberta Education Diploma - Mathematics 30-2: Exam Prep & Study Guide, National Entrance Screening Test (NEST): Exam Prep, Common Core Math - Geometry: High School Standards, Common Core Math - Functions: High School Standards. (Click on "Corresponding Angles" to have them highlighted for you.) However, if the image is drawn to scale, the angles in question should be greater than 90 degrees. Two lines are parallel if and only if the two angles of any pair of corresponding angles of any transversal are congruent (equal in measure). Angles are congruent, <2 alt int. Explain. Assume the lines are parallel. d. Similar Similar polygons have corresponding angles congruent, but we don't know if the sides are congruent, or we know that they are not congruent. If the transversal intersects non-parallel lines, the corresponding angles formed are not congruent and are not related in any way. The angles that are formed in the same position, in terms of the transversal, are corresponding angles. Corresponding angles are pairs of angles that lie on the same side of the transversal in matching corners. In this lesson, you will learn how to identify corresponding angles. So the lines are parallel if and only if the angles shown are 35 degrees. Two triangles are congruent if both their corresponding sides and angles are equal. Log in or sign up to add this lesson to a Custom Course. Angles are congruent 5. m<1 = 50 angles that form a linear pair are supplementary, m<2 = 130, corresponding angles are congruent 6. m<1 = 79, alt ext. She has recently earned a Master's degree. \| {{course.flashcardSetCount}} c. Symmetric When A polygon is symmetric, half of it is a mirror image of the other half. Subtracting 3x from both sides of the equation gives us 7 = 4x - 5. Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure. But this is not the answer to the question; we need to find the measure of the angles. Anyone can earn The angle is formed by the distance between the two rays. Get access risk-free for 30 days, After you've finished, you should be able to: To unlock this lesson you must be a Study.com Member. Corresponding angles are formed when a transversal passes through two lines. Congruent triangles. Let’s use congruent triangles first because it requires less additional lines. Find the magnitude of a corresponding angle. If the two lines are parallel then the corresponding angles are congruent. This means that the corresponding sides are equal and the corresponding angles are equal. A line that passes through two distinct points on two lines in the same plane is called a transversal. If two figures are similar, their corresponding angles are congruent (the same). Subtracting 3x from both sides gives x - 50 = 50. In completing the examples, students will solidify their knowledge of corresponding angles and understand that corresponding angles are equal if and only if the lines intersected by the transversal are parallel. Thus, 6y-14 = 4y + 6 6y – 4y = 6 + 14 2y = 20 y = 10 … Sometimes the two other lines are parallel, and the transversal passes through both lines at the same angle. Please update your bookmarks accordingly. Did you know… We have over 220 college The full form of CPCT is Corresponding parts of Congruent triangles. Corresponding Angles Theorem The Corresponding Angles Theorem states: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Parallel lines are two lines on a two-dimensional plane that never meet or cross. Angle corresponds to angle, so they are congruent. ∠3 and ∠4 form a straight angle, so∠4=120°. Adding 50 to both sides yields x = 100. If you're given information about two triangles and asked to prove parts of the triangles are congruent, see if you can show the two triangles are congruent. 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811	Bones are hard, rigid structures that form the human skeletal system. They are dynamic organs that undergo constant changes in response to environmental stimuli. They can fuse together (human babies are born with around 300 bones while there are only about 206 bones in an adult human), increase or decrease in size, thin down or thicken, or strengthen further as needed. When broken, such as in the case of injury, the bones can regenerate without leaving scar tissue. The bones important functions including the following: - Provide support, structure, and shape to the human body - Protect vital, delicate organs - Produce white and red blood cells as well as other marrow-derived cells produced by stem cells, which lie in the innermost layer of the bone (marrow) - Facilitate respiration - Store fat and minerals that are made available as the body needs them, thereby playing an important role in homeostasis Allows motor and locomotive function, in coordination with muscles and joints that connect bones together Bones are made of four types of cells: osteocytes, osteoclasts, osteoblasts, and lining cells. Osteoblasts are the ones that make new bones as the body grows. Osteoblasts, when come together, create a flexible material called osteoid, which when fused with minerals, become hard and strong. Osteoblasts are also the ones responsible for rebuilding existing bones in case of fracture or injury. Osteocytes, which are commonly found on compact bones that support the body and muscles, are star-shaped cells that exchange minerals with other cells in the area. Lining cells – These cover the outside surface of the bones which main function is to control the movements of molecules found inside and outside of the bone. Osteoclasts – These are cells that break down to reabsorb existing bone. They also work together with osteoblasts to reshape bones in cases of bone problems such as fracture. Common Bone Problems/Conditions Osteoporosis – A condition common in older adults characterized by loss of bone mass and weakening of the bone structure. This leads to the loss of bone density making bones more susceptible to breakage and fractures. Paget’s disease – A condition wherein bones become abnormally enlarged and thickened yet brittle and fragile. This is caused by a disorder that prevents bone cells from rebuilding and re-molding bone tissue. Rickets/Osteomalacia – A bone disease also characterized by brittle and weak bones and is linked to vitamin D deficiency Acromegaly – A bone disease caused by the overproduction of growth hormone in the body, leading to overgrown bone structures in the face and extremities Bone cancer – A cancer that affects the bones either directly or through metastasis from cancer in another part of the body Osteomyelitis – A bacterial infection leading to the inflammation of the bone or bone marrow Common Bone Procedures and Surgeries Bone fracture repair – Minor bone fractures are usually treated with splinting or casting. However, major bone fractures may require a surgical procedure called open reduction and internal fixation (ORIF) surgery that involves the use of metal screws, rods, pins or plates to hold the affected bone in place. Bone grafting - This refers to a range of surgical methods performed to stimulate the formation of new bones. This minimally invasive procedure involves adding tissues to specific areas to induce bone-forming cells. Surgical cleansing – Bone infections and osteomyelitis are often treated surgically wherein bones are accessed through the skin and cleansed with antibiotics. Bone rebuilding – Bones that are severely damaged or infected are rebuilt through a variety of ways, including bone grafting, bone transport or the use of an external fixator. Bone biopsy – Usually done to confirm bone cancer diagnosis or to investigate abnormal growths, a bone biopsy is a procedure where a small sample of a bone is removed and examined.<\|endoftext\|>	4.40625
359	Permanent magnet step motors: The principle of operation of permanent magnet motor illustrated below: 1) In this type of motor rotor is magnetized to produce a permanent magnet poles and stator contains two phase windings that should be excited by either polarity currents viz 3) The magnetic poles produced by stator current cause rotor to move as shown below: 4) Due to force of attraction for the excitation shown in above fig: 5) The step angle in the phses would be of 45 degrees from the movement point P 6) The direction of the rotation will be decided by the sequence in which motor will be excited: 1) Motor torque is generated due to the interaction between the magnetic fields produced by the stator and rotor. 2) The step angle of permamant magnet is large and it is difficult to manufacture motors with smaller step angles. 3) The inertia of this type of motor is very high as compare to the variable reluctance type. 4) This type of motor needs the bipolar excitation viz current which is flowing through the winding needs the reverse direction or polarity for an Ex A , B which we can't see in variable reluctance types of motor. 5) Variable reluctance motor require unipolar voltage drive alongwith the less number of power devices per phase but in case of permanent magnet the power devices will be more as compared to variable reluctance. Advantages of permanent magnet motor: 1) Due to inbuilt permanent magnet these are having lower power requirements. 2) Due to permanent magnet these are having higher detent torque. 1) Due to use of permanent magnet the motor gives lower torque per unit volume 2) Step size is negatively large 3) Bipolar excitation is used<\|endoftext\|>	3.90625
408	Microsoft Office Tutorials and References In Depth Information Referring to Cells in Other Worksheets: Using Them in Formulas Referring to Cells in Other Worksheets: Using Them in Worksheets are, as the name suggests, analogous to pages or sheets in a topic. But unlike the hard-copy variety, the information contained on these sheets can be referred to and actively used by other sheets in formulas. For example, you may want to add the salaries of two or more employees in your company, and that information may be stored in different sheets. How then would you go about adding them in one formula? The problem is, as already indicated, every sheet has the same bundle of 16 billion addresses; so if each salary is placed in cell its respective cell A3, how is each distinguished from the other and utilized in the same formula besides? This short exercise will show you how: On a blank spreadsheet, type 23000 in cell A3 on Sheet1, and type 32500 in A3 on Sheet2 (don’t worry about formatting here). Now, we want to add these two values—let’s say in a formula in D5 on Sheet1. Click in D5 and type , = and then, as usual, click cell A3 and type , as shown in Figure 9–18. So + far, we haven’t done anything new. Figure 9–18. Starting to write the formula on Sheet1. First we use cell A3 from Sheet1. Then click the Sheet2 tab and click cell A3 on that sheet, as shown in Figure 9–19. Now Sheet2. Note the formula in the 4. Then press Enter, or click the check mark by the formula bar. The result, 55500, appears in D5 of Sheet1—where you wanted it to go. The<\|endoftext\|>	3.90625
966	(krō'mӘsfēr´´) [Gr.,=color sphere], layer of rarefied, transparent gases in the solar atmosphere; it measures 6,000 mi (9,700 km) in thickness and lies between the photosphere (the sun's visible surface) and the corona (its outer atmosphere). The flash spectrum has been a valuable tool in the study of the chromosphere. This spectrum is obtained before a solar eclipse reaches totality and is formed from the thin arc of the sun disappearing behind the moon's disk. An analysis of the emission lines gives information about the heights of the chromosphere and the heights at which various elements exist in it. Using the flash spectrum, scientists have found that the chromosphere is composed primarily of hydrogen, which causes its visible pinkish tint, and of sodium, magnesium, helium, calcium, and iron in lesser amounts. The chromosphere consists of three distinct layers that, moving outward from the sun's surface, decrease in density and increase abruptly in temperature. The lower chromosphere is about 10,800 degrees Fahrenheit (6,000 degrees Celsius), the middle rises to 90,000 degrees Fahrenheit (50,000 degrees Celsius), and the upper part, merging into the lower corona, reaches 1,800,000 degrees Fahrenheit (1,000,000 degrees Celsius). At 600 mi (1,000 km) above the photosphere, the chromosphere separates into cool, high-density columns, called spicules, and hot, low-density material. The spicules, each about 500 mi (800 km) in diameter, shoot out at 20 mi per sec (32 km per sec) and rise as high as 10,000 mi (16,000 km) before falling back. Any point on the sun will erupt a spicule about once every 24 hr and there may be up to 250,000 of them at any instant. Other types of solar activity are found to occur in the chromosphere. The elements of each layer are sometimes distributed in bright, cloudlike patches called plages, or flocculi, and in general are located along the same zones as sunspots and fluctuate with the same 11-yr cycle; the relationship between the two is not yet understood. Most spectacular of the solar features are the streams of hot gas, called prominences, that shoot out thousands or even hundreds of thousands of miles from the sun's surface at velocities as great as 250 mi per sec (400 km per sec). Two major classifications are the quiescent and the eruptive prominences. Quiescent prominences bulge out from the surface about 20,000 mi (32,000 km) and can last days or weeks. Eruptive prominences are thin flames of gas often reaching heights of 250,000 mi (400,000 km); they occur most frequently in the zones containing sunspots. Dark strandlike objects called filaments were discovered on the disk and were originally thought to be a special kind of feature. These are now known to be prominences seen against the bright background of the photosphere. Until the middle of the 19th cent. prominences could be viewed extending from the edge of the sun's disk only during a solar eclipse. However, in 1868 a method of observing them with a spectroscope at any clear time of day was developed, and in 1930 the invention of the coronagraph allowed them to be continuously photographed. Another phenomenon occurring in the chromosphere is the solar flare, a sudden and intense brightening in a plage that rises to great brilliance in a few minutes, then fades dramatically in a half hour to several hours. This feature is also associated with sunspots and is thought to be triggered by the sudden collapse of the magnetic field in the plage. A flare releases the energy equivalent of a billion hydrogen bombs and is the most energetic of solar events. The ultraviolet and X-ray radiation from larger flares can disrupt magnetic compasses and navigation and radio signals as well as affect the electrical grid on the earth and can damage satellites and space probes. Cosmic rays and solar wind particles from some flares interact in the polar regions, creating brilliant auroral displays (see aurora). Layer or region of the solar atmosphere lying above the PHOTOSPHERE and beneath the CORONA . The name chromosphere comes from the Latin... The layer of a star's atmosphere between the photosphere and the corona . It is much less dense than the photosphere. The sun's... Sudden intense brightening of a small part of the Sun’s surface, often near a sunspot group. Flares develop in a few minutes and may last several h<\|endoftext\|>	4
1,094	# KIM MALTMAN'S MINI-COURSE ON TRIG BASICS ## CHAPTER 4, SECTION A: HOW DO THE TRIG FUNCTIONS CHANGE IN BETWEEN THE 16 SPECIAL DIRECTIONS FOR WHICH WE KNOW THEIR EXACT VALUES? OVERVIEW: This section of Chapter 4 deals with the general question of how the trig functions sin(θ), cos(θ), tan(θ), sec(θ), csc(θ) and cot(θ) change as the angle θ is changed. This information will allow you to easily figure out how to generate (reproduce) the graphs of the six trig functions as functions of θ which most of you will have seen in your high school textbooks. Past experience suggests that a significant number of you, even if you have been shown these graphs, will not have had explained to you where it is that they come from. After working your way through this section of the course, you will be able to easily generate such graphs for yourself, and will no longer have to try to memorize them. The key pieces of information which will be used in figuring out how the trig functions change as θ is changed are: • the unit circle picture for the sine and cosine functions (in its general form, where it applies to ALL directions in the plane); and • the algebraic relations which express the tangent, cotangent, secant and cosecant as ratios involving the sine and cosine, and which define the generalizations of these functions to non-first-quadrant directions. We will find out, in the end, that • Each of the trig functions behaves very simply in each of the four quadrants, either steadily increasing or steadily decreasing as one increases θ through the quadrant in question. • Whether a particular trig function increases or decreases as one increases θ through a particular quadrant depends on which function it is we are considering, and which quadrant it is we are focussing on. • It is easy to use the basic geometry provided by the unit circle picture to find out whether the sine or cosine increases or decreases in a given quadrant. • The information about how the sine and cosine behave (increasing or decreasing) in the quadrant of interest, together with the information about how the tangent, cotangent, secant and cosecant are expressed in terms of the sine and/or cosine, make it easy to find out how the tangent, cotangent, secant or cosecant behave (increasing or decreasing) in the same quadrant. Note that we already know the values of all six trig functions for three angles in each quadrant (as well as along the four coordinate directions), so, once we know that each of the functions either steadily increases or steadily decreases as we move through each of the quadrants, we can also use our knowledge of the exact values in the quadrant of interest to help remind ourselves of whether the function of interest is decreasing or increasing in that quadrant. It is, however, a good idea to go through the discussion of the underlying geometry once, as done in this section of the course, so you acquire a good intuitive understanding of the geometrical origin of this behavior of the trig functions, and can reproduce the results for yourself in future if you need to. A list of the background material needed to follow the contents of this section is given below. If you are unfamiliar with one of the background topics, or feel you would like to review it, click on the relevant link. If you are already familar with all of the background material, you can proceed directly to the contents of the current section of the course by clicking HERE. BACKGROUND MATERIAL (AND LINKS): For this section of the course, you should be familiar with • the idea and geometrical meaning of angles and angular measure (reviewable by reading (or re-reading) the various sections of Chapter 1 of the course, which can be accessed by clicking HERE); • the algebraic relations amongst the various trig functions of the same angle, especially those which allow the tangent, cotangent, secant and cosecant to be written in terms of the sine and cosine (reviewable by reading (or re-reading) Section (a) of Chapter 3 of the course); • the unit circle picture for the sine and cosine functions (reviewable by reading (or re-reading) Section (b) of Chapter 3 of the course); • the generalization of the trig functions to angles corresponding to non-first-quadrant directions using the unit circle picture for the sine and cosine functions and the algebraic expressions for the tangent, cotangent, secant and cosecant functions as ratios involving the sine and cosine (reviewable by reading (or re-reading) Section (c) of Chapter 3 of the course) Knowledge (or a brief review) of the geometrical constructions and special triangles which allow you to reconstitute the values of all six trig functions for the sixteen special directions around the plane for which these values are known exactly will also be helpful. This material can be reviewed by reading (or re-reading) Section (d) of Chapter 2, Section (e) of Chapter 3 and Section (f) of Chapter 3 of the course. CLICK HERE TO PROCEED TO THE CURRENT SECTION OF THE COURSE CLICK HERE FOR TO SEE THE ANSWERS TO THE EXERCISES FOR ALL SECTIONS OF CHAPTER 4 INCLUDING THIS SECTION<\|endoftext\|>	4.4375
1,421	# What is 100% as a ratio? Among every 100 community college students, 57 are female. Similarly, 25% means a ratio of 25100,3% 25 100 , 3% means a ratio of 3100 and 100% means a ratio of 100100 . In words, “one hundred percent” means the total 100% is 100100 , and since 100100=1 100 100 = 1 , we see that 100% means 1 whole. ## How do you find 75 percent of a number? To find out 75% of any number,you just multiply that number by 3/4. Or first you can divde that number by 4 and mutiply your result by 3. You get 75% of any no. 100 halved=50. ## What is 125 Simplified? If we divide both the top and the bottom by 5 since we know 125 is a multiple of 5, we get 25/200. Divide both by 5 again to get 5/40. Divide by 5 one more time to get 1/8, and since the numerator is 1, the fraction cannot be simplified any further. Therefore, the answer in its simplest form is 1/8. See also Where is wanted drained captain? ## What is the ratio of 75 100? 75/100 expressed in the simplest fraction is 3/4. If a fraction is not in ‘lowest terms,’ it’s possible to divide the numerator and denominator by a… ## How do I calculate a ratio? If you are comparing one data point (A) to another data point (B), your formula would be A/B. This means you are dividing information A by information B. For example, if A is five and B is 10, your ratio will be 5/10. Solve the equation. ## Is a 75 A good grade in high school? 75 is a fine grade. If that’s your full potential, that is a great grade and you should be very proud. ## Is a 75 passing? This is an above-average score, between 80% and 89% C – this is a grade that rests right in the middle. C is anywhere between 70% and 79% D – this is still a passing grade, and it’s between 59% and 69% ## How do you find the value of 100? If you are given a percentage (other than 100%) then just divide the amount by the percentage (so that you find “1% of it”) and then multiply by 100 (so that you find 100%). For example, if it is said 15 equals 20%, then divide 15 by 20 to get 1% (which equals 3/4 or 0.75), then multiply it by 100 which equals 75. ## How do you find 100 percent of a number? 100 percent is 100/100, which is equal to 1. To find the percentage of a number, you multiply the percentage to the number, so for example if we use 50, and we want 100 percent of that, we’d do 50 * 100/100 or 50 * 1 which is just 50. ## What is the percent of 1 out of 100? What is this? Now we can see that our fraction is 1/100, which means that 1/100 as a percentage is 1%. ## What is the simplest form of 57 100? As you can see, 57/100 cannot be simplified any further, so the result is the same as we started with. ## What is 75 squared simplified? Simplifying square roots So 75 = 5 3 sqrt{75}=5sqrt{3} 75 =53 square root of, 75, end square root, equals, 5, square root of, 3, end square root. ## What is 0.5 as a simplified fraction? Answer: 0.5 as a fraction is written as 1/2 Let us see how to convert a decimal number into a fraction. ## What is 77 as a fraction? 2. What is 77% in the fraction form? 77% in the fraction form is 77/100. If you want you can simplify it further as 77/100. ## What is 95 100 as a decimal? As you can see, in one quick calculation, we’ve converted the fraction 95100 into it’s decimal expression, 0.95. ## How do you simplify a ratio? To simplify a ratio, divide each number in the ratio by the same amount. To fully simplify a ratio, divide each number in the ratio by their highest common factor. Always keep the numbers in the ratio as whole numbers. For example, the ratio 9:3 simplifies to 3:1 by dividing by 3. ## Is a 75 on a test good? This is an above-average score, between 80% and 89% C – this is a grade that rests right in the middle. C is anywhere between 70% and 79% D – this is still a passing grade, and it’s between 59% and 69% See also What is Connecticut known for? ## Is a 75 average good? 75% is a decent average in high school because it gives you an option of getting into both university (some programmes, not all, look at requirements for programmes) and college. ## How much is 75% off? For example, a TV set might originally set you back \$5000. Determine the percentage discount – in our example store, everything is 75% off. The sum that stays in your pocket – your savings – is simply these two values multiplied by each other: 75% * \$5000 = 0.75 * \$5000 = \$3750 . ## How do you calculate 75 percent? To find 75% of a number, quarter it then multiply by 3. 75% is the same as three-fourths or three-quarters. To find 10% of a number, divide it by 10! 10% is the same as one-tenth. ## What is 10% of an amount? 10 percent means one tenth. To calculate 10 percent of a number, simply divide it by 10 or move the decimal point one place to the left. For example, 10 percent of 230 is 230 divided by 10, or 23. ## What is a 66 out of 75? What is this? Now we can see that our fraction is 88/100, which means that 66/75 as a percentage is 88%.<\|endoftext\|>	4.53125
1,232	Design of Square Footing: Footing or foundation is defined as the part of substructure, which transmits the loads from the super-structure to surrounding soil stratum safely. Foundation are classified as two types, 1. Shallow foundation 2. Deep foundation The depth of the foundation is less than or equal to the width of the foundation then the foundation is said to be shallow foundation. If the depth of the foundation is greater than width of the foundation then the foundation is said to be Deep foundation. Design of footing mainly depends on the safe bearing capacity of the soil on which the footing rests and the load coming from the superstructure. Footings may be isolated, combined. Isolated or independent footings are the footings that support the individual columns. They distribute and spread the load over a sufficiently large of the soil stratum to minimize the bearing pressure. Isolated footings may be square, rectangular or circular. In general, it is assumed that the soil behaves elastically that is the strain in the soil is proportional to applied stress and strain distribution in the soil immediately under the base of the footing is linear. Stress distribution is different soils. For analysis purpose, a footing can be compared with a rigid body in equilibrium subjected to loads. Like other structural members, a footing is designed to resist shear forces and bending moments. In design, for any soil the pressure distribution is assumed to uniform. In design, the critical section for one way shear (beam shear) is at a distance equal to the effective depth, d from the face of column footing. The critical section for two way shear or slab type shear shall be at a distance d/2 from the periphery of column, perpendicular to the plane of the slab. The critical section for bending moment is at the face of the column. Generally the footing is sensitive to punching shear. IS-CODE PROVISIONS FOR DESIGN OF FOOTINGS: 1. Footings shall be designed to sustain the applied loads, moment and forces. And safe bearing capacity is not exceeded. 2. In R.C.C. footing, the thickness at the edge shall not be less than 15cm for footing on soil. 3. The greatest bending moment to be used in the design of an isolated concrete at the face of the column. The critical section for diagonal cracking is taken at a distance equal to the effective depth from the face of the column in hard soils and shall not exceed nominal shear stress. No-26 1) Type of footing = Square footing 2) Size of the column = 230 X 450 mm 3) Load on footings: Axial Load on footing (ETABS) (P) = 850 KN = 250kN/m2 (Assume) S.B.C of soil STEP – 1: Self-weight of footing = 10 % of axial load. = 850 X 0.10 = 85 KN Total load transmitted to the soil = axial load + self-weight = 850 + 85 = 935 KN = 250kN/m2 S.B.C of soil STEP – 2: Area of footing (A) = Total load/SBC of soil = 935 / 250 =3.74 ≅ 4.0 m2 Size of the square footing = √𝐵 = √4 = adopt = 2m x 2m ULTIMATE BEARING CAPACITY; qu = 𝑃𝑢 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑓𝑜𝑜𝑡𝑖𝑛𝑔 = 850 2𝑥2 = 220 N-M ≅ 0.22 𝐾𝑁 − 𝑀 BENDING MOMENT: M U = qu x B x ( B−b )2 = 0.22 x 2000 x 8 ( 2000−230)2 8 = 172.3 x 106 N-mm Calculating depth (d) ∴ MU = 0.138 fck bd2 172.3 x 106 = 0.138 x 25 x 2000 x d2 d = 158 mm ∴ So provide twice the depth (Take 350mm) Assume Cover as 50mm Overall depth = 350 + 50 = 400 mm d = depth – half dia of bar – effective cover d = 400 – 12/2 – 50 = 344mm (Provide 12mm bar) Area of Steel:- Ast = 0.5 x fck x fy 4.6MU [1 − √1 − fckbd2 ] x bd 4.6 x 172.3 x 106 25 = 0.5 x 415 x [1 − √1 − 25 x 2000 x3442 ] x 2000x344 = 1437.84 mm2 SPACING: S = ast x B 𝑎𝑠𝑡 = 113.09 x 2000 1437.84 = 157 mm ≅ 160 c/c Provide 12mm @ 160 c/c in both direction Check for one way shear: Critical section for one way shear is‘d’ from face of the column. ( B b) d 2 Shear Force Vu Pu xB (2000 230) 350 2 Shear Force Vu 0.22 x 2000 Shear Force Vu = 235400 N. Nominal Shear stress v= 235400 = 0.33 2000𝑥350 N/mm2 Percentage of steel:Pt = 𝜋 𝑥122 𝑥100 100 Ast =4 = 2.01 spacingxdepth 160𝑋350 From IS 456-2000 Table No-19 Shear stress in concrete = τc= 0.82 N/mm2 τv<τc Hence it is safe 1) Check for two-way shear: 2000 b+d 230 2000 2000 450 450 b+d b+d 230 2000 b+d<\|endoftext\|>	3.6875
655	Year 5 Year 5 # Rounding to estimate ## Switch to our new maths teaching resources Slide decks, worksheets, quizzes and lesson planning guidance designed for your classroom. ## Lesson details ### Key learning points 1. In this lesson, we will apply our understanding of rounding to the nearest multiples of 10 000 and 1000 to estimate the answer to addition equations. ### Licence This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated. ## Video Share with pupils ## Worksheet Share with pupils ## Starter quiz Share with pupils ### 5 Questions Q1. Which equation demonstrates how the number 346 000 can be partitioned? 30 000 + 4000 + 600 300 + 40 + 6 Correct answer: 300 000 + 40 000 + 6000 3000 + 400 + 60 Q2. 2. Use partitioning to solve the equation: 723 000 + 15 000 = 728 000 748 000 758 000 Q3. 3. Use partitioning to solve the equation: 456 000 - 132 000 = 234 000 235 000 325 000 Q4. 4. Use partitioning to solve the equation: 158 000 + 26 000 = 164 000 174 000 178 000 Q5. 5. Use partitioning to solve the equation: 738 000 - 156 000 = 482 000 682 000 782 000 ## Exit quiz Share with pupils ### 5 Questions Q1. What does 'rounding to estimate' mean? Adding and subtracting using the column method. Correct answer: Rounding numbers in an equation to provide an approximate answer before calculation. Using decimal numbers in addition equations. Using whole numbers in addition equations. Q2. Round '456 244' to the nearest multiple of 10 000. 400 000 450 000 500 000 Q3. 3. Using rounding to the nearest multiple of 10 000 to estimate the answer to: 341 782 + 456 913 = 300 000 + 500 000 = 800 000 340 000 + 450 000 = 790 000 Correct answer: 340 000 + 460 000 = 800 000 342 000 + 457 000 = 799 000 Q4. 4. Use rounding to the nearest multiple of 1000 to estimate the answer to: 187 221 + 243 891 = 187 000 + 243 000 = 430 000 Correct answer: 187 000 + 244 000 = 431 000 190 000 + 240 000 = 430 000 200 000 + 200 000 = 400 000 Q5. What is one of the problems associated with rounding to estimate? Rounding to estimate always takes much longer than actually calculating the answer.<\|endoftext\|>	4.71875
837	# Find λif the vectors ˆi − ˆj + ˆk, 3ˆi+ ˆj + 2ˆk and ˆi+λˆj+ˆ3kare coplanar Video Solution Text Solution Generated By DoubtnutGPT ## To find the value of λ such that the vectors ^i−^j+^k, 3^i+^j+2^k, and ^i+λ^j+3^k are coplanar, we need to use the condition that the scalar triple product of these vectors is zero.1. Define the Vectors: Let: a=^i−^j+^k=(1,−1,1) b=3^i+^j+2^k=(3,1,2) c=^i+λ^j+3^k=(1,λ,3)2. Set Up the Scalar Triple Product: The scalar triple product a⋅(b×c) must equal zero for the vectors to be coplanar. We can compute b×c using the determinant of a matrix formed by the vectors.3. Calculate the Cross Product b×c: b×c=∣∣∣∣∣^i^j^k3121λ3∣∣∣∣∣ Expanding this determinant: =^i∣∣∣12λ3∣∣∣−^j∣∣∣3213∣∣∣+^k∣∣∣311λ∣∣∣ Calculating the 2x2 determinants: - For ^i: 1⋅3−2⋅λ=3−2λ - For ^j: 3⋅3−2⋅1=9−2=7 - For ^k: 3⋅λ−1⋅1=3λ−1 Therefore, b×c=(3−2λ)^i−7^j+(3λ−1)^k4. Calculate the Dot Product a⋅(b×c): Now we compute: a⋅(b×c)=(1,−1,1)⋅((3−2λ),−7,(3λ−1)) This gives: =1(3−2λ)+(−1)(−7)+1(3λ−1) Simplifying: =3−2λ+7+3λ−1 =9+λ5. Set the Scalar Triple Product to Zero: For coplanarity, we set: 9+λ=0 Solving for λ: λ=−9Final Answer:λ=−9 \| Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS Aggarwal, Manohar Ray, Cengage books for boards and competitive exams. Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi medium and English medium for IIT JEE and NEET preparation<\|endoftext\|>	4.53125
644	Research conducted by Rebecca Waite, 2009 Rwanda is a population-dense, landlocked country lacking many natural resources.1 Overwhelmed by a high population growth rate of 2.8 percent, the small nation is particularly susceptible to environmental conflicts.2 Eighty-seven percent of Rwandans are farmers, who depend on biomass, particularly wood, for their energy needs.3 Many rely on slash-and-burn agriculture, and are forced to return to areas that should remain fallow longer for better nutrient retention due to the immense competition for land. This is an especially serious problem in areas where the family land is divided for the next generation of males — as families expand, the amount of space allotted to each son shrinks. When these small land plots are overused, the soil is rapidly depleted of essential nutrients. As a result, crops become difficult to grow, and malnutrition drives families to forested areas, where a newly burned plot can provide for the season. An emerging trend of changing grassland where herds graze into cropland has also decreased levels of nutrients in soil, since less manure is available to use as fertilizer.4 Much of the forest has also been cleared to build homes for returnees from the genocide. Other causes of deforestation include the extraction of products such as charcoal, medicine, food, and gold.5 Deforestation in Rwanda has caused increased levels of erosion, poor water retention in soils, and the loss of tremendous biodiversity. Moreover, the competition for land and accessible natural resources required for people's day-to-day lives causes social tension within families and the community at-large.6 Domestic violence and sexual exploitation are prevalent in parts of Rwanda, and competition for limited supplies of firewood can trigger such violence. Without the necessary fuel source, meals may not be prepared on time, provoking men to beat their wives. Additionally, children may be asked to wander deep into the receding woods for firewood, where they are easy targets for rape. The Be Ready project team has noted that domestic violence and sexual exploitation of young children due to the limited wood supply are prevalent in their community in southwest Rwanda. By training women and men about women's rights and domestic violence issues, constructing more fuel efficient stoves, and planting trees for firewood, they believe that men will show less violent tendencies, and children will be at decreased risk for sexual exploitation. The team hopes to reduce the number of domestic abuse victims and violated children, and facilitate peace for the community's 250 families. Global Grassroots' Conscious Social Change Programs mobilize women to advance grassroots solutions benefiting women. Projects like Be Ready demonstrate that addressing critical environmental issues like deforestation can also help mitigate several of the underlying factors of violence against women. 5http://www.rwandagateway.org/article.php3?id_article=92 and http://www1.american.edu/ted/ice/rwanda.htm 1950 Lafayette Road Suite 200, Box 1 Portsmouth, NH 03801 USA Tel (+1) 603.643.0400 © 2019 Global Grassroots 501(c)(3) Non-Profit<\|endoftext\|>	3.96875
548	# Positive and Negative Acceleration ## Description This is a simple physics tutorial aimed at high school students in grades 11 and 12 (IB, HSC) that runs through the concept of positive and negative acceleration and how it compares to 'acceleration and deceleration'. Video tutorial to come shortly. ## Introduction In previous material (and in everyday language), we've learned that: • Any increase in speed = acceleration • Any decrease in speed = deceleration Now we're going to move into more intricate definitions, and use the terms 'positive' and 'negative' acceleration. ### What is Positive Acceleration? Positive acceleration refers to any object that is experiencing a force acting on it in the same direction it is travelling in. In this scenario, the object increases speed, but continues to travel in the same direction. ### What is Negative Acceleration? Negative acceleration refers to any object that has a force acting on it in a direction that is opposite to what it is initially travelling in. In this scenario the object may: • travel at a greater speed, but in the opposite direction, OR • travel at a slower speed, in the opposite direction, OR • travel at a slower speed and in the same direction (known as deceleration), OR • have simply changed direction (to opposite direction) (but is travelling at the same speed). ## Determining positive and negative acceleration ### Step 1: Assign Signs We can describe directions either with words (e.g. up/north or down/south) or with symbols (e.g. using the minus or plus symbols). Conventionally, we attribute positive signs to the north and east direction; and negative signs to the south and west. ### Step 2: Perform Calculation Use this equation to calculate acceleration. ### Step 3: Determine if acceleration is positive or negative Instinctively you'd think that if the calculated acceleration is positive, it means that the object is undergoing positive acceleration, and vice versa. This is not the case. Instead, we need to compare the sign of the calculated acceleration with the sign of the initial velocity. What sign? I hear you ask. The sign that tells you what direction the initial velocity and the acceleration are in. Let's look at how we can define directions with signs. • When an object is undergoing positive acceleration, the acceleration sign will be the same sign as the initial velocity. • When an object is undergoing negative acceleration, the acceleration sign will be the opposite sign as the initial velocity. If you're still not sure why we can't assume that the calculated acceleration sign is the same as the type of acceleration the object is undergoing, check out this tutorial.<\|endoftext\|>	4.40625
435	The outer part of a tree’s trunk and branches is the bark. The term bark refers to all of the tissue outside the cambium, a layer of actively dividing cells that causes the tree to grow thicker. The cambium forms the wood as well as the inner layer of bark. This inner layer of bark consists of living tissue called phloem, which carries the food that is made in the leaves to all the other parts of the tree. Phloem generally remains functional for about a year. As the inner bark grows, the older phloem gets pushed outward and is eventually sloughed off. The outer layer of bark is dead tissue, consisting of older, nonfunctional phloem and dead cells called cork cells. Even though the outer layer of bark is dead, it is still very useful to the tree. It protects the tree from injury by people, insects, and other animals and helps to keep out germs and fungi, which can cause diseases. The outer bark also minimizes water loss from the trunk and branches. It protects the tree from the weather and can be very effective in resisting fire. Most tree species have bark that is unique in structure and appearance. In fact, many trees can be identified by the characteristics of their bark alone. The pattern of cork development is the main factor in determining how the bark looks. Some trees, such as pines and pear trees, have a scaly type of bark, while trees such as paper birch and cherry have smooth sheets of bark. Birch bark peels because it has alternating layers of thick- and thin-walled cork cells. Bark varies from the smooth, copper-colored covering of the gumbo-limbo tree to the thick, soft, spongy bark of the punk, or cajeput, tree. Other types of bark include the commercial cork of the cork oak and the rugged, fissured outer coat of many other oaks. Sycamores have flaking, patchy-colored barks, while the shagbark hickory has a rough shinglelike outer covering.<\|endoftext\|>	3.90625
395	The Origins of Hispanic Heritage Month Hispanic Heritage Month began as Hispanic Heritage Week, established by legislation sponsored by Rep. Edward R. Roybal (D-Los Angeles) and signed into law by President Lyndon Johnson in 1968. The commemorative week was expanded by a bill sponsored by Rep. Esteban E. Torres (D-Pico Rivera) and implemented by President Ronald Reagan in 1988 to cover a 30-day period (September 15 – October 15). It was enacted into law on August 17, 1988, on the approval of Public Law 100-402. September 15 was chosen as the starting point for the celebration because it is the anniversary of the independence of five Latin American countries: Costa Rica, El Salvador, Guatemala, Honduras, and Nicaragua. All declared independence in 1821. In addition, Mexico, Chile, and Belize celebrate their independence days on September 16, September 18, and September 21, respectively. Hispanic Heritage Month (HHM) also celebrates the long and significant presence of Hispanic and Latino Americans in North America. A map of late 18th-century North America shows this presence, from the small outpost of San Francisco founded in Alta California in 1776, through the Spanish province of Texas with its vaqueros (cowboys), to the fortress of St. Augustine, Florida — the first colonial settlement in North America, founded in 1513, ninety-four years before the English landed in Jamestown, Virginia. Hispanic Heritage Month at Volunteer State Community College Hispanic Heritage Month activities will commence in October with our program entitled Understanding the Faces of DACA on October 4th. There will also be a Festival on October 20th featuring Aztec Dances, Latin American Food Tasting, games for kids, and health check-ups. Moreover, throughout the year the Office of Diversity and Inclusion will sponsor and co-sponsor programming for the Hispanic and Latinx students. Please visit this site often for programming updates.<\|endoftext\|>	3.6875
1,415	Along with the sensational news out of South Africa this morning that researchers on the Rising Star Expedition have discovered a new hominin species, Homo naledi, comes a second article from the team on the geological context in which the fossil remains were found. Buried within the heavily scientific text is one very interesting claim: that Homo naledi deliberately disposed of the bodies of the dead. While the authors ably make the case that these 15 individuals didn't arrive in the cave by accident, they stop short of speculating on behavior, which is notoriously difficult to reconstruct from bones alone. But studies of our chimpanzee cousins and their reactions to death could hold answers to the question of whether Homo naledi engaged in burial of its dead. The study of what happens to bodies and bones after death is known as taphonomy, and it encompasses all post-mortem processes from decomposition to burial to recovery. Writing in the journal eLIFE today, Paul Dirks and colleagues detail the geological context of the Dinaledi cave in which Homo naledi was found and explain in depth the processes that helped form the deposits in which the fossils were found. Most notably, the researchers found that: there was no evidence the bones had been exposed on the surface of the ground prior to finding their way into the chamber of the cave; all of the fractures in the bones were post-mortem, meaning the bodies probably did not fall into the cave; there was no evidence that the bones were transported into the cave by water; and the arrangement of the bones suggested the individuals did not all die at once. Even more interesting is the fact that the researchers found no easy ancient access to the Dinaledi Chamber. Today, the chamber is difficult to reach, and the only Rising Star Expedition team members to enter it were small women scientists, who were able to squeeze through very narrow passages to recover the fossil remains. So what can account for the discovery of intact portions of skeletal remains deep within an ancient cave? Dirks and colleagues outline five major hypotheses, but quickly reject all but two. Homo naledi does not appear to have lived in the cave, so that explanation can be ruled out. The bones do not show evidence of water transportation, so it's unlikely a flooding event moved the bodies into the cave. Since there is no evidence of predator bones anywhere near the bodies, and since there is no evidence of other species within the cave, it's also not likely that a large carnivore put the remains of its meal there. One plausible explanation for the collection of the bones of one species in the cave is a mass fatality event or "death trap." Berger and colleagues reported on the age-at-death and sex of the H. naledi collection, which included: 3 infants, 3 young juveniles, 1 old juvenile, 1 subadult, 4 young adults, and 1 old adult. In normal, ancient populations, the assemblage of dead individuals is often skewed towards the very old and the very young, who are more at risk of death. But in catastrophic events, such as the eruption of Their preferred explanation, though, is deliberate disposal of the dead. In this hypothesis, they write, "bodies of the individuals found in the cave would either have been carried into, or dropped through an entrance similar to, if not the same as, the one presently used to enter the Dinaledi Chamber." Dropping the bodies into a soft cave surface may have happened, but the hominins might also have entered the chamber, "carrying the bodies or dying there, which would explain not only the absence of green fractures but the presence of delicate, articulated remains in the excavation pit deep in the chamber, well away from the entrance point." If Dirks and colleagues are right that the Dinaledi remains represent deliberate and repeated body disposal, this begs the question of whether this constitutes burial, which is a complex behavior not seen in very early hominins. "Every previously known case of cultural deposition," they note, "has been attributed to species of the genus Homo when cranial capacities near the modern human range." Did Homo naledi have brains too small to understand death and burial? The answer to this may lie in observations of our chimpanzee cousins, who also seem to mourn the loss of their compatriots. Research by primatologists over the last five years has produced an array of examples of chimpanzees' reactions to death. Three chimpanzee mothers in Bossou, Guinea, for example, were observed carrying the corpses of their dead offspring for weeks after death. Not only did the mothers keep the babies with them, but they were seen caring for the bodies as if they were still alive, grooming them and sharing nighttime sleep with them. This behavior may have facilitated mummification of the infant chimps. Eventually, the mothers abandoned the babies' bodies, and researchers speculate it may be related to hormonal changes as the post-partum mothers returned to a normal estrus cycle. Could the bodies of the six H. naledi infants and young children have been brought to the cave following a similar period of mourning? Chimpanzees have also been seen to care for members of their troop who are dying, and protecting the bodies of their dead. Since our closest living relatives, to whom researchers often look for clues about ancient hominin behavior, respond to death in startlingly human ways, it is not unreasonable to think that H. naledi with its larger brain had an even more complex process involving the dead. Dirks and colleagues conclude that "deliberate disposal of bodies in the Dinaledi Chamber implies that morphologically primitive hominins like H. naledi may have had their own distinctive patterns of behavioral complexity, even though the reason why H. naledi may have ventured deep into the cave system remains unresolved." That is, although the evidence does not add up to burial of the dead in the sense we use the phrase today, it is likely that these hominins understood death to some extent and took action to deal with it. As the age of the Homo naledi remains is not yet known, it is unclear if this discovery will rewrite our understanding of purposeful burial within the Homo genus. Currently, we are fairly confident that Neandertals buried their dead around 40,000 years ago, and probably as long as 100,000 years ago. If the Homo naledi remains end up being considerably older than that, anthropologists will be discussing burial, behavior, and brain size for years to come, to try to make sense of the increasingly complicated hominin family tree and the mental abilities of our earliest ancestors. The open-access article by Paul Dirks and colleagues, "Geological and taphonomic context for the new hominin species Homo naledi from the Dinaledi Chamber, South Africa," was published today eLIFE (2015;4:e09561).<\|endoftext\|>	3.734375
437	Time for our weekly Playful Preschool Series . This week we are exploring lots of activities to do with balls. Playing in general is a wonderful way to develop language skills. We are excited to show you a language activity that works on learning about prepositions. Ball and Magna Tile Language Play Full Disclosure: This post contains affiliate links. Why We Chose This Language Activity My 3 year-old’s favorite toy is Magna Tiles. They are a bit of an investment but one I’m so glad we made. Both of my kids play with them on a daily basis. I love to listen to them describe their creations and problem solve when something doesn’t work quite the way they want it to. To add a new dynamic to the mix one day, I set out a tray of small balls for them to incorporate into their play. Set-up for this play is super easy! You will need a set of Magna Tilesand a collection of small balls. I let the boys just explore for a bit with the tiles and balls. Then, I told them that I had some challenges for them. They were pretty excited! I said, “One ball needs to fit inside a box. Can you build a box for one ball?” They quickly began building boxes with the tiles. My youngest preschooler enjoying seeing how many balls he could fit inside the box. Then I said, “A ball wants to travel over a bridge. Can you build a bridge for the ball?” My 4 year-old enjoyed trying to make a trestle bridge. The boys played for over an hour with this activity. Here are some more challenge suggestions. Place two balls under a bridge. Build two boxes. Place a ball between the houses. Build a fence with tiles. Place a ball behind the fence. Roll a ball down a ramp. This activity is a great way to develop language skills with your preschoolers. Keep reading. Visit the links below for more great ball ideas from the #playfulpreschool team!<\|endoftext\|>	3.875
157	Grade 1: Money Math - Coins and Dollars worksheet \| Kids Academy This engaging video features coins and dollars to strengthen money math knowledge, while an experienced teacher to guides your child through this critical Grade 1 skill! There’s no better way to learn this real-world concept than with the support and expertise of a real teacher guiding your child every step of the way! As your child watches, he or she will practice: • Understanding the value of coins and dollars • Counting and comparing money amounts • Using important math vocabulary, like “greater than”, “less than”, and “equal to” Open in the app this useful money math worksheet and start watching to practice this important Grade 1 skill!<\|endoftext\|>	3.8125
679	Dear friends, Please read our latest blog post for an important announcement about the website. ❤, The Socratic Team # Circle A has a radius of 2 and a center at (3 ,1 ). Circle B has a radius of 4 and a center at (8 ,3 ). If circle B is translated by <-2 ,4 >, does it overlap circle A? If not, what is the minimum distance between points on both circles? Then teach the underlying concepts Don't copy without citing sources preview ? #### Explanation Explain in detail... #### Explanation: I want someone to double check my answer 1 Jim G. Share Mar 1, 2018 $\text{no overlap "~~0.71" units}$ #### Explanation: $\text{What we have to do here is to "color(blue)"compare ""the}$ $\text{distance (d) between the centres with the "color(blue)"sum of radii}$ • " if sum of radii">d" then circles overlap" • " if sum of radii"< d" then no overlap" $\text{Before calculating d we require to find the centre of B}$ $\text{under the given translation}$ $\text{under the translation } < - 2 , 4 >$ $\left(8 , 3\right) \to \left(8 - 2 , 3 + 4\right) \to \left(6 , 7\right) \leftarrow \textcolor{red}{\text{new centre of B}}$ $\text{to calculate d use the "color(blue)"distance formula}$ •color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2) $\text{let "(x_1,y_1)=(3,1)" and } \left({x}_{2} , {y}_{2}\right) = \left(6 , 7\right)$ $d = \sqrt{{\left(6 - 3\right)}^{2} + {\left(7 - 1\right)}^{2}} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71$ $\text{sum of radii } = 2 + 4 = 6$ $\text{Since sum of radii"< d" then no overlap}$ $\text{min. distance "=d-" sum of radii}$ $\textcolor{w h i t e}{\text{min. distance }} = 6.71 - 6 = 0.71$ graph{((x-3)^2+(y-1)^2-4)((x-6)^2+(y-7)^2-16)=0 [-20, 20, -10, 10]} • 54 minutes ago • An hour ago • An hour ago • An hour ago • 5 minutes ago • 13 minutes ago • 28 minutes ago • 29 minutes ago • 48 minutes ago • 48 minutes ago • 54 minutes ago • An hour ago • An hour ago • An hour ago<\|endoftext\|>	4.625
586	Ragwort (Senecio jacobea) is often found in pasture throughout the UK and contains a poisonous substance (toxin). This toxin (Pyrillozidine) causes damage to the liver of a number of animals including horses and donkeys. Most animals tend to avoid eating Ragwort as it is not very palatable. Poisoning generally occurs when horses ingest Ragwort in dried hay, which has been contaminated with the plant. Although if food is scarce or there are a large number of plants present within the pasture, horses may be forced to eat it. There are two types of poisoning with Ragwort – acute (immediate) and chronic (long term). The acute form is rarely seen as large quantities need to be eaten but when it occurs it is manifested as sudden death. Chronic poisoning is the most common. The signs of poisoning are usually not seen until 4 weeks to 6 months after eating the plants. Small doses of the poison gradually accumulate in the horse’s liver where it causes damage to the liver cells and scarring. Eventually the liver shrinks in size. The liver has large functional reserves and so it is only once these reserves have been exhausted that signs of poisoning develop. These signs can often come on suddenly, although in some horses and ponies mild illness can precede more severe symptoms. Signs of chronic disease include loss of appetite, depression, diarrhoea, weight loss, sensitivity to sunlight and jaundice (yellow colour to skin or eyes). The liver is responsible for filtering the blood of many substances so when it stops functioning correctly these compounds can affect the brain. Animals can develop neurological symptoms such as weakness, circling and head pressing. Ragwort is a biennial plant, which in its first year forms flat rosettes. In the second year it becomes much taller and produces yellow flowers. The only reliable method of prevention is to remove the weed from pasture. The plants should be pulled up by their roots and disposed of away from livestock. It is important to ensure that animals have no access whatsoever to any plants even dried as they can still be poisonous. The poison can also be absorbed through the skin of humans so it is important that impervious gloves are worn. Plants on adjacent land should be removed to avoid the spreading of seed back into your paddocks. Always ensure that there is adequate grazing or alternative food sources such as hay, so that your horse or pony is not tempted to eat any ragwort, which may have been missed. Sprays are available for the control of ragwort and advice can be sort from your local farm merchant on appropriate ones for you. There is a DEFRA Code of Practice to prevent the spread of ragwort. Those who disregard the need for the weed’s control do face prosecution by the government (Ragwort Control Act 2004):DEFRA<\|endoftext\|>	3.71875
378	The battle had begun on February 21, after the Germans—led by Chief of Staff Erich von Falkenhayn—developed a plan to attack the fortress city of Verdun, on the Meuse River in France. Falkenhayn believed that the French army was more vulnerable than the British, and that a major defeat on the Western Front would push the Allies to open peace negotiations. From the beginning, casualties mounted quickly on both sides of the conflict, and after some early gains of territory by the Germans, the battle settled into a bloody stalemate. Among the weapons in the German arsenal was the newly-invented flammenwerfer, or flamethrower; that year also saw the first use by the Germans of phosgene gas, ten times more lethal than the chlorine gas they previously used. As fighting at Verdun stretched on and on, German resources were stretched thinner by having to confront both a British-led offensive on the Somme River and Russia’s Brusilov Offensive on the Eastern Front. In July, the Kaiser, frustrated by the state of things at Verdun, removed Falkenhayn and sent him to command the 9th Army in Transylvania; Paul von Hindenburg took his place. By early December, under Robert Nivelle, who had been appointed to replace Philippe PÉtain in April, the French had managed to recapture much of their lost territory, and in the last three days of battle took 11,000 German prisoners before Hindenburg finally called a stop to the German attacks. The massive loss of life at Verdun—143,000 German dead out of 337,000 casualties, to France’s 162,440 out of 377,231—would come to symbolize, more than that of any other battle, the bloody nature of trench warfare on the Western Front.<\|endoftext\|>	3.703125
1,050	# Maximum Area Property of Equilateral Triangles A particular case of the Isoperimetric Theorem tells us that among all triangles with the same perimeter, the equilateral one has the largest area. A related theorem concerning the triangles inscribed into a given circle is also true: Among all triangles inscribed in a given circle, the equilateral one has the largest area. Proof Among all triangles inscribed in a given circle, the equilateral one has the largest area. ### Proof The proof depends on the following ### Lemma Among all triangles inscribed in a given circle, with a given base, the isosceles one has the largest area. The assertion of the lemma is quite obvious: Among all inscribed triangles with a given base, the tallest one is isosceles and, therefore, it has the largest area, due to the standard formula A = b×h/2, where A, b, and h are the area, the base and the altitude of a triangle. The lemma shows that for a triangle that has two unequal sides, there is another triangle (an isosceles one at that) with the same circumcircle but larger area. The only triangle for which no improvement is possible is equilateral. The lemma also shows that in order to prove the statement we only need to look among isosceles triangles. Consider the diagram below: Let α be the base angle of an isosceles triangle ABC. Then half the apex angle at C equals 90° - α. Let O be the circumcenter of the triangle and D and E the midpoints of sides BC and AB, respectively. Thus, in the diagram, we successively obtain ∠ACB = 180° - 2α, ∠BCE = ∠OCD = 90° - α, OB = OC, BD = CD, ∠OBD = ∠OCD = 90° - α, ∠OBE = α - (90° - α) = 2α - 90°,90° - 2α,2α - 90°,2α OE = R sin(2α - 90°) = R -cos,sin,-cos,cos,tan(2α), CE = R(1 - cos(2α)), AB = 2 EB = 2R cos(2α - 90°) = 2R sin,sin,cos,tan(2α), Area( ΔABC) = AB×CE / 2 = R² (1 - cos(2α)) sin(2α). So to maximize the area of triangle ABC we need to find the maximum of function f(β) = sin(β)(1 - cos(β)), where β = 2α. The simplest way to do that is to compute and equate to 0 the derivative: f'(β) = cos(β)(1 - cos(β)) + sin(β) sin(β) = -cos²(β) + cos(β) + (1 - cos²(β)). Letting x = cos(β) we are left with solving the quadratic equation f(x) = 2x² - x - 1 = 0. Using the quadratic formula, the equation has two roots: x = -1/2 and x = 1, the latter being a spurious byproduct of the applied technique, i.e., looking for the roots of a derivative. Derivatives vanish at internal extrema and saddle points, whereas, for this problem we are only interested in a maximum. In the range of interest (0 < β < 180°), the equation cos(β) = -1/2 admits only one solution, viz., β = 120°, hence 2α = 120°, and α = 60°, making ΔABC indeed equilateral,scalene,equilateral,right,obtuse. (The second root x = 1, leads to cos(2α) = 1. Allowing for an expanded angle range that includes 0, α = 0. A triangle with a 0° angle is called degenerate. Its area is 0 and, therefore, it serves an example of an inscribed triangle with the least area.) It goes without saying (see the discussion of the general Isoperimetric Theorem) that our statement admits an equivalent formulation: Among all triangles with the given area, the equilateral one has the smallest circumscribed circle. • Equilateral Triangles on Sides of a Quadrilateral • Euler Line Cuts Off Equilateral Triangle • Four Incircles in Equilateral Triangle • Problem in Equilateral Triangle • Problem in Equilateral Triangle II • Sum of Squares in Equilateral Triangle • Triangle Classification • Isoperimetric Property of Equilateral Triangles • Angle Trisectors on Circumcircle • Equilateral Triangles On Sides of a Parallelogram • Pompeiu's Theorem • Circle of Apollonius in Equilateral Triangle<\|endoftext\|>	4.5625
675	# 2nd Class Mathematics Patterns Patterns ## Patterns Category : 2nd Class LEARNING OBJECTIVES • identify patterns in shapes, numbers and letters, observe and extend given patterns. • create block patterns by stamping thumb prints, leaf prints, vegetable prints, etc. • create patterns from shapes, numbers and letters. • describe given patterns. QUICK CONCEPT REVIEW A pattern shows the same thing again and again. It is an arrangement of objects; numbers or letters with some fixed rule. Look at the patterns made with circles and tiles. In the first pattern you can see one shaded and one unshaded circle getting repeated. In the second pattern a square is divided diagonally and shaded half. KINDS OF PATTERNS • Patterns are of different kinds – repeating patterns and increasing or decreasing patterns. 1. Repeating Pattern In repeating pattern same things appear again and again. (a) (b) A B A B A B A B 2. Increasing Pattern In increasing pattern things increase in size or number. (a) X XX XXX XXXX (b) 3. Decreasing Pattern In decreasing pattern things decrease in size or number. (a) (b) ABCD ABC AS A NUMBER PATTERNS • Various patterns can be made by numbers. • We can guess the next number in a pattern. Examples: Increasing Pattern: In increasing pattern numbers increase in a certain manner, for example. In the above example numbers are increasing by two. Historical Preview Geometric patterns were the great source of decoration in Islamic art which include calligraphy and vegetal patterns. Decreasing Pattern: In decreasing pattern numbers decrease in a certain manner, for example. In the above example numbers are decreasing by one. Repeating Pattern: In repeating pattern numbers are repeated in a certain manner, for example. In the above example 1 and 2 are appearing one by one. SHAPE PATTERNS • We can make a pattern with shapes. • In shape pattern shapes are repeated in a particular manner. Examples: Patterns given above are based on repetition of shapes. Turning Shapes A shape can be turned up, down, left, right again and again. We can see turning shape patterns in art and fabrics. Examples: TILES PATTERNS The tiles which cover the floor of your house creates a pattern. The method of putting the tiles together without gaps is called tessellation. Examples: Amazing Facts • Patterns are found in the natural world. • Natural patterns include trees, waves, symmetries and stripes. • We can see beautiful patterns on the skin of various animals, like Tiger, Zebra, Leopard etc. THUMB PATTERNS Patterns which are made with the help of thumb and fingers are called thumb patterns. Examples: VEGETABLE AND FRUIT PATTERNS Patterns which are created with the help of vegetables and fruits, are called vegetable and fruit patterns. Examples: Real Life Examples • We can see different types of patterns on the curtain, bed sheet, floor and bag. We can see patterns in buildings, monuments and so on. #### Other Topics You need to login to perform this action. You will be redirected in 3 sec<\|endoftext\|>	4.8125
579	Utah Math 3 - 2020 Edition 3.09 Polynomial identities and complex numbers Lesson We can extend our knowledge of complex numbers and polynomial identities to find complex factors of polynomials. With this type of factorization, the key is to introduce the term $i^2=-1$i2=1 in the given expression whenever required. Let's work through a few examples. #### Worked examples ##### Question 1 Factor the expression $x^2+9$x2+9. Think: The expression $x^2+9$x2+9 can only be factored if we rewrite the second term using the fact that $i^2=-1$i2=1. Once it is rewritten, we can apply the difference of two squares polynomial identity. Do: $x^2+9$x2+9 $=$= $x^2-9i^2$x2−9i2 Rewrite the second term using $i^2=-1$i2=−1 $=$= $\left(x+3i\right)\left(x-3i\right)$(x+3i)(x−3i) Factor using the difference of two squares polynomial identity ##### Question 2 Factor the expression $4x^2-12ix-9$4x212ix9. Think: The middle term has a coefficient of $-12i$12i. If we rewrite the last term using $i^2=-1$i2=1, we might be able to factor the expression. Do: $4x^2-12ix-9$4x2−12ix−9 $=$= $4x^2-12i+9i^2$4x2−12i+9i2 Using $i^2=-1$i2=−1, we get $-9=+9i^2$−9=+9i2 $=$= $\left(2x-3i\right)^2$(2x−3i)2 Now, factor using the identity $\left(A-B\right)^2=A^2-2AB+B^2$(A−B)2=A2−2AB+B2. #### Practice questions ##### Question 3 Complete the factoring by filling in the empty box. 1. $9ix-54=9i\left(\editable{}\right)$9ix54=9i() ##### Question 4 Factor the expression $3x^2+108$3x2+108. Leave your answer in terms of $i$i. ##### Question 5 Factor the expression $x^2+12ix-36$x2+12ix36. ### Outcomes #### III.N.CN.8 Extend polynomial identities to the complex numbers.<\|endoftext\|>	4.875
2,707	<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" /> # Measures of Central Tendency and Dispersion ## Mean, median, mode, range Estimated11 minsto complete % Progress Practice Measures of Central Tendency and Dispersion MEMORY METER This indicates how strong in your memory this concept is Progress Estimated11 minsto complete % Measures of Central Tendency and Dispersion ### Measures of Central Tendency The word “average” is often used to describe the general characteristics of a group of unequal objects. Mathematically, an average is a single number which can be used to summarize a collection of numerical values. In mathematics, there are several types of “averages” with the most common being the mean, the median and the mode. #### Mean The arithmetic mean of a group of numbers is found by dividing the sum of the numbers by the number of values in the group. In other words, we add all the numbers together and divide by the number of numbers. #### Finding the Mean Find the mean of the numbers 11, 16, 9, 15, 5, 18. There are six separate numbers, so the mean=11+16+9+15+5+186=746=1213\begin{align}\text{mean} = \frac{11 + 16 + 9+ 15+ 5+18}{6}=\frac{74}{6}=12\frac{1}{3}\end{align}. The arithmetic mean is what most people automatically think of when the word average is used with numbers. It’s generally a good way to take an average, but it can be misleading when a small number of the values lie very far away from the rest. A classic example would be when calculating average income. If one person (such as former Microsoft Corporation chairman Bill Gates) earns a great deal more than everyone else who is surveyed, then that one value can sway the mean significantly away from what the majority of people earn. #### Real-World Application: Annual Income The annual incomes for 8 professions are shown below. Form the data, calculate the mean annual income of the 8 professions. Profession Annual Income Farming, Fishing, and Forestry $19,630 Sales and Related$28,920 Architecture and Engineering $56,330 Healthcare Practitioners$49,930 Legal $69,030 Teaching & Education$39,130 Construction $35,460 Professional Baseball Player$2,476,590 (Source: Bureau of Labor Statistics, except ()-The Baseball Players' Association (playbpa.com)). There are 8 values listed, so the mean is 19630+28920+56330+49930+69030+39130+35460+24765908=346,877.50\begin{align}\frac{19630+28920+56330+49930+69030+39130+35460+2476590}{8}= \346,877.50\end{align} As you can see, the mean annual income is substantially larger than the income of 7 out of the 8 professions. The effect of the single outlier (the baseball player) has a dramatic effect on the mean, so the mean is not a good method for representing the ‘average’ salary in this case. #### Median The median is another type of average. It is defined as the value in the middle of a group of numbers. To find the median, we must first list all the numbers in order from least to greatest. #### Finding the Median Find the median of the numbers 11, 21, 6, 17, 9. We first list the numbers in ascending order: 6, 9, 11, 17, 21. The median is the value in the middle of the set (in bold). The median is 11. There are two values higher than 11 and two values lower than 11. If there is an even number of values, then the median is the arithmetic mean of the two numbers in the middle (in other words, the number halfway between them). The median is a useful measure of average when the data set is highly skewed by a small number of points that are extremely large or extremely small. Such outliers will have a large effect on the mean, but will leave the median relatively unchanged. #### Mode The mode can be a useful measure of data when that data falls into a small number of categories. It is simply a measure of the most common number, or sometimes the most popular choice. The mode is an especially useful concept for data sets that contains non-numerical information, such as surveys of eye color or favorite ice-cream flavor. Of course, a data set can contain more than one mode; when it does, it is called multimodal. In fact, every value in a data set could be a mode, if every value appears an equal number of times. However, this situation is quite rare. You might encounter data sets with two or even three modes, but more than that would be unlikely unless you are working with very small sample sets. #### Real-World Application: Age of Customers Jim is helping to raise money at his church bake sale by doing face painting for children. He collects the ages of his customers, and displays the data in the graph below. Find the mean, median and mode for the ages represented. License: CC BY-NC 3.0 By reading the graph we can see that there was one 2-year-old, three 3-year-olds, four 4-year-olds, etc. In total, there were 1+3+4+5+6+7+3+1=30\begin{align}1 + 3 + 4 + 5 + 6 + 7 + 3 + 1 = 30\end{align} customers. The mean age is found by adding up all the ages multiplied by the number of times each age appears, and then dividing by 30: 2(1)+3(3)+4(4)+5(5)+6(6)+7(7)+8(3)+9(1)30=17030=523\begin{align}\frac{2(1)+3(3)+4(4)+5(5)+6(6)+7(7)+8(3)+9(1)}{30} = \frac{170}{30} = 5 \frac{2}{3} \end{align} Since there are 30 children, the median is half way between the 15th\begin{align}15^{th}\end{align} and 16th\begin{align}16^{th}\end{align} oldest (that way there will be 15 younger and 15 older than the median age). Both the 15th\begin{align}15^{th}\end{align} and 16th\begin{align}16^{th}\end{align} oldest fall in the 6-year-old range, therefore the median is 6. The mode is given by the age group with the highest frequency. Reading directly from the graph, we see that the mode is 7; there are more 7-year-olds than any other age. Watch this video for assistance with the example above: ### Example #### Example 1 Find the mean, median and mode of the numbers 2, 17, 1, -3, 12, 8, 12, 16. Mean=2+17+1+(3)+12+8+12+169=7.22¯¯¯¯¯\begin{align}\text{Mean}=\frac{2+17+1+(-3)+12 + 8 + 12 +16}{9}= 7.\overline{22}\end{align} We first list the numbers in ascending order: -3, 1, 2, 8, 12, 12, 16, 17. The median is the value in the middle of the set, so the median lies between 8 and 12. Halfway between 8 and 12 is 10, so 10 is the median. The mode is the most frequent number or numbers. The only number that repeats is 12, so 12 is the mode. ### Review 1. Find the median and mode of the numbers given in Example A, specifically: 11, 16, 9, 15, 5, and 18. 2. Find the median and mode of the salaries given in Example B, specifically:19,630, $28,920,$56,330, $49,930,$69,030, $39,130,$35,460, and $2,476,590. 3. Find the mean, median and mode of the data set: 14, 9, 3, 14, 2, 7, 13, 6. 4. Find the mean, median and mode of the data set: 5, 3, 5, 0, 1, 5, 3, 4, 4, 4. 5. Find the mean, median and mode of the data set: 8, 5, 10, 4, 4, 10, 6, 4, 7, 8, 2, 8, 10, 9, 2, 1, 6, 10, 5, 3. 6. Find the mean, median and mode of the following numbers. Which of these will give the best average? 15, 19, 15, 16, 11, 11, 18, 21, 165, 9, 11, 20, 16, 8, 17, 10, 12, 11, 16, 14 7. Ten house sales in Encinitas, California are shown in the table below. Find the mean, median and mode of the sale prices. Explain, using the data, why the median house price is most often used as a measure of the house prices in an area. Address Sale Price Date Of Sale 643 3RD ST$1,137,000 6/5/2007 911 CORNISH DR $879,000 6/5/2007 911 ARDEN DR$950,000 6/13/2007 715 S VULCAN AVE $875,000 4/30/2007 510 4TH ST$1,499,000 4/26/2007 415 ARDEN DR $875,000 5/11/2007 226 5TH ST$4,000,000 5/3/2007 710 3RD ST $975,000 3/13/2007 68 LA VETA AVE$796,793 2/8/2007 207 WEST D ST \$2,100,000 3/15/2007 For 8-10, determine which measure of central tendency (mean, median, or mode) would be most appropriate for the following. 1. The life expectancy of store-bought goldfish. 2. The age in years of audience for a kids TV program. 3. The average actual weight of sacks of potatoes labeled as 5-pound bags. To view the Reviwe answers, open this PDF file and look for section 13.9. ### Notes/Highlights Having trouble? Report an issue. Color Highlighted Text Notes ### Vocabulary Language: English TermDefinition arithmetic mean The arithmetic mean is also called the average. descriptive statistics In descriptive statistics, the goal is to describe the data that found in a sample or given in a problem. inferential statistics With inferential statistics, your goal is use the data in a sample to draw conclusions about a larger population. measure of central tendency In statistics, a measure of central tendency of a data set is a central or typical value of the data set. Median The median of a data set is the middle value of an organized data set. Mode The mode of a data set is the value or values with greatest frequency in the data set. multimodal When a set of data has more than 2 values that occur with the same greatest frequency, the set is called multimodal . Outlier In statistics, an outlier is a data value that is far from other data values. Population Mean The population mean is the mean of all of the members of an entire population. resistant A statistic that is not affected by outliers is called resistant. Sample Mean A sample mean is the mean only of the members of a sample or subset of a population.<\|endoftext\|>	4.8125
840	87 percent of 633 Here we will show you how to calculate eighty-seven percent of six hundred thirty-three. Before we continue, note that 87 percent of 633 is the same as 87% of 633. We will write it both ways throughout this tutorial to remind you that it is the same. 87 percent means that for each 100, there are 87 of something. This page will teach you three different methods you can use to calculate 87 percent of 633. We think that illustrating multiple ways of calculating 87 percent of 633 will give you a comprehensive understanding of what 87% of 633 means, and provide you with percent knowledge that you can use to calculate any percentage in the future. To solidify your understanding of 87 percent of 633 even further, we have also created a pie chart showing 87% of 633. On top of that, we will explain and calculate "What is not 87 percent of 633?" Calculate 87 percent of 633 using a formula This is the most common method to calculate 87% of 633. 633 is the Whole, 87 is the Percent, and the Part is what we are calculating. Below is the math and answer to "What is 87% of 633?" using the percent formula. (Whole × Percent)/100 = Part (633 × 87)/100 = 550.71 87% of 633 = 550.71 Get 87 percent of 633 with a percent decimal number You can convert any percent, such as 87.00%, to 87 percent as a decimal by dividing the percent by one hundred. Therefore, 87% as a decimal is 0.87. Here is how to calculate 87 percent of 633 with percent as a decimal. Whole × Percent as a Decimal = Part 633 × 0.87 = 550.71 87% of 633 = 550.71 Get 87 percent of 633 with a fraction function This is our favorite method of calculating 87% of 633 because it best illustrates what 87 percent of 633 really means. The facts are that it is 87 per 100 and we want to find parts per 633. Here is how to illustrate and show you the answer using a function with fractions. Part 633 = 87 100 Part = 550.71 87% of 633 = 550.71 Note: To solve the equation above, we first multiplied both sides by 633 and then divided the left side to get the answer. 87 percent of 633 illustrated Below is a pie chart illustrating 87 percent of 633. The pie contains 633 parts, and the blue part of the pie is 550.71 parts or 87 percent of 633. Note that it does not matter what the parts are. It could be 87 percent of 633 dollars, 87 percent of 633 people, and so on. The pie chart of 87% of 633 will look the same regardless what it is. What is not 87 percent of 633? What is not 87 percent of 633? In other words, what is the red part of our pie above? We know that the total is 100 percent, so to calculate "What is not 87%?" you deduct 87% from 100% and then take that percent from 633: 100% - 87% = 13% (633 × 13)/100 = 82.29 Another way of calculating the red part is to subtract 550.71 from 633. 633 - 550.71 = 82.29 That is the end of our tutorial folks. We hope we accomplished our goal of making you a percent expert - at least when it comes to calculating 87 percent of 633. Percent of a Number Go here if you need to calculate the percent of a different number. 87 percent of 634 Here is the next percent tutorial on our list that may be of interest.<\|endoftext\|>	4.65625
787	# FAQ: What Is The Symbol For Absolute Value? ## What is the symbol for absolute? Absolute Value Examples and Equations. The most common way to represent the absolute value of a number or expression is to surround it with the absolute value symbol: two vertical straight lines. \|6\| = 6 means “the absolute value of 6 is 6.” \|–6\| = 6 means “the absolute value of –6 is 6.” ## What is the absolute value of – \| – 3? Absolute Value means and “−6” is also 6 away from zero. More Examples: The absolute value of −9 is 9. The absolute value of 3 is 3. ## What is the absolute value of – \| – 7? The absolute value of a number is its distance from zero on the number line. For example, – 7 is 7 units away from zero, so its absolute value would be 7. And 7 is also 7 units away from zero, so its absolute value would also be 7. ## What is the absolute value of – \| – 4? Absolute (denoted by the vertical bars) means that everything between them is converted to non-negative. So \|− 4 \|= 4 and so is \| 4 \|= 4. You might be interested: Often asked: How To Put Absolute Value In Word? ## What is the absolute value of 8? Absolute value is always nonnegative, since distance is always nonnegative. For example, the absolute value of 8 is 8, since 8 is 8 units from 0 on the number line. The absolute value of − 8 is also 8, since − 8 is also 8 units from 0 on the number line. ## What are the rules of absolute value? Absolute value equations are equations where the variable is within an absolute value operator, like \|x-5\|=9. The challenge is that the absolute value of a number depends on the number’s sign: if it’s positive, it’s equal to the number: \|9\|=9. If the number is negative, then the absolute value is its opposite: \|-9\|=9. ## How do you do absolute value inequalities? You begin by making it into two separate equations and then solving them separately. An absolute value equation has no solution if the absolute value expression equals a negative number since an absolute value can never be negative. You can write an absolute value inequality as a compound inequality. ## Why do we use absolute value? When you see an absolute value in a problem or equation, it means that whatever is inside the absolute value is always positive. Absolute values are often used in problems involving distance and are sometimes used with inequalities. That’s the important thing to keep in mind it’s just like distance away from zero. ## What is the absolute value of 20? On a number line it is the distance between the number and zero. The symbol for absolute value is to enclose the number between vertical bars such as \|- 20 \| = 20 and read “The absolute value of -20 equals 20 “. You might be interested: Quick Answer: How To Absolute Value In Calculator In Casio? ## What is the opposite of 4? For example, the opposite of 4 is -4, or negative four. On a number line, 4 and -4 are both the same distance from 0, but they’re on opposite sides. This type of opposite is also called the additive inverse. ## What does the absolute mean? adjective. free from imperfection; complete; perfect: absolute liberty. not mixed or adulterated; pure: absolute alcohol. complete; outright: an absolute lie; an absolute denial. free from restriction or limitation; not limited in any way: absolute command; absolute freedom.<\|endoftext\|>	4.625
719	A Guide to Reading Levels A popular method used by schools to measure a student reader’s ability is Lexile level or a Lexile Measure. A Lexile measure is a valuable tool for teachers, parents, and students. It serves two unique functions: it is the measure of how difficult a text is OR a student’s reading ability level. The Lexile Framework was developed by MetaMetrics©, an educational assessment and research team, funded originally by the National Institute of Child Health and Human Development. How It Works A student receives his or her Lexile measure from one of two ways: taking a school-administered SRI (Scholastic Reading Inventory) test, which is specifically designed to measure Lexile or reading ability OR by taking a standardized reading test which converts the reader’s results to a Lexile measure. If a student gets a 550L then he or she is a 550 level Lexile reader. 550L is the measure of his or her readability level. It is important to note it is never called a score! This encourages student achievement. Determining Your Child’s Lexile Level The Lexile level will always be shown as a number with an “L” after it — for example 770L = 770 Lexile. The higher the Lexile measure, the higher the student’s reading level. The reader’s Lexile Framework works in intervals of five with 5L being the lowest. The highest possible measure is 2000L. Anything below 5L is assessed as a BR or Beginning Reader. Determining a Book’s Lexile Level A book’s Lexile measure is analyzed by MetaMetrics©. After a text is assessed, it is given a measure like that of a student’s readability level, 600L for example. In this measure, MetaMetrics© is assessing the text’s difficulty level. A book or magazine at a 500L has a Lexile Level of 500. MetaMetrics© predicts and assesses how difficult a text will be for a reader to comprehend. The two main criteria it tests are word frequency and sentence strength. A text’s Lexile Framework works in increments of 10 with 10L being the lowest. Measures below 10L are classified as BR or Beginning Reader. Lexile Levels in Practice The ideal for both reader and text is to match both their assessed Lexile measure. For example a book or magazine with a 770L and a reader assessed at a Lexile level of 770. The reading levels per classroom are wide-ranging and varied. There are many factors that go into matching a student to his or her ideal text. The Lexile Framework is a good place to start in picking the right book at the right Lexile level as it targets areas in need of intervention and encourages achievement across grade levels and curricula. How to Find Books on Your Child’s Lexile Level Lexile levels are scientifically and mathematically assigned based on the difficulty and readability of a book. Once you know your child’s Lexile level, you can search for books that match this level to expand your home library and encourage daily reading practice in your own home. Use the Lexile database to search by Lexile level, title, or subject to find books your child will enjoy and be able to read without becoming discouraged at his or her reading achievement. Use the chart below to compare Lexile Levels with other leveled reading systems:<\|endoftext\|>	4.125
198	Plants require mineral salts such as nitrates for growth. The concentration of nitrates is higher on plant root cell than it is in the soil solution surrounding it. The plant cannot rely on diffusion as the nitrates would diffuse out of root cell into the soil.` For an organism to function, substances must move into and out of cells. Three processes contribute to this movement – diffusion, osmosis and active transport. Substances are transported passively down concentration gradients. Often, substances have to be moved from a low to a high concentration - against a concentration gradient. Active transport is a process that is required to move molecules against a concentration gradient.The process requires energy. Active transport in plants For plants to take up mineral ions, ions are moved into root hairs, where they are in a higher concentration than in the dilute solutions in the soil. Active transport then occurs across the root so that the plant takes in the ions it needs from the soil around it.<\|endoftext\|>	3.734375
377	The series of conflicts that wracked the kingdom of England between 1455 and 1487 are today collectively known as the Wars of the Roses. Although the first clashes were fought for control of the king, the saintly but weak-minded Henry VI, by the time of Towton the kingdom itself was at stake, with two kings vying for the throne. The first six years of the conflict, between the First Battle of St Albans and the Battle of Towton, witnessed a blood feud as horrible as any seen in English history, immortalised by Shakespeare in his play Henry VI. The two opposing factions that fought the Wars of the Roses are today characterised as ‘Yorkist’ and ‘Lancastrian’, though it is doubtful that they would have referred to themselves in these terms. Similarly, neither army fought under the emblem of a single rose: the adoption of a white rose for the Yorkists (and later for Yorkshire itself) and a red rose for the Lancastrians (and Lancashire) is a much later development. In fact Yorkshire (and the city of York) was overwhelmingly Lancastrian in its allegiances. The Lancastrians take their name from John of Gaunt, Duke of Lancaster, whose son (Henry IV), grandson (Henry V) and great-grandson (Henry VI) had reigned in succession from 1399, the year in which Henry IV succeeded his cousin, Richard II. The Yorkists are named after the House of York, the dynasty established by Richard, Duke of York, whose sons eventually ruled as Edward IV and Richard III. All were descended from King Edward III (d. 1377), and were therefore related by blood. The closeness of the family ties between some of the main protagonists in the Wars can only have increased the horror and bitterness of the struggle.<\|endoftext\|>	3.828125
2,879	# Solving The Convex Hull Problem using Divide and Conquer Algorithm Akshat Chaturvedi Last Updated: May 13, 2022 ## Introduction Convex Hull is a convex polygon having the smallest area and containing all points in the 2-D plane. The problem states that we are given a set of points in a 2-D plane, and we have to find the convex hull of those points (i.e., return the vertices of the convex hull). We’ll try to come up with a divide and conquer solution for this problem. I strongly recommend you to check out this blog if you want to know more about the working of Divide and Conquer algorithms. Moreover, the idea of this problem is very similar to the Merge Sort algorithm, which I have discussed in detail in that blog. For an overview, the Divide and Conquer algorithm (or DAC) solves a very big task or a problem by breaking it into smaller sub-tasks or sub-problems. ## Algorithm The steps that we’ll follow to solve the problem are: 1. First, we’ll sort the vector containing points in ascending order (according to their x-coordinates). 2. Next, we’ll divide the points into two halves S1 and S2. The set of points S1 contains the points to the left of the median, whereas the set S2 contains all the points that are right to the median. 3. We’ll find the convex hulls for the set S1 and S2 individually. Assuming the convex hull for S1 is C1, and for S2, it is C2. 4. Now, we’ll merge C1 and C2 such that we get the overall convex hull C. Pretty straightforward, right!! Here the only tricky part is merging the two convex hulls C1 and C2. ### Let’s see how we merge the sub-results. The idea here is that the two tangents of the left and the right convex hulls (C1 and C2) will make the final convex hull C. The base case in the recursion will be when the number of points is less than or equal to 5 (i.e., n<=5); we will use the Brute-Force approach to find the solution. (Remember in merge sort, our base case was when the size of the array is one, we consider it as sorted) ## Finding the upper tangent and lower tangent. Let L1 be the line that joins the rightmost point of the left convex hull C1 and the leftmost point of the right convex hull C2. As L1 passes through the polygon C2, we take the anti-clockwise next point on C2 and label the line L2. The line is above the polygon C2, but it crosses the polygon C1, so we move to the clockwise next point, making the line L3. This again crosses the left polygon, so we move to line L4. This line is crossing the right polygon, so we move to line L5. Now, this final line is crossing neither of the points. Hence we get the upper tangent for the given two convex hulls. For finding the lower tangent, we need to move inversely through the hulls, i.e., if the line is crossing the convex hull C2, we move to clockwise next and to anti-clockwise next if the line is crossing the convex hull C1 Here, PQRSUVW is the convex hull of the set of given points. ## Implementation ``````#include<bits/stdc++.h> using namespace std; pair<int, int> mid; // Function to find the quadrant where the point p lies { if (p.first >= 0 && p.second >= 0){ return 1; } if (p.first <= 0 && p.second >= 0){ return 2; } if (p.first <= 0 && p.second <= 0){ return 3; } else{ return 4; } } // Function to check whether the line is crossing the polygon or not int orient(pair<int, int> a, pair<int, int> b, pair<int, int> c) { int r = (b.second-a.second)(c.first-b.first) - (c.second-b.second)(b.first-a.first); if (r == 0){ return 0; } if (r > 0){ return 1; } else{ return -1; } } // Sorting of the points in Brute Force solution is dont using this function bool mycompare(pair<int, int> point1, pair<int, int> point2) { pair<int, int> p = make_pair(point1.first - mid.first, point1.second - mid.second); pair<int, int> q = make_pair(point2.first - mid.first, point2.second - mid.second); if (o != t){ return (o < t); } return (p.secondq.first < q.secondp.first); } // Finds upper tangent of two polygons 'c1' and 'c2' represented as two vectors. vector<pair<int, int>> merger(vector<pair<int, int> > c1, vector<pair<int, int> > c2) { int n1 = c1.size(), n2 = c2.size(); int ia = 0, ib = 0; for (int i=1; i<n1; i++){ if (c1[i].first > c1[ia].first){ ia = i; } } // ib -> leftmost point of b for (int i=1; i<n2; i++){ if (c2[i].first < c2[ib].first){ ib=i; } } // finding the upper tangent int inda = ia, indb = ib; bool done = 0; while (!done) { done = 1; while (orient(c2[indb], c1[inda], c1[(inda+1)%n1]) >=0) inda = (inda + 1) % n1; while (orient(c1[inda], c2[indb], c2[(n2+indb-1)%n2]) <=0) { indb = (n2+indb-1)%n2; done = 0; } } int uppera = inda, upperb = indb; //finding the lower tangent inda = ia, indb=ib; done = 0; int g = 0; while (!done) { done = 1; while (orient(c1[inda], c2[indb], c2[(indb+1)%n2])>=0) indb=(indb+1)%n2; while (orient(c2[indb], c1[inda], c1[(n1+inda-1)%n1])<=0) { inda=(n1+inda-1)%n1; done=0; } } int lowera = inda, lowerb = indb; vector<pair<int, int>> ret; int ind = uppera; ret.push_back(c1[uppera]); while (ind != lowera) { ind = (ind+1)%n1; ret.push_back(c1[ind]); } ind = lowerb; ret.push_back(c2[lowerb]); while (ind != upperb) { ind = (ind+1)%n2; ret.push_back(c2[ind]); } return ret; } // Brute force algorithm to find convex hull for a set of less than 6 points vector<pair<int, int>> bruteHull(vector<pair<int, int>> c) { set<pair<int, int> >s; for (int i=0; i<c.size(); i++) { for (int j=i+1; j<c.size(); j++) { int x1 = c[i].first, x2 = c[j].first; int y1 = c[i].second, y2 = c[j].second; int a1 = y1-y2; int b1 = x2-x1; int c1 = x1y2-y1x2; int pos = 0, neg = 0; for (int k=0; k<c.size(); k++) { if (a1c[k].first+b1c[k].second+c1 <= 0){ neg++; } if (a1c[k].first+b1c[k].second+c1 >= 0){ pos++; } } if (pos == c.size() \|\| neg == c.size()) { s.insert(c[i]); s.insert(c[j]); } } } vector<pair<int, int>>ret; for (auto e:s){ ret.push_back(e); } mid = {0, 0}; int n = ret.size(); for (int i=0; i<n; i++) { mid.first += ret[i].first; mid.second += ret[i].second; ret[i].first = n; ret[i].second = n; } sort(ret.begin(), ret.end(), mycompare); for (int i=0; i<n; i++){ ret[i] = make_pair(ret[i].first/n, ret[i].second/n); } return ret; } // Function to find the convex hull for a set of points vector<pair<int, int>> divide(vector<pair<int, int>> a) { if (a.size() <= 5){ return bruteHull(a); } vector<pair<int, int>> left, right; for (int i=0; i<a.size()/2; i++){ left.push_back(a[i]); } for (int i=a.size()/2; i<a.size(); i++){ right.push_back(a[i]); } // Finding convex hull for the left and right sets vector<pair<int, int>> left_convex_hull = divide(left); vector<pair<int, int>> right_convex_hull = divide(right); // Merging the convex hulls to get the final result return merger(left_convex_hull, right_convex_hull); } int main() { vector<pair<int, int> > a; int n; cin>>n; for(int i=0; i<n; i++){ int x, y; cin>>x; cin>>y; a.push_back(make_pair(x, y)); } // sorting the set of points according to the x-coordinate sort(a.begin(), a.end()); vector<pair<int, int>> ans = divide(a); cout << "The points in the convex hull are:\n"; for (auto e:ans) cout << e.first << " " << e.second << endl; return 0; }`````` Sample Input ``````10 0 0 1 -4 -1 -5 -5 -3 -3 -1 -1 -3 -2 -2 -1 -1 -2 -1 -1 1`````` Sample Output ``````The points in the convex hull are: -5 -3 -1 -5 1 -4 0 0 -1 1`````` ## Complexity Analysis ### Time Complexity The best-case, worst-case, and average-case time complexity of this approach is O(nlogn) because the merging of the two convex hulls takes O(n) time, and we are diving the set of points into left and right halves, so it takes O(logn) time, hence in total, the algorithm will take O(nlogn) time. ### Space Complexity The space complexity of this divide and conquer approach for solving the convex hull problem will be O(n) because we are utilizing extra space. 1). What are the applications of the convex hull problem? Answer: The problem has various applications, which include: • Geographic Information Systems • Image Processing • Pattern Recognition • Game Theory 2). What is the time complexity to solve the convex hull problem? Answer: It takes O(n3) time using the brute force approach, whereas the divide and conquer approach takes O(n) time to find the convex hull. The divide and conquer algorithm is preferred here because it solves the problem in linear time and saves a lot of computation cost. 3). What are the Divide and Conquer algorithm? Answer: The Divide and Conquer algorithm (or DAC) solves a very big task or a problem by breaking it into smaller sub-tasks or sub-problems; after solving, we combine all the sub-tasks in a specific manner so that we get the result for the big task. 4). List the advantages of the Divide and Conquer approach for the Convex Hull Problem. • The Divide and Conquer method solves the problem in O(n*logn) time, whereas the Brute-Force solution takes up O(n3) time to solve the Convex Hull problem. • The DAC algorithm is ideal for multi-processing systems because it inhibits parallelism. It is also fast because it makes excellent use of cache memory without shifting too much computation to main memory, which is relatively slower. • The Divide and conquer algorithm divides the problem into smaller sub-problems; hence it is recommended for more complex problems like finding a convex hull for a set of points. The sub-problems can be executed on different processors making it more time-efficient. ## Key Takeaways If you are worried about the upcoming Campus Placements, then don’t worry. Coding Ninjas has got you covered. A precisely crafted and designed course on interview preparation awaits you; just click on this link. With this course, you can: • Find out where you stand and get recommendations on moving towards your dream job. • Understand your strengths & weaknesses with our 360-degree evaluation. • Prepare yourself for non-tech topics and soft skills. Happy Learning!<\|endoftext\|>	4.53125
485	# What is the equation of the tangent line of f(x)=x^3+2x^2-3x+2 at x=1? Dec 24, 2015 $y = 4 x - 2$ #### Explanation: Step 1: Find derivative of the equation $f \left(x\right) = {x}^{3} + 2 {x}^{2} - 3 x + 2$ $f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left({x}^{3}\right) + \frac{d}{\mathrm{dx}} \left(2 {x}^{2}\right) - \frac{d}{\mathrm{dx}} \left(3 x\right) + \frac{d}{\mathrm{dx}} \left(2\right)$ $f ' \left(x\right) = 3 {x}^{2} + 4 x - 3$ Step 2: Find the slope of the tangent line at $x = 1$ $m = f ' \left(1\right) = 3 \left({1}^{2}\right) + 4 \left(1\right) - 3 = 4$ Step 3: Find the $y$ coordinate of the function when $x = 1$ $f \left(1\right) = {\left(1\right)}^{3} + 2 {\left(1\right)}^{2} - 3 \left(1\right) + 2$ $f \left(1\right) = 2$ So, the original point of the graph is $\left(1 , 2\right)$ Step 4: Find the equation of the tangent line using point slope formula $y - {y}_{0} = m \left(x - {x}_{0}\right)$ $m = 4 \text{ " " } \left(1 , 2\right)$ $y - 2 = 4 \left(x - 1\right)$ $\implies y - 2 = 4 x - 4$ $\implies y = 4 x - 4 + 2$ $\implies y = 4 x - 2$<\|endoftext\|>	4.6875
423	The difference between success and failure for students with LD and ADHD often comes down to how effectively the curriculum is adapted to individual needs. Accommodations and modifications are the tools used by the IEP team to achieve that end. Accommodations allow a student to complete the same tasks as their non-LD peers but with some variation in time, format, setting, and/or presentation. The purpose of an accommodation is to provide a student with equal access to learning and an equal opportunity to show what he knows and what he can do. Accommodations are divided into four categories: - Variations in time: adapting the time allotted for learning, task completion, or testing - Variation of input: adapting the way instruction is delivered - Variation of output: adapting how a student can respond to instruction - Variation of size: adapting the number of items the student is expected to complete Unlike accommodations, which do not change the instructional level, content, or performance criteria, modifications alter one or more of those elements on a given assignment. Modifications are changes in what students are expected to learn, based on their individual abilities. Examples of modifications include use of alternate books, pass/no pass grading option, reworded questions in simpler language, daily feedback to a student. - Can your child participate in the activity in the same way as her peers? - If not, can she do the same activity with adapted materials? - If not, can she do the same activity with adapted expectations and materials? - If not, can she accomplish the goals of the lesson by working with a partner or small group? - If not, can she do the same activity with intermittent assistance from an adult? - If not, can she do the same activity with direct adult assistance? - If not, can she do a different, parallel activity? Eve Kessler, Esq., a criminal appellate attorney with The Legal Aid Society, NYC, is co-founder of SPED*NET Wilton and a Contributing Editor of Smart Kids.<\|endoftext\|>	4.03125
873	# 78°F equals 25°C in Celsius Welcome to Warren Institute! In today's article, we'll delve into the fascinating world of temperature conversion in Mathematics education. Have you ever wondered how to convert 78 degrees Fahrenheit to Celsius? Join us as we explore the step-by-step process and unlock the mystery behind this common conversion. Understanding temperature scales is crucial for everyday life and scientific endeavors, so let's dive in and discover the mathematical relationship between Fahrenheit and Celsius. Get ready to expand your mathematical knowledge and gain valuable insights into real-world applications of temperature conversion. ## The Fahrenheit and Celsius Scales In this section, we will explore the relationship between the Fahrenheit and Celsius temperature scales and how they are used in Mathematics education. ## Converting Fahrenheit to Celsius Converting temperatures from Fahrenheit to Celsius is a fundamental skill in Mathematics education. To convert 78 degrees Fahrenheit to Celsius, we can use the formula C = (F - 32) * 5/9, where C represents the temperature in Celsius and F represents the temperature in Fahrenheit. Plugging in 78 degrees Fahrenheit into the formula gives us the conversion to Celsius. ## Understanding the Meaning of Temperature Scales It's important for students to understand the significance of different temperature scales and how they are used in real-world contexts. Explaining the practical implications of 78 degrees Fahrenheit in terms of Celsius can help students grasp the concept of temperature conversion and its relevance in Mathematics education. ## Real-World Applications in Mathematics Education Teachers can incorporate real-world examples, such as weather forecasts or cooking temperatures, to demonstrate the application of temperature conversion in Mathematics education. By relating 78 degrees Fahrenheit to Celsius in these contexts, students can see the practical utility of mathematical concepts in everyday life. ## frequently asked questions ### How can I convert 78 degrees Fahrenheit to Celsius using the formula? To convert 78 degrees Fahrenheit to Celsius using the formula, you can use the formula C = (F - 32) * 5/9. Plugging in 78 for F, you get C = (78 - 32) * 5/9, which simplifies to C = 46 * 5/9, resulting in C ≈ 25.6 degrees Celsius. ### What is the mathematical relationship between Fahrenheit and Celsius temperature scales? The mathematical relationship between Fahrenheit and Celsius temperature scales is given by the formula: Celsius = (Fahrenheit - 32) * 5/9 ### How can I use mental math strategies to quickly convert temperatures from Fahrenheit to Celsius? You can use mental math strategies to quickly convert temperatures from Fahrenheit to Celsius by using the formula: subtract 32 from the Fahrenheit temperature, then multiply by 5/9. ### What are some real-life examples or applications of converting Fahrenheit to Celsius in a mathematics lesson? Converting Fahrenheit to Celsius can be applied to everyday situations such as cooking, weather forecasting, and understanding temperature settings on household appliances. This application helps students understand the practicality of temperature conversion and its relevance in their daily lives. ### How can I explain the concept of temperature conversion between Fahrenheit and Celsius to elementary students in a mathematics education setting? You can explain the concept of temperature conversion between Fahrenheit and Celsius to elementary students by using real-life examples and visual aids. This can help them understand the relationship between the two scales and how to convert temperatures from one to the other using simple formulas. In conclusion, understanding the conversion between Fahrenheit and Celsius is an essential skill in mathematics education. With the formula F = (C x 9/5) + 32, we can easily convert 78 degrees Fahrenheit to Celsius, which equals 25.6 degrees Celsius. This knowledge not only enhances our mathematical abilities but also allows us to better comprehend the world around us. See also Finding the Original Price: Discounted Item Calculations If you want to know other articles similar to 78°F equals 25°C in Celsius you can visit the category General Education. Michaell Miller Michael Miller is a passionate blog writer and advanced mathematics teacher with a deep understanding of mathematical physics. With years of teaching experience, Michael combines his love of mathematics with an exceptional ability to communicate complex concepts in an accessible way. His blog posts offer a unique and enriching perspective on mathematical and physical topics, making learning fascinating and understandable for all. Go up<\|endoftext\|>	4.53125
811	West Nile Virus What is West Nile Virus (WNV)? West Nile Virus (WNV) is a mosquito-borne virus that primarily infects birds. The bite of an infected mosquito however can also spread the virus to humans. WNV virus can cause high fevers, encephalitis (inflammation of the brain), or even meningitis (inflammation of the lining of the brain and spinal cord). In addition to humans it can infect other wildlife including horses and other mammals. Experts believe it is a seasonal epidemic in North America that flares up in the summer and continues into the fall. In a small number of people infected by the virus, the disease can be serious, even fatal. How is WNV spread? - WNV is most often spread to humans by the bite of an infected mosquito that gets infected by biting a bird that carries the virus. Horses and other mammals bitten by mosquitoes that carry the virus can also become infected. Whenever mosquitoes are active, there is a risk of getting WNV. The risk is highest from late July through September. - In a very small number of cases, WNV has been spread through blood transfusions, organ transplants, breastfeeding and during pregnancy from mother to baby. - WNV is not spread through casual contact such as touching or kissing a person with the virus. What are the symptoms of West Nile Virus? If symptoms develop, they usually appear 3 to 14 days after the bite of an infected mosquito. - Serious Symptoms - About one in 150 people infected with WNV will develop severe illness. Symptoms can include high fever, headache, neck stiffness, stupor, disorientation, coma, tremors, convulsions, muscle weakness, vision loss, numbness and paralysis. These symptoms may last several weeks, and neurological effects may be permanent. - Milder Symptoms - Up to 20% of the people who become infected display symptoms such as fever, headache, body aches, nausea, vomiting, and sometimes swollen lymph glands or a skin rash on the chest, stomach and back. Symptoms can last for as short as a few days; though, even healthy people have been sick for several weeks. - No Symptoms - Most people (80%) infected with WNV do not have any symptoms. What is the treatment for WNV? There is no treatment for WNV infection. Illness may last weeks to months, even in healthy persons. In more severe cases people may need hospital care for supportive treatment such as intravenous fluids, help with breathing, or nursing care. What should I do if I think I have WNV? Milder WNV illness usually improves without medical attention. A person may choose to see their doctor. Seek medical attention immediately if symptoms of severe WNV illness develop, such as unusually severe headaches or confusion. Severe WNV illness usually requires hospitalization. Pregnant women and nursing mothers who develop symptoms that could be WNV are encouraged to see their doctor. How can West Nile Virus be prevented? The best way to avoid WNV infection is to prevent mosquito bites: - Use insect repellent. The Centers for Disease Control and Prevention (CDC) recommends the use of insect repellents containing active ingredients registered with the U.S. Environmental Protection Agency (EPA). Always follow manufacturer's directions carefully. - Be careful using repellent on the hands of children because repellents may irritate the eyes and mouth. - Wear protective clothing such as long sleeved shirts and pants. - Limit outdoor activity from dusk to dawn when mosquitoes are most active. - Avoid areas where mosquitoes may be present (i.e. shaded and wooded areas). - Maintain window and door screens to keep mosquitoes out of buildings. - Get rid of mosquito breeding sites by emptying standing water that collects in birdbaths, boats, buckets, tires, unused pools, roof gutters and other containers.<\|endoftext\|>	3.90625
1,624	# How to Calculate Significant Difference Between Two Means in Excel Get FREE Advanced Excel Exercises with Solutions! This article highlights 2 ways to calculate the significant difference between two means in Excel. Calculating the significant difference between two means may become necessary on many occasions. For example, to determine which team performs better when two teams follow two different methods to perform the same task. Or, to determine if the increase in the weight of fruits of a tree for using chemical fertilizers is significant. It will be very tedious to do this manually. The following picture highlights the purpose of this article. Have a quick look through it to do that. ## What Is the Significant Difference and T-Test? Significant Difference: The significant difference simply means that two sets of data are significantly different. Because, in statistics, we consider that there is a 5% probability of this happening by chance. If this is not the case then the datasets are different. A Null Hypothesis assumes that two means are the same. Now, a p-Value (probability) of less than 0.05 rejects this hypothesis. This way, we can conclude that the two sets of data or means are different and not the same. T-Test: T-Test is a term from statistics that allows us to do a comparison of two data populations and their means. The test is used to see if the two sets of data are significantly different from one another. A Null Hypothesis is used to test for the significant difference. In addition, the datasets usually follow the normal distribution curve. However, the variances are unknown and assumed to be equal. William Sealy Gosset devised the T-Test in around 1908 while working at the Guinness Brewery in Ireland. He used the T-Test to monitor the quality of the stout brewed by Guinness. A T-Test analysis could be used to test the results from two different portfolios that are managed under two different investment strategies. In this analysis, a Null Hypothesis can be created, for instance, where the means of the returns for the two portfolios do not differ. The test then looks at the T-Statistic and T-Distribution to determine a p-Value (probability) that can be used to either validate or refute the Null Hypothesis. The T-Test is one of several different types of statistical tests used for hypothesis testing. Others include An Analysis of Chi-Square Test, F-Test, Variance Test, and Z-Test. The T-Test is considered a more conservative approach and better suited for smaller groups of data. ## How to Calculate Significant Difference Between Two Means in Excel Imagine you have an orange orchard with 10 trees. The soil has enough natural fertilizer in it. Now you want to find out if using chemical fertilizer will increase the weight of the oranges. Then, measure the average weights of the oranges from each of the trees. This time do not apply any chemical fertilizers. Then, do the same in the next season. But, this time, apply some fertilizers to those trees. Suppose, you get the following result. Now, it seems that using fertilizers has increased the average weight of the oranges. But, you canâ€™t be sure if this was just by chance without calculating the significant difference. Follow the methods below to do that. ### 1. Calculate Significant Difference with T.TEST Function You can easily calculate the significant difference between two means using the T.TEST function in Excel. Just enter the following formula in cell D16. Then, you will get the following result. `=T.TEST(C5:C14,D5:D14,2,1)` The formula returns a p-value of less than 0.05. So, the Null Hypothesis is rejected. Therefore, we can say that using chemical fertilizers has increased the weight of the oranges. ðŸ”ŽÂ How Does the Formula Work? The T.TEST function has four arguments. You need to enter them properly to avoid faulty results. Syntax: T.TEST(array1, array2, tails, type) Arguments: array1: Required. Refers to the first range of data. array2: Required. Refers to the second range of data. tails: Required. Enter 1 for One-tailed distribution and 2 for Two-tailed distribution. One-tailed distribution allows for effect in one particular direction, either positive or negative, good or bad while two-tailed distribution allows both. type: Required. Enter 1 for Paired, 2 for Two-sample equal variance (homoscedastic), and 3 for Two-sample unequal variance (heteroscedastic). Paired type indicates that the datasets are from two dependent or related or the same sources. Use Two-sample equal variance for two independent datasets with equal variances and Two-sample unequal variance for unequal variances. ### 2. Calculate the Significant Difference Using Data Analysis in Excel You can also calculate the significant difference between two means in Excel using the Analysis Toolpak. Follow the steps below to be able to do that. ðŸ“Œ Steps • First, select the Data tab to see if your Excel has the Analysis Toolpak installed. Is the Data Analysis feature visible? If yes then go to step 4. • Otherwise, press ALT+F+T to open the Excel Options window. Then, go to the Add-ins tab. Next, select Go as shown in the picture below. • Next, check the checkbox for Analysis Toolpak on the Add-ins window. Then select OK. After that, you will be able to access the Data Analysis feature from the Data tab. • Now, select Data >> Data Analysis. This will open the Data Analysis window. Next, choose the proper analysis tool from the list. Here, we will select t-Test: Paired Two Sample for Means for our dataset. Now select OK. This will take you to that analysis tool window. • Now enter \$C\$4:\$C\$14 and \$D\$4:\$D\$14 for Variable 1 Range and Variable 2 Range respectively using the upward arrows. • Next, check the checkbox for Labels. You could keep this unchecked if you did not include the headers in the above step. • Now, enter 0.05 in the textbox for Alpha. Then, mark the radio button for Output Range. Next, select the location where you want to get the results. Excel shows the results in a new worksheet by default. Then, select OK. Finally, you will see the following result. The highlighted value indicates the same result obtained in the earlier method. ## Things to Remember • Make sure you are using the proper arguments that correspond to your dataset while using the function. Otherwise, you might end up with erroneous results. • Select the proper analysis tool based on your dataset while using the Data Analysis window. • The TTEST function is the earlier version of the T.TEST function. Download Practice Workbook You can download the practice workbook from the download button below. ## Conclusion Now you know how to calculate the significant difference between two means in Excel. Please let us know if this article has provided you with the desired solution to your problem. You can also use the comment section below for further queries or suggestions. Stay with us and keep learning. ## What is ExcelDemy? ExcelDemy Learn Excel & Excel Solutions Center provides free Excel tutorials, free support , online Excel training and Excel consultancy services for Excel professionals and businesses. Feel free to contact us with your Excel problems. Md. Shamim Reza Hello there! This is Md. Shamim Reza. Working as an Excel & VBA Content Developer at ExcelDemy. We try to find simple & easy solutions to the problems that Excel users face every day. Our goal is to gather knowledge, find innovative solutions through them and make those solutions available for everybody. Stay with us & keep learning. We will be happy to hear your thoughts Advanced Excel Exercises with Solutions PDF<\|endoftext\|>	4.40625
1,096	# Quick Answer: What Is The Quotient Rule For Limits? ## What is the quotient law for limits? The limit of a quotient is equal to the quotient of the limits. The limit of a constant function is equal to the constant. The limit of a linear function is equal to the number x is approaching. , if it exists, by using the Limit Laws.. ## How do you find limits? Find the limit by rationalizing the numerator In this situation, if you multiply the numerator and denominator by the conjugate of the numerator, the term in the denominator that was a problem cancels out, and you’ll be able to find the limit: Multiply the top and bottom of the fraction by the conjugate. ## What are the limit properties? A General Note: Properties of LimitsConstant, klimx→ak=kQuotient of functionslimx→af(x)g(x)=limx→af(x)limx→ag(x)=AB,B≠0Function raised to an exponentlimx→a[f(x)]n=[limx→∞f(x)]n=An l i m x → a [ f ( x ) ] n = [ l i m x → ∞ f ( x ) ] n = A n , where n is a positive integer6 more rows ## What is the importance of limits? Limits allow us to study a number from afar. That is, we can study the points around it so we can better understand the given value we want to know. Especially in derivatives, where change in position is purely relative, the points around a given value are critically important. ## What is infinity minus infinity? ∞ – ∞ = 1. Woops! It is impossible for infinity subtracted from infinity to be equal to one and zero. Using this type of math, we can get infinity minus infinity to equal any real number. Therefore, infinity subtracted from infinity is undefined. ## What is the product rule for limits? The Product Law basically states that if you are taking the limit of the product of two functions then it is equal to the product of the limits of those two functions. [f(x) · g(x)] = L · M. The proof of this law is very similar to that of the Sum Law, but things get a little bit messier. ## Can you subtract limits? Limits can be added and subtracted, but only when those limits exist. ## What is infinity divided by infinity? You can’t really say that infinity divided by infinity is anything. For all intents and purposes, it is undefined. This is because infinity is seen as a concept, not a number – and its symbol merely represents the concept. ## Can limits be multiplied? The multiplication rule for limits says that the product of the limits is the same as the limit of the product of two functions. That is, if the limit exists and is finite (not infinite) as x approaches a for f(x) and for g(x), then the limit as x approaches a for fg(x) is the product of the limits for f and g. ## Can 0 be a limit? Typically, zero in the denominator means it’s undefined. However, that will only be true if the numerator isn’t also zero. … However, in take the limit, if we get 0/0 we can get a variety of answers and the only way to know which on is correct is to actually compute the limit. ## How do you know if a limit is one sided? A one-sided limit is the value the function approaches as the x-values approach the limit from one side only. For example, f(x)=\|x\|/x returns -1 for negative numbers, 1 for positive numbers, and isn’t defined for 0. The one-sided right limit of f at x=0 is 1, and the one-sided left limit at x=0 is -1. ## What is the limit? In mathematics, a limit is the value that a function (or sequence) “approaches” as the input (or index) “approaches” some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. ## What does the quotient rule mean? In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. ## What makes a limit not exist? Limits typically fail to exist for one of four reasons: The one-sided limits are not equal. The function doesn’t approach a finite value (see Basic Definition of Limit). The function doesn’t approach a particular value (oscillation). ## When should you use the quotient rule? You want to use the quotient rule when you have one function divided by another function and you’re taking the derivative of that, such as u / v. And you can remember the quotient rule by remembering this little jingle: Lo d hi minus hi d low, all over the square of what’s below. ## How does quotient rule work? What is the Quotient rule? Basically, you take the derivative of f multiplied by g, subtract f multiplied by the derivative of g, and divide all that by [ g ( x ) ] 2 [g(x)]^2 [g(x)]2open bracket, g, left parenthesis, x, right parenthesis, close bracket, squared.<\|endoftext\|>	4.625
844	Research published this week in Proceedings of the Royal Society B: Biological Sciences solves a mystery that has long shrouded our understanding of white oaks: where did they come from? The approximately 125 white oak species in the Americas and 25 in Eurasia—including the massive bur oak of American prairies and savannas, the valley oak of California and the eponymous white oak of eastern North American forests—are important in forests and savannas throughout much of the northern hemisphere. Yet, despite their economic and ecological importance, not much was known about the evolutionary history of the white oak group until now. This paper, co-authored by Andrew Hipp of The Morton Arboretum and Paul Manos and John McVay of Duke University, reveals for the first time that Europe and east Asia have eastern North America to thank for their white oaks. Coupling genomic data with fossil records and novel analytical methods, the research suggests that the Eurasian white oaks arose from a North American ancestor that migrated to Europe, perhaps by way of the North Atlantic land bridge. This is a story that has long been hidden by ancient hybridization among the Eurasian white oaks. The research implements new analytical tools to tease apart hybridization from evolutionary history to tell the full white oak story. The study, funded by a collaborative four-year National Science Foundation grant to five institutions led by The Morton Arboretum, also shows that two oak species found on opposite ends of the globe, the Armenian oak (Quercus pontica) found in the Caucasus mountains and Sadler’s oak (Quercus sadleriana) of California and Oregon, are each other’s closest relatives. These species, the authors argue, are the last remnants of a widespread white oak lineage that stretched at least from Europe to the Pacific Northwest, of which all are extinct except for these two species. “Understanding even the most basic questions—how many oak species are there? Where do they live?—rests on our understanding of oak evolution,” said study co-author Andrew Hipp, Ph.D. of The Morton Arboretum. “This is the first paper to conclusively separate the role of gene flow and divergence to recover a holistic portrait of the white oak tree of life. It is a crucial step toward understanding why white oaks became so important to the ecology of temperate forests and savannas in the northern hemisphere.” The importance of oaks White oaks are just one of many types of more than 450 species of oaks which live around the world. Considered a keystone species, oaks support our planet’s ecosystem like very few other tree species do. These stately trees are fundamental to the health of our forests, providing critical food, habitat and shelter for animals, birds and insects, and have the highest amount of biomass compared to any other tree species in the forest, working harder to clean our air than many of their fellow tree species. In the Chicago region alone, oak ecosystems provide more than $2 billion worth of flood control and other water management services. Today, oaks need our help. Around the world, due to pests, disease, forest loss and low rates of reproduction, oak forests are now a fraction of what they once were, upsetting the delicate balance of our forest ecosystems and leaving humans without their benefits. With only 17 percent of ancient oak forest area remaining in the Chicago region and similar oak loss throughout the world, human intervention is critical to ensure we don’t lose this important species. Researchers at The Morton Arboretum are committed to ensuring oaks remain, studying the ways they regenerate, encouraging the protection of our oak forests, improving land management approaches and planting new trees to extend existing populations. Learn more about what The Morton Arboretum is doing to conserve oaks globally. The study was funded by National Science Foundation awards to A.L.H. (1146488) and P.S.M. (1146102). Read the paper, published in the Proceedings of the Royal Society B: Biological Sciences, in its entirety.<\|endoftext\|>	3.84375
810	The Stern-Brocot Tree As with the Farey series, the Stern-Brocot tree is built using mediant fractions. Place two starting terms 0/1 and 1/0 some way apart and their mediant fraction 1/1 in between and a little down. This creates two gaps: one between 0/1 and 1/1 and another between 1/1 and 1/0. Compute two corresponding mediants and place them below 1/1. Continue in this manner. The next step will add a row of four fractions, then eight, and so on. When building the Farey series, mediants sometimes have to be reduced to lowest terms. For example, in F5, (2 + 3)/(5 + 5) = 1/2. This never happens with the Stern-Brocot tree. The fact is proved by induction based on the following property of the tree: any two fractions m1/n1 and m2/n2 whose mediants will be computed at any stage of the construction, satisfy (1) m2n1 - m1n2 = 1 A mediant fraction generated by two terms that satisfy (1) stands in the same relation with both of its progenitors. From this we observe that the rows of numerators and denominators of the terms in the Stern-Brocot tree are computed independently of each other. They may be dealt with separately. The row of numerators starts with the pair of integers 0,1. The row of denominators starts with the pair of integers 1,0. They are just symmetric reflections of each other. Let denote the first tree as [0,1] and the second [1,0]. We may generalize and consider trees that grow from an arbitrary pair of integers [x,y]. Since, at any stage of construction, we only carry out linear operations (actually only addition), we get a whole vector space of trees. The space is 2-dimensional as we can write [x,y] = x[1,0] + [0,1]. Each tree [x,y] consists of two parts: the left tree [x,x+y] =x[1,1] + y[0,1] and the right tree [x+y,y] = x[1,0] + y[1,1]. The tree [1,1] is a mirror image of itself. In particular, all its rows are palindromic. Let's record several observations: 1. The tree of the denominators [1,0] has [1,1] as its left part. 2. The left part of the tree of the numerators [0,1] is still [0,1] while [1,1] is now the right part. 3. Fractions on the left from 1/1 are less than 1; fractions on the right from 1/1 are greater than 1 Combining all together we see that 1. The left part of the tree of the denominators is palindromic. 2. The right part of any row of numerators coincides with the left part of the corresponding row of denominators. 3. The left part of the row of numerators coincides with the previous row of numerators. Surprisingly, the Stern-Brocot tree contains all non-negative fractions. Therefore, its left subtree contains all fractions between 0 and 1. Somehow it must be possible to pluck off the tree the Farey series of any order. Whatever the exact process, since the only criterion for a fraction to belong to the Farey series is the magnitude of its denominator, and, as we know, the right part of any row of denominators is palindromic, the same will be true for the denominators in the Farey series.<\|endoftext\|>	4.5
737	\| Goals and Objectives\| It is often said, “games are a metaphor for life,” as such, successful mastery of this course will develop more than a student’s mathematical capabilities, it will develop lifelong learning skills that will be used in all aspects of life. - The course is designed to help students particularly grades 2 through 5 understand, develop and solve basic arithmetic equations. - Students will learn the principles of addition, subtraction, multiplication and division in a fun game environment. - Students will represent and solve problems involving addition and subtraction (Math Common Core Standards - Grade 2). - Students will understand and identify place value (Math Common Core Standards - Grade 2&3). - Effectively using and understanding the use of the number line. - Students will represent and solve problems involving multiplication and division (Math Common Core Standards - Grade 3&4). - Understand properties of multiplication and the relationship between multiplication and division (Math Common Core Standards - Grade 3). - Understanding the use of probability and its effectiveness in everyday life. - The course will drive students to gradually speed up their mathematical calculations. - Exhibiting respect for self and others (English Language Arts - Common Core Standards - Grade 2&3). - Students will learn to work individual and as a member of a team. - Understanding and apply patience and self-control. - Students will provide motivation to their peers. - Students are encouraged to take what they are learning outside the confines of their classroom and apply it directly to their everyday lives. After successful completion of this 8-9 week course students should have gained the following: - Student will know how to represent and solve problems involving addition, subtraction, multiplication and division correctly with numbers under 100 (Math Common Core Standards - Grade 3). - Student will know how to identify place value from ones, tens, hundred and thousand (Math Common Core Standards - Grade 3). - Using the number line, students will correctly solve equations during game play. - Students will understand the Strategic use of probability within the game and demonstrate how it is used to have their scores either reduced, increase or remain unchanged. Students will know and explain how probability applies to everyday life and the decisions people make. - The student will be able to utilize strategic thinking to make proper decisions on DOMATICS tile placement within games. Students will be able to explain the correlation between formulating a strategy and successfully reaching one's goals - Students will understand and demonstrate proper sportsmanship before, during and after their games. Ability to explain the importance of sportsmanship within everyday life. - Students will know how to demonstrate self-control by resisting the urge to answer problems out of turn. Students will demonstrate patience by providing others an opportunity to learn and play at a slower pace without feeling rushed and or ridiculed. Students will possess the ability to explain the importance of self-control and patience in everyday life (English Language Arts - Common Core Standards - Grade 2&3). - Students will be able to identify the pros and cons when working individually and working as a member of a team and how it relates to everyday life. - Students will demonstrate leadership qualities by providing motivation to other players during the game through encouragement. Student will be able to explain the importance of motivation within everyday life. - Successfully showing respect to other players by providing them an opportunity to work out equations, ask questions without ridicule and being attentive throughout the game. Ability to explain how showing respect is important in gaining respect in everyday life (English Language Arts - Common Core Standards - Grade 2&3).<\|endoftext\|>	4.1875
2,379	What is Food Protein-Induced Enterocolitis Syndrome or FPIES? This very problematic immune system response involves an overreaction of the body to a specific food protein that happens typically 1 to 4 hours after consuming it. This condition develops in babies within the first few weeks of life after being born and sometimes lingers until adolescence or even adulthood. Thankfully, FPIES is commonly cured for most children so you can look forward to your child seeing relief from this bothersome allergic condition. Symptoms of FPIES include vomiting, diarrhea, and acidification of the blood. However, these are the most noteworthy and visible symptoms and individuals will experience unique reactions differently from person-to-person. Other characteristics of suffering from this conditions that can help identify it in your baby include: - Bloody diarrhea or stools - Lethargy or fatigue - Abdominal distension (Belly sticking out) - Cyanosis or blue discoloration of the skin (From lack of proper circulation) - Decreased body temperature or hypothermia - High numbers of blood cell and platelet counts - Low blood pressure - Hypovolemic shock (Severe symptoms from rapid fluid loss that requires immediate hospitalization) - Weight loss - Failure to grow or gain weight - Development of atopic dermatitis, or an allergic inflammatory skin disorder - Environmental allergies leading to asthma or allergic rhinitis (hay fever) - In addition to the most common vomiting, diarrhea, and acidosis Is there an FPIES cause? Right now, there is not a definitive cause of FPIES that researchers can declare without a reasonable doubt. However, there are some ideas that get thrown around and considered as acceptable hypotheses. This condition can arise due to an inappropriate response of the immune system to the proteins in particular foods. When the specific food passes through the stomach and into the small intestine, the undigested protein causes localized inflammation in the digestive tract. It has been determined that the inflammation that is caused lets your bodily fluids leak through the intestinal wall to be excreted out as diarrhea. In addition, small food particles and nutrients may also pass through the small intestinal wall and be removed through waste along with essential cellular water. This is particularly dangerous when you think about it because the body’s cells need water and nutrients to function properly. FPIES Food List to Avoid The most common food triggers for causing an FPIES reaction are soy and cow’s milk. In young babies, this is what they consume 40% of the time for their initial reaction. What generally causes the symptoms is a liquid form of food, like formula or breastmilk. Usually, babies are fed formula with cow’s milk and soy ingredients. Feeding a baby a particular formula that causes the reaction in a child usually gives it away that they have FPIES. Solid foods have also been shown to cause FPIES symptoms directly, but foods that are generally not known to be allergenic are also not of concern. The most common solid foods that are frequently associated with these reactions are: - Rice, wheat, turkey, fish, chicken, gluten grains, eggs, peanuts, and potatoes Recently, some parents are insisting that their baby’s consumption of fruit also causes the same immune system reaction. In adults with FPIES, the most common cause of the condition is the ingestion of shellfish. FPIES Safe Foods List to Consume It’s not medically safe to say that there are foods that will certainly not cause an allergic reaction because, for any unique individual, any type of food allergies are possible. With some knowledge of chemistry, you could consider the vast amount of compounds present in one particular food and all of these compounds have some potential to set off your immune system. However, there are some foods that parents on the internet have agreed to have worked very well for their children who suffer from FPIES. It seems that babies respond best to fruits like apples and bananas as one of their first foods, especially if you suspect that there is an allergy potential or if your older children have allergies as well. If you already know your child suffers from this allergic condition, mashing up fruits like apples, pears, berries, and peaches could benefit them greatly. For vegetables, a more healthy option, you could try to mash up carrots, which seem to be the least likely to cause an FPIES reaction while being very edible for young children. One very popular mom who has many children with FPIES recommends that you mash or blend spinach, carrots, or leafy greens so that you can provide an adequate serving of vegetables to your child, which is essential for proper nutrition and bodily function. A common mistake that many parents make is only feeding their child fruit. Despite fruit being a great source of nutrition, it is also very high in sugar and should only be consumed in moderation, especially for children and babies. Is FPIES Hereditary or Genetic? Yes, Food Protein-Induced Enterocolitis Syndrome is passed down hereditarily through your genes to your children. Evidence of this being true can be seen in mothers who have multiple children with FPIES and other types of allergies and allergic conditions. For many different types of allergies and conditions, the root cause lies in genetics. Even though parents may not obviously display the allergy or presence of it being in your genetic code, it can still be passed down to children by chance. Will My Baby Outgrow FPIES? Fortunately, Food Protein-Induced Enterocolitis Syndrome is not usually a condition that you must suffer with for your entire life. Babies and children typically outgrow FPIES by age 4 at the latest. Sometimes, it will even be resolved by age 2. If you know that you have a child with FPIES, then you should be visiting a doctor regularly to get their opinion, referrals, and resources that you need to help treat them. Once the professional notices that your child’s allergy is improving, they could recommend that you slowly start to reintroduce the offending foods back into your child’s diet. This will help them obtain the proper nutrition that they need for better cellular function. The more diversity that an individual has in their diet, the better their nutrition should be by theory. Parents who are frantically asking “Does FPIES go away for children?” should be reassured that it does. The best treatment for a child that already has FPIES would be to completely eliminate the offending foods in the first place so that healing can start, free from unnecessary inflammation. FPIES and Breastfeeding Connection For some parents, they may find that breastfeeding their baby can cause symptoms of FPIES in their child as well. This is relatively uncommon but could make a lot of sense because food is chemically similar when it is used to make breast milk to provide for your baby’s nutrition. Basically, the food protein is transferred through your breastmilk and into your baby, so this can cause an FPIES reaction in babies as well. The best FPIES breastfeeding diet would be one that takes into account all the possibilities for allergenicity for your baby. You would need to avoid the same foods that your baby reacts to because it can be transmitted to them through breastmilk. Generally, the best diet to consume while you are breastfeeding them would be a healthy diet filled with foods that are gentle and nutritious, like the fruits and vegetables your baby is not allergic to. In addition, healthy protein foods like fish and eggs should benefit the quality of your breast milk as well. Gluten-free grains like rice and tree nuts should not bother an FPIES baby, but should help them obtain the healthy nutrients that they need to heal and develop free from systemic inflammation. How Common is FPIES? Food Protein-Induced Enterocolitis Syndrome is a newly-developing condition that researchers aren’t too sure about at the moment. It is considered a rare disease for young children, and even more rare in adults. However, it seems that instances of FPIES cases are increasing with each year. In 2013, only about 60 babies in the United States were reported to suffer from this condition while in 2017 there are about 3,000 recorded instances. Is there Natural Treatment Available? If you suspect that your child has FPIES, then you should not be relieved by these numbers because the scientific uncertainty surrounding this allergic condition is still unraveling. If you have even come this far to research the possibility of your child having a severe allergic reaction that could lead to death in some cases, you should definitely see a professional for their opinion and network of medical connections that your child can benefit from. An immunologist or allergist may be able to provide excellent advice and information because they have extensively studied conditions like these and did homework about it and stuff like that for years. If you are a more organic, holistic parent then you could consider your child’s nutrition to speed up the recovery process of their body so that they can outgrow the illness. Many parents love formula because it is easy and convenient while supplying many types of nourishment to their baby’s body. Some infant formulas on the market now boast surprisingly advanced and effective, with much more of a benefit for a child than older formulas. For a child with FPIES or with any allergies whatsoever, this PurAmino Hypoallergenic Amino Acid Based Infant Formula Powder does an amazing job at providing bio-absorbable vitamins and minerals, but also essential fats like DHA and ARA found in fresh-caught salmon and algae. This formula is very impressive and can benefit any child’s nutrition and development greatly. Consider Sulfur Deficiency as an Emerging Problem for New Generations I personally like to consider how possible nutrient deficiencies could be affecting my health without me knowing it. A few years ago, I had found out that a sulfur deficiency was affecting my immunity and worsened my allergies intensely. It seems that sources of this stuff in our foods is deficient because of the way civilization has abused fertile soils in agriculture over the decades of industry expansion. Simply, keeping up with having to feed billions of people, and counting, doesn’t allow enough time for soil to naturally replenish sulfur through geological decomposition processes. We are just constantly leaching the soil of its minerals and we aren’t putting them back with fertilizer. The benefits of obtaining enough sulfur in our diet are immense. It’s necessary for proper functioning of the brain, the immune system, and digestive system. Chemicals involved in preventing allergic reactions require sulfur, so this may explain the visible rise in allergies amongst newborns. Doesn’t it seem like they are getting severe and more prevalent? Like every mom is turning into a Pinterest allergy mom with a story about their child’s struggle with a bothersome severe allergy. Thankfully, a bioavailable source of cellular sulfur is available on the market known as organic MSM crystals made of natural sulfur. Perhaps you could benefit your baby’s health by supplementing small amounts of this sulfur supplement in their water. It is tasteless and offers the body’s cells an immediate source of natural sulfur because the chemical form of natural sulfur in your cells is literally MSM itself. Do some research on it if you don’t believe me because it has done wonders for my health, well-being and allergies. If you have a question or something to add about Food Protein-Induced Enterocolitis, then please leave me a comment below! Talk to you soon,<\|endoftext\|>	3.734375
541	# GMAT: Math 49 \| Problem solving \| GMAT \| Khan Academy \| Summary and Q&A 12.1K views December 18, 2008 by GMAT: Math 49 \| Problem solving \| GMAT \| Khan Academy ## TL;DR This video discusses solving GMAT math problems on topics such as coordinate geometry, reverse digits of a number, circle tangents, and parallel resistances, as well as probability calculations. ## Install to Summarize YouTube Videos and Get Transcripts ### Q: How do you find the equation of a line in a coordinate system? To find the equation of a line, you need to determine the slope using the change in y over the change in x, and the y-intercept using a point on the line. The equation is typically written in the form y = mx + b, where m is the slope and b is the y-intercept. ### Q: How do you find the radius of a circle tangent to both axes? Drawing a line from the center of the circle to the x-axis creates a right triangle. The distance from the center to the x-axis is equal to the radius. Using the Pythagorean theorem, you can find the radius in terms of the distance from the origin to the center of the circle. ### Q: What is the formula for calculating combined resistance in parallel resistances? The formula for calculating combined resistance in parallel resistances is 1/r = 1/x + 1/y, where r is the combined resistance and x and y are the individual resistances. ### Q: How do you calculate the probability of a specific event occurring? To calculate the probability of a specific event occurring, you multiply the probabilities of each independent event together. If there are events that should not occur, you subtract the probability of those events from 1 to find the probability of the desired event. ## Summary & Key Takeaways • The video begins with a discussion of how to find the equation of a line in a coordinate system using slope and the y-intercept. • It then moves on to solving a problem involving reversing the digits of a two-digit number and finding the difference between the original and reversed numbers. • The next problem involves finding the radius of a circle tangent to both axes, using the Pythagorean theorem and the distance from the origin to the center of the circle. • The video then covers the concept of parallel resistances in an electric circuit and how to calculate the combined resistance. • Finally, it concludes by solving a probability problem involving three individuals attempting to solve a problem, and finding the probability that two individuals solve it while the third does not.<\|endoftext\|>	4.65625
1,692	Courses Courses for Kids Free study material Free LIVE classes More # Properties of Subtraction of Integers ## An Introduction to Subtraction of Integers Last updated date: 19th Mar 2023 Total views: 33.9k Views today: 0.13k Subtracting integers is the process of finding the difference between two integers. It may result in an increase or a decrease in value, depending on whether the integers are positive or negative. We can subtract one integer from another. This is the easiest way. We can also subtract the sum of two integers, which is also easy to do. But in many cases, we need to do subtraction operations involving more than two numbers. A negative number can be subtracted from a positive number or added to a positive number, for instance. In this article, we are going to learn about the subtraction of integers and their properties. Subtraction of Integers ## What are Integers? An integer is a number with no decimal or fractional part and it includes negative and positive numbers, including zero. A few examples of integers are -5, 0, 1, 5, 8, 97, and 3,043. Adding integers is the process of finding the sum of two or more integers where the value might increase or decrease depending on the integer being positive or negative. Example: Add the given integers: 2 + (-5) Ans: Here, the absolute values of 2 and (-5) are 2 and 5, respectively. Their difference (larger number - smaller number) is 5 - 2 = 3 Now, among 2 and 5, 5 is the larger number, and its original sign is “-”. Hence, the result gets a negative sign, "-”. Therefore, 2 + (-5) = -3 ## Subtraction of Integers Definition The subtraction of integers is an arithmetic operation performed on integers with the same sign or with different signs to find the difference. Subtraction of Integers Example: Subtract the given integers: 7 - 10 Ans: 7 - 10 can be written as (+ 7) - (+)10 Convert the given expression into an addition problem and change the sign of the subtrahend, so, we get: 7 + (-10) Here, the absolute values of 7 and (-10) are 7 and 10, respectively. Their difference (larger number - smaller number) is 10 - 7 = 3. Now, among 7 and 10, 10 is the larger number, and its original sign is “-”. Hence, the result gets a negative sign, "-”. Therefore, 7 - 10 = -3 ## Properties of Addition and Subtraction of Integers This section will be about the properties of addition and subtraction of integers. We will discuss the following properties: • If we subtract 0 from any integer, the answer will be the integer itself. Example: 15 - 0 = 15 • If we subtract any integer from 0, we will find the additive inverse or the opposite of the integer. Example: 0 - 15 = -15 • Subtraction of integers is done by changing the sign of the subtrahend. After this step, if both numbers are of the same sign, then we add the absolute values and attach the common sign. If both the numbers are of different signs, then we find the difference between the absolute numbers and place the sign of the bigger number in the result. Example: (-1) - (-6) = -1 + 6 = 5 • For addition: When both integers have the same signs, add the absolute values of integers, and give the same sign as that of the given integers to the result. Example: 5 + 2 = 7 • For addition: When one integer is positive and the other is negative, find the difference between the absolute values of the numbers and then give the sign of the larger of these numbers to the result. Example: 2 + (-5) = -3 The principles for subtracting integers are displayed in the table below along with examples. Properties of Addition and Subtraction of Integers ## Properties of Addition and Subtraction of Integers Related Solved Examples Q1. Solve: -90 - (-56) Ans: This question is based on subtracting two integers with the same sign. From the question, we have -90 - (-56). This can be written as -90 + 56. Let us find the difference between the absolute values. So, 90 - 56 is 34. Since 90 > 56, the answer sign will be the same as the sign of 90 which is negative. Therefore, -90 - (-56) = -34. Q2. Subtract -7 from -12 using the rules of subtracting integers. Ans: This question is based on subtracting integers with the same sign. Here, we have to subtract two integers with the same sign, -12 and -7. -12 - (-7) = -12 + 7 = -5 Therefore, the difference between -12 and -7 is -5. Q3. Add the given integers: (-2) + 5 Ans: Here, the absolute values of (-2) and 5 are 2 and 5, respectively. Their difference (larger number - smaller number) is 5 - 2 = 3. Now, among 2 and 5, 5 is the larger number, and its original sign is “+”. Hence, the result will be a positive value. Therefore,(-2) + 5 = 3 Q4. Add the following integers: (-9) + (-5) Ans: According to the rules of integers in addition, when both the integers have the same signs, we add the absolute values of integers and give the same sign as that of the given integers to the result. The absolute values of the given integers are 9 and 5. So we will add 9 + 5 = 14 and the sign of the sum will be negative. Therefore, (-9) + (-5) = -14 ## Properties of Addition and Subtraction of Integers Worksheet Here is a worksheet based on the property of addition and subtraction of integers, which will help in increasing the maths confidence of students. Q1. $-10-(-6)=$ Ans: -4 Q2. $18-(2)=$ Ans: 16 Q3. $-20+15=$ Ans: -5 Q4. $-13+(-10)=$ Ans: -23 Q5. $18+(-17)=$ Ans: 1 ## Summary In this article, we have learned about the addition and subtraction of integers. Here, we have seen how by using certain properties we can easily solve problems based on addition and subtraction. As for subtraction, we have to check the sign and then solve it. By applying the rules, it can be solved easily. The same thing can be done while solving the addition of integers. At the end of the article, we have added some solved examples and practice problems to have command over the topic. ## FAQs on Properties of Subtraction of Integers 1. What are the characteristics of subtraction? Subtraction has three properties: • Subtraction has the identity property. • Subtraction's order property. • The property of subtracting a number by itself. 2. What three components makeup subtraction? The components of a subtraction issue are the minuend, subtrahend, and difference. In the subtraction problem, 16 – 7 = 9, the number 16 is the minuend, the number 7 is the subtrahend and the number 9 is the difference. 3. What is minuend and subtrahend? The number being subtracted from is formally referred to as the minuend, while the number being subtracted is known as the subtrahend. The distinction is the outcome.<\|endoftext\|>	4.9375
1,258	Anne Trafton \| MIT News Office Timing is critical for playing a musical instrument, swinging a baseball bat, and many other activities. Neuroscientists have come up with several models of how the brain achieves its exquisite control over timing, the most prominent being that there is a centralized clock, or pacemaker, somewhere in the brain that keeps time for the entire brain. However, a new study from MIT researchers provides evidence for an alternative timekeeping system that relies on the neurons responsible for producing a specific action. Depending on the time interval required, these neurons compress or stretch out the steps they take to generate the behavior at a specific time. “What we found is that it’s a very active process. The brain is not passively waiting for a clock to reach a particular point,” says Mehrdad Jazayeri, the Robert A. Swanson Career Development Professor of Life Sciences, a member of MIT’s McGovern Institute for Brain Research, and the senior author of the study. MIT postdoc Jing Wang and former postdoc Devika Narain are the lead authors of the paper, which appears in the Dec. 4 issue of Nature Neuroscience. Graduate student Eghbal Hosseini is also an author of the paper. One of the earliest models of timing control, known as the clock accumulator model, suggested that the brain has an internal clock or pacemaker that keeps time for the rest of the brain. A later variation of this model suggested that instead of using a central pacemaker, the brain measures time by tracking the synchronization between different brain wave frequencies. Although these clock models are intuitively appealing, Jazayeri says, “they don’t match well with what the brain does.” No one has found evidence for a centralized clock, and Jazayeri and others wondered if parts of the brain that control behaviors that require precise timing might perform the timing function themselves. “People now question why would the brain want to spend the time and energy to generate a clock when it’s not always needed. For certain behaviors you need to do timing, so perhaps the parts of the brain that subserve these functions can also do timing,” he says. To explore this possibility, the researchers recorded neuron activity from three brain regions in animals as they performed a task at two different time intervals — 850 milliseconds or 1,500 milliseconds. The researchers found a complicated pattern of neural activity during these intervals. Some neurons fired faster, some fired slower, and some that had been oscillating began to oscillate faster or slower. However, the researchers’ key discovery was that no matter the neurons’ response, the rate at which they adjusted their activity depended on the time interval required. At any point in time, a collection of neurons is in a particular “neural state,” which changes over time as each individual neuron alters its activity in a different way. To execute a particular behavior, the entire system must reach a defined end state. The researchers found that the neurons always traveled the same trajectory from their initial state to this end state, no matter the interval. The only thing that changed was the rate at which the neurons traveled this trajectory. When the interval required was longer, this trajectory was “stretched,” meaning the neurons took more time to evolve to the final state. When the interval was shorter, the trajectory was compressed. “What we found is that the brain doesn’t change the trajectory when the interval changes, it just changes the speed with which it goes from the initial internal state to the final state,” Jazayeri says. Dean Buonomano, a professor of behavioral neuroscience at the University of California at Los Angeles, says that the study “provides beautiful evidence that timing is a distributed process in the brain — that is, there is no single master clock.” “This work also supports the notion that the brain does not tell time using a clock-like mechanism, but rather relies on the dynamics inherent to neural circuits, and that as these dynamics increase and decrease in speed, animals move more quickly or slowly,” adds Buonomano, who was not involved in the research. The researchers focused their study on a brain loop that connects three regions: the dorsomedial frontal cortex, the caudate, and the thalamus. They found this distinctive neural pattern in the dorsomedial frontal cortex, which is involved in many cognitive processes, and the caudate, which is involved in motor control, inhibition, and some types of learning. However, in the thalamus, which relays motor and sensory signals, they found a different pattern: Instead of altering the speed of their trajectory, many of the neurons simply increased or decreased their firing rate, depending on the interval required. Jazayeri says this finding is consistent with the possibility that the thalamus is instructing the cortex on how to adjust its activity to generate a certain interval. The researchers also created a computer model to help them further understand this phenomenon. They began with a model of hundreds of neurons connected together in random ways, and then trained it to perform the same interval-producing task they had used to train animals, offering no guidance on how the model should perform the task. They found that these neural networks ended up using the same strategy that they observed in the animal brain data. A key discovery was that this strategy only works if some of the neurons have nonlinear activity — that is, the strength of their output doesn’t constantly increase as their input increases. Instead, as they receive more input, their output increases at a slower rate. Jazayeri now hopes to explore further how the brain generates the neural patterns seen during varying time intervals, and also how our expectations influence our ability to produce different intervals. The research was funded by the Rubicon Grant from the Netherlands Scientific Organization, the National Institutes of Health, the Sloan Foundation, the Klingenstein Foundation, the Simons Foundation, the Center for Sensorimotor Neural Engineering, and the McGovern Institute.<\|endoftext\|>	3.8125
562	# Physics Class 12 NCERT Solutions: Chapter 9 Ray Optics and Optical Instruments Part 7 (For CBSE, ICSE, IAS, NET, NRA 2022) Glide to success with Doorsteptutor material for CBSE/Class-12 : get questions, notes, tests, video lectures and more- for all subjects of CBSE/Class-12. Q: 15. Use the mirror equation to deduce that: (A) An object placed between and of a concave mirror produces a real image beyond . (B) A convex mirror always produces a virtual image independent of the location of the object. (C) The virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole. (D) An object placed between the pole and focus of a concave mirror produces a virtual and enlarged image. [Note: This exercise helps you deduce algebraically properties of images that one obtains from explicit ray diagrams.] (A) For a concave mirror, the focal length (f) is negative. When the object is placed on the left side of the mirror, the object distance (u) is negative. For image distance v, we can write the lens formula as: The object lies between and . ( and f are negative) Using equation (1) , we get: is negative, i.e.. , is negative. Therefore, the image lies beyond . (B) For a convex mirror, the focal length (f) is positive. When the object is placed on the left side of the mirror, the object distance (u) is negative. For image distance v, we have the mirror formula: Using equation (2) , we can conclude that: Thus, the image is formed on the back side of the mirror. Hence, a convex mirror always produces a virtual image, regardless of the object distance. (c) For a convex mirror, the focal length (f) is positive. When the object is placed on the left side of the mirror, the object distance (u) is negative, For image distance v, we have the mirror formula: But we have Hence, the image formed is diminished and is located between the focus (f) and the pole. (D) For a concave mirror, the focal length (f) is negative. When the object is placed on the left side of the mirror, the object distance (u) is negative. It is placed between the focus (f) and the pole. For image distance v, we have the mirror formula: The image is formed on the right side of the mirror. Hence, it is a virtual image. For and , we can write: Magnification, Hence, the formed image is enlarged. Developed by:<\|endoftext\|>	4.4375
506	# Dividing Integers 2 teachers like this lesson Print Lesson ## Objective SWBAT recognize and apply rules when dividing integers. #### Big Idea Do you see anything familiar here? Students will use their knowledge of multiplication to help them with the rules of division. ## Launch 5 minutes As student enter the room, they will have a seat, take out their Problem of the Day (POD) sheet and begin to work on the question on the SMARTboard. The prompt for today's lesson is: Are there any similarities between multiplying and dividing? We just developed rules that students can use when multiplying signed integers. I want them to begin to think about the connection between multiplying and dividing as we move to division. If they see similarities, we can make predications about the rules they foresee. If they don't see any simliarities, I will have them write down the differences they indentify in their journals so we can address them as we develop the rules for division (MP7). ## Explore 25 minutes We will continue class with notes on dividing integers-I want to focus on student understanding of operations with signed integers. I want students to recognize the difference between 4/ -2 and 4/2. Do they have a conceptual understanding of what it means to divide by a negative integer? Do students understand what the opposite means? I need to ensure as they work independently that they don't randomly assign a negative value to the quotient of any problem that has a negative sign. A common misconception my students usually have is that if they see a negative sign, the answer is negative. It usually doesn't matter if there is one negative sign, two, or many. If they see a negative sign, the answer is negative. I want to help them move away from memorizing rules to understanding what they remember. Focusing on dividing by negative integers and dividing negative integers will help magnify the rules we are creating and solidify their understanding. I also included some problems in the practice that have answers that are not whole numbers. This will prompt discussion on dividing rational numbers and expressing them as decimals. ## Landing 5 minutes Exit Ticket Is the quotient of 10/ (-3) positive or negative? Explain how you know. This exit ticket will be a formative assessment that will tell me how much the students understand about applying the signs to the quotient. We worked on problems like this when we multiplied integers, I want to see if they are making any connections.<\|endoftext\|>	4.6875
787	has been on Earth for a very long time. Long before the dinosaurs who walked the earth 100 million years ago and even longer than jellyfish who have floated around the oceans for 550 million years . This group of animals are sessile aquatic creatures and for a long time were classified as plants. If you’ve been swimming in the sea or even walked along the beach surveying what the ebbing tide has left behind, you have surely seen one. They are sponges. Simple multicellular organisms that have successfully evolved to colonise all the seas and oceans of the world, including the deep-sea. They can even be found in freshwater aquatic environments such as ponds, lakes, and streams. Sponges are filter feeders who use flagella lined cells to create a current that pushes water into their internal canals, where they remove small organic particles from the water. Today there are over 11,000 described species of sponges and as many as 8,500 thought to be awaiting discovery . The true masters of survival A study by a team of scientist led by researcher Dr. Gordon Love from the University of California analysed the fossil record for traces of a steroid biomarker produced by a common class of sponges . This class of sponges is the demosponges and they make up over 80% of known sponge species. The presence of this steroid biomarker in fossils of known ages confirmed that sponges are one of the evolutionary oldest animals present on our planet today, having survived at least the last 635 million years . These ancient creatures have survived so long because of their simple structure which allows them to adapt and evolve quickly, to many different environments. The cells of sponges are capable of differentiating into functional cell types, however, sponges lack the ‘true’ tissues, organs, and systems associated with more complex animals . Given their long evolutionary history, coupled with a simple organisational structure, it is unsurprising that sponges have developed a large number of symbiotic relationships with aquatic microbes . Aquatic microbes are microscopic organisms present in the water such as bacteria, microalgae, and fungi. These mutually-beneficial relationships have allowed sponges to evolve into many new distinct species with an astonishing variety of shapes, sizes, and forms. The relationship between sponges and microbes is mutually-beneficial as the microbes receive a protected environment and in return carry out a whole range of survival functions for their host sponge, including metabolic function and removal of waste products. The microbe species community within a single sponge is incredibly diverse and abundant, with microbes often accounting for over 40% of the sponge’s volume. There is growing interest among scientists in both sponges and their symbiotic microbes, as these relationships often lead to pharmacologically important substances being produced. The reason why sea sponges are threatened today It is known that the main threat to our oceans and the marine animals is climate change. But given that sponges have survived through countless fluctuations in climates over millions of years, why do these future changes in our oceans pose such a threat? It is, in fact, these tiny microbes, who the sponges have come to so heavily depend on, that are susceptible to slight rises in temperature . Research by Dr. Nicole Webster from the Australian Institute of Marine Science highlights how the delicate equilibrium of the symbiotic relationship between marine hosts and their microbes will be affected by rising sea-surface temperatures. Sea-surface temperatures of 33 degrees Celsius or higher cause a breakdown in the relationship between the sponges and their microbes, where these once helpful microbes become harmful disease causing parasites to their sponge hosts . Based on current climate change projections and the resulting degeneration of microbe-host relationships, the future survival of sponges is an uncertain one at best.<\|endoftext\|>	3.96875
778	###### Norm Prokup Cornell University PhD. in Mathematics Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule. #### Next video playing in 10 Introduction to Polar Coordinates - Problem 1 # Introduction to Polar Coordinates - Concept Norm Prokup ###### Norm Prokup Cornell University PhD. in Mathematics Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule. Share Another form of plotting positions in a plane is using polar coordinates. We are used to using rectangular coordinates, or xy-coordinates. Polar coordinates use a graphing system based on circles, and we specify positions using the radius and angle of a point on a circle centered at the origin. We must also know how to convert from rectangular to polar coordinates and from polar coordinates to rectangular. I want to talk about the polar coordinate system. First, just a review of the rectangular coordinate system. Remember that the rectangular coordinate system has 2 axes, an x axis and a y axis and every point in the plane can be described by a pair of coordinates that gives the x and y coordinates of the point. So every point can be described with a sort of an address. And the polar coordinates system does something similar, only instead of using the x and y axis, it uses the polar axis. This is the polar axis. And it calls the origin the pole. Now, a point let's call it p would be described by r and theta where r is its distance from the origin or the pole. This is r and theta is an angle measured from the polar axis. So this is theta. So every point in the plane can be described this way too. Now one of the interesting things about polar coordinates is that the polar coordinates of a particular point aren't necessarily unique. For example this point 3 coma pi over 2, right. It's 3 units away from the pole and it's an angle of pi over 2 from the polar axis, this point can also be located by 3 coma 5 pi over 2, right? It's 3 units away from the pole but you go round pi over 2 and then another 2 pi and get to the same point. You could also describe this point as -3, negative pi over 2 right? You go negative pi over 2 and then go backwards 3 to get here. So what's important to know is that the same point in the polar coordinate system can be described multiple ways. Right? So that's something to be aware of when you're dealing with polar coordinates. Now another thing to be aware of is how to convert from polar coordinates to rectangular and to do that, we need a little Trigonometry. So let's call this this distance x and this distance y. And then the point p can also be described by the coordinates x and y if this is the origin. Well, power x, y, r and theta are related. Well, for one thing they have the Pythagorean Theorem x squared plus y squared equals r squared. For another, tangent of theta equals y over x. And you could also say that x over r equals cosine theta so x equals r cosine theta and y over r equals sine theta. So y equals r sine theta. These equations are going to become really important in the next couple of lessons where we convert back and forth between rectangular and polar coordinates. So once again, the polar coordinate system, the origin's called the pole. We have one axis called the polar axis and we describe points by listing their distance from the pole and their angle from the polar axis. © 2023 Brightstorm, Inc. All Rights Reserved.<\|endoftext\|>	4.46875
508	This week we experiment with color! We are doing such a FUN experiment. It is called crashing colors. We bet you already know that some colors are known as primary colors (meaning first or most important). These colors are red, blue and yellow. When combined (mixed together) they make other colors. Do you know any colors that are made from any of these pairs of primary colors? Blue + Yellow Makes: Red + Blue Makes: Yellow + Red Makes: Write down what you think these color pairs make in your journal! In this experiment, we will see what happens when the three primary colors “bump” into each other. But here is the thing to remember…DON’T STIR! We’re not going to mix the colors ourselves we are going to let a little detergent do that job for us. Check it out in action: What you will need: Red, Blue, and Yellow liquid food coloring, Liquid Dish Detergent Result timing: 5 minutes What to do: - First fill up the cereal bowl with milk, not quite to the top. - Gently add a few drops of red color into the milk in a small spot at the edge of the dish. - Repeat with blue and yellow, moving around the dish’s edge so that each color is as far from the other two colors as you can make it. DO NOT stir the milk or jiggle the bowl! What do you think is going to happen when you add the detergent? - Write down your prediction in your journal. - Slowly pour a little detergent into the middle of the bowl. - Draw and color a picture in your journal of what you see. How does it work? Just like water, milk is made of molecules, tiny pieces that stick together. They stick so closely that when you put in the food coloring, the food coloring (for the most part) just sits on top of the milk. Scientist call this surface tension, the molecules stick together as if there is an invisible skin across the top of the milk. When you add detergent to the milk, it pulls the milk molecules apart so that the surface tension is weakened. The milk and detergent molecules move around, and so does the food coloring! Where the colors mix, you may see a little green, orange, and purple. Let us know! We would love to see you doing the experiments!<\|endoftext\|>	3.65625

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