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What is CNC Machining: Key Concepts & What to Look for in a Machine Shop
Organizations across industries can enhance their manufacturing processes by leveraging computer-numerical control (CNC) machining. The objective of CNC machining is to create a prototype by cutting a block of material into a specific shape.
CNC machining boasts both financial and production advantages over manufacturing alternatives like conventional machining and 3D printing. It is more cost-effective, more accurate, and a faster process overall.
But the appeal of the process doesn’t end there. Rather than relying on live operators to control the manufacturing functions, CNC machining features pre-programmed software and consoles that oversee the movement of the factory tools at play. The process can be deployed to monitor a broad range of machinery—including but not limited to grinders, mills, and routers—through a specific set of prompts.
CNC machining is especially useful because the manufacturing industry tends to use large quantities of materials, which come in complex shapes and a variety of sizes. Many CNC machines include multiple axes that can accommodate different angles and difficult-to-cut materials.
Developed in the 1940s, the first CNC machines used punched-paper technology instead of the digital software we see today. That said, the process has consistently produced large-scale results with great precision, no matter the application. The computerization component of CNC machining guarantees comprehensive, consistent outcomes.
I. How Does CNC Machining Work?
CNC machining is common in projects that require a high level of precision and repetition. The process can accommodate 3D shapes that—in many cases—are too complex to create via conventional machining. The instructions are written in G-code, which is generated using a form of CAD or CAM software before being fed to computers.
The code is written and revised by programmers, and as such, it can be updated as needed to produce the correct prototype shape and quantity. Upon activating a CNC machining system, the desired cuts and shapes are programmed into the software and relayed to the machinery that will carry out the tasks. The pre-programmed CNC machines feature the exacting, high-speed movements required to customize the prototype.
When the program is loaded, an operator will conduct a test of the code to make sure it’s error-free. This trial run is called “cutting air,” and is meant to protect the CNC machine by reducing the risk of damage. Even the smallest mistakes involving speed or positioning can scrape the CNC machine.
To this end, while the design details are automated in CNC machining—this helps to ensure consistency throughout the production process—a design may require cutters and drills to meet the exact specifications of the prototype. (We’ll go over this in more detail shortly.) A router or spindle will then turn the cutting implement and cut the material.
As the program runs, the process can be repeated with the highest precision. Here are two things of note in terms of how CNC machining works:
1. The CNC machining process involves open- and closed-loop systems.
There are two systems used in CNC machining to manage position control. Open-loop systems run a signal in one direction from the controller to the motor of the machine. Conversely, closed-loop systems are capable of receiving feedback signals, and can therefore correct errors regarding velocity and position.
This means that with minimal force and speed, the process can run through an open-loop control. With more force and greater speed, closed-loop control is needed to oversee the pace, consistency, and precision of the CNC machine. An open-loop system is best used in small-scale projects, while a closed-loop system is ideal for industrial applications.
2. There are a number of different types of CNC machines.
Using a CNC machine, complicated cuts can be achieved in mere minutes. CNC machining is a popular way to customize prototypes made from a variety of materials, allowing the manufacturer to generate complex shapes that would be challenging or even impossible to create manually.
Basic machines tend to move in one or two axes, while more advanced machines may have up to five axes. Multi-axis machines can turn and flip the material automatically, eliminating the need for manual intervention. They are usually more accurate as well.
The different CNC machine types are as follows:
Mills are the most common type of CNC machine. Frequently used in industrial manufacturing, they rely on G-code programs and shape the end prototype by moving across the X, Y, and Z axes.
Lathes are similar to mills, only the tools operate in a circular motion. Like mills, lathes can produce complex designs that would be impossible to create with a manual machine.
- Plasma Cutters
Plasma cutters use a plasma torch to cut hard surfaces like metals. The process involves a combination of compressed-gas air and electrical arcs.
- Water Jet Cutters
Like plasma cutters, water jet cutters shape hard materials like granite and metals—this time through the high-pressure application of water. Water jet cutters offer a cooler alternative for materials that can’t withstand a high-heat process and are often used in the mining and aerospace industries.
- Electrical Discharge Machines
Also known as die-sinking or spark machines, electrical discharge machines use electrical sparks to discharge a current and remove pieces of the material to achieve the desired shape.
Both CNC machining and 3D printing are used to create prototypes from a digital file. CNC machining is considered subtractive manufacturing—that is, a process whereby three-dimensional objects are created by cutting away from a solid block of material.
Meanwhile, 3D printing is known as additive manufacturing and involves the use of digital instructions to create new layers. The process generally uses fused-deposition (FDM) software to create the product.
CNC machining offers the following benefits over 3D printing:
1. CNC machining is more precise than 3D printing.
Many CNC mills have accuracy rates of approximately 0.001 inch. In nearly all cases, CNC machines offer greater precision than 3D printers, in large part because the machines feature a higher tolerance for heat.
3D printers may produce distorted products in high heat, and as such, they cannot guarantee precision in hotter conditions. Meanwhile, CNC machining can accommodate the production of complex prototypes in a variety of shapes and sizes, in nearly any production environment.
2. CNC machining is faster than 3D printing.
CNC machines can operate around the clock when they’re properly maintained, and represent a much faster solution than 3D printing. Conversely, 3D printers may need to slow their pace to achieve the right design. They are less efficient than CNC machines when producing large quantities of prototypes.
Ultimately, CNC machines are better equipped to test prototypes because they can quickly build a design for developers to test. (It would take a 3D printer significantly longer to conduct the same test.) And the differences in speed don’t end there, because 3D printing requires extra work after the prototypes are built. In 3D printing the products must be washed, polished, and sealed before they’re available for people to use, which simply isn’t the case in CNC machining. With a CNC machine, the prototype becomes ready for use on completion of the fabrication process.
3. CNC machining is more versatile than 3D printing.
CNC machines are much more versatile than 3D printers. First of all, they feature a range of quality settings, which—in contrast to 3D printers—can create prototypes with rougher designs in some parts of the material and smooth designs in others.
The machines can also accommodate heavier materials than 3D printers, making them ideal for engines, aircraft, and other machines that require durability. In turn, most cutting tools can be used in any CNC machine, while 3D printers struggle to produce products that need to withstand extreme conditions.
Further, 3D printers cannot switch between materials—yet CNC machines can handle all kinds of materials, including soft and hardwoods, acrylic, thermoplastics, machining wax, metal alloys such as aluminum, steel, brass, and copper. Granted, different materials require different tools, but the tools involved in CNC machining can be swapped with ease.
In summary, CNC machines offer greater precision, speed, and versatility in large-scale production environments than 3D printers. When it comes to efficiency and accuracy, CNC machines offer a compelling manufacturing solution.
Companies across many industries outsource their CNC machining needs—usually to save time, money, and stress. These are just some of the industries that use precision CNC machining:
If you are involved in one of these industries—or perhaps another industry, such as publishing or hospitality, or even the military—you may have CNC machining work you would like to outsource.
You wouldn’t want to waste money investing in the wrong items or spend valuable time on a complex process when you could be focusing on your core business. By hiring a machine shop, experienced machinists will fulfill your precision machining needs for you.
When you outsource your CNC machining work, you can avoid the costly, time-consuming setup and maintenance the process entails. A company like Nexus Automation can produce exactly what you need in the right quantities as your organization works to bring its products to market.
Here are three reasons to outsource your precision CNC machining:
1. Save money.
The costs associated with in-house CNC machining are accompanied by marked operating expenses and high housing prices. For one-off or short-term projects in particular, outsourcing ought not be overlooked. Hiring a precision CNC machine shop will allow you to invest your capital where it matters most, and create a quality product as efficiently as possible.
By working with a machine shop, you can cut your CNC machining costs in the long term—specifically by not having to purchase the equipment, tools, or even the materials you need to get the job done.
2. Save time.
Hiring a CNC machining company will save you time because the shop will know exactly what resources they need to bring your project to fruition.
On the other hand, when using CNC machines in-house, you will need to make certain you have the right machinery, software, and materials to create your prototype. This is easier said than done.
Say you intend to use a CNC mill to transform aluminum into your desired shape. What you might not know is that you will need a specific set of tools and applications to produce your prototype. And if for your next prototype you intend to swap materials and use hardwood instead, you’ll need an entirely different set of tools and applications.
3. Enjoy expert support.
If you are working in-house and the CNC machining parts you’ve designed come out flawed, you’ll have to figure out a solution. However, if you’ve hired a precision machine shop to take the reins, they’ll work through any issues that arise and offer excellent customer service along the way.
When you hire a CNC machining company, you are paying for constant support, skilled labor, and tailored service. At Nexus Automation, our expert machinists have the knowledge, skills, and best practices to deliver a high-quality product in a short time period.
IV. 5 Things to Look for in a CNC Machine Shop
So we’ve discussed the benefits of outsourcing your CNC machining needs. Now let’s explore what to look for in a precision machine shop. While references and testimonials can be very helpful, and the company’s portfolio will offer deep insights into the quality of their work, there are other factors you should pay attention to when searching for a CNC machining company.
Five of these factors are as follows:
1. Variety in machines.
Does the company have the machines you need to complete your prototype? Make sure the shop you hire has the right in-house machinery—you wouldn’t want to hire one company, only to realize you need to outsource a different part of the job to another organization later down the road. If you work with both steel and plastic, for instance, search for a machine shop that is equipped to handle both materials. Nexus Automation has our clients covered in this way.
2. Organized workflow.
Make sure the CNC machine shop you hire has an organized, productive labor force. State-of-the-art equipment doesn’t mean much without a team of talented laborers. The right machine shop will include experienced machinists who are trained to work in a quality, streamlined, customer-oriented manner. At Nexus Automation, our machinists have been meticulously trained in the CNC machining process—not only do they have decades of experience, but they can also employ our cutting-edge equipment with ease.
3. Engineering capabilities.
You may create your own designs. Nonetheless, the CNC machine shop you work with should have engineering capabilities of its own. Companies should search for a supplier with extensive engineering, design, and manufacturing experience so the contractor can make improvements and define any practical limitations that may arise during the machining process. Collaboration and knowledge are key here—and an industry-leading precision machine shop like Nexus Automation will have you covered.
4. ISO certification.
Check that your CNC machine shop is ISO-certified, which will guarantee they’re equipped to provide the level of quality you deserve. Nexus Automation, for instance, features ISO 9001:2015 and ISO 13485:2016 certifications, which we believe speaks to our quality management approach. By definition, ISO certification means the machine shop exceeds all expectations in the way of quality. And at Nexus, we streamline our processes and accelerate our clients’ projects without compromising the work we deliver.
5. Capacity for quick-turn and small production runs.
Quick-turn and small production runs reflect the machine shop’s dedication, flexibility, and speed. We, at Nexus Automation, take pride in our ability to execute quickly and efficiently, no matter the project scope. Through our CNC machining processes and people, we guarantee seamless, defect-free prototypes in small volumes—delivered as fast as we can handle. Our world-class project management also includes an encrypted secure server, SolidWorks PDM, and EPICOR ERP, all of which translate to seamless integration in our clients’ supply chains and timeframes.
V. Precision Machining with Nexus Automation
If you are searching for a CNC machine shop, look no further than Nexus Automation. We are a small business—not to mention a certified minority- and woman-owned business—that works locally with clients throughout the Bay Area.
Our CNC machines are designed to fulfill a variety of custom services, ranging from machined components to entire assemblies. We guarantee quick time-to-market and compliance for even the most demanding specs, and produce sophisticated parts from a range of materials.
Not only that, but our expert machinists will ensure your custom parts fit and function exactly as you designed them. The design will remain in our system indefinitely, and can be updated at any time you choose.
Our clients—who range from early-stage startups to Fortune 500 companies in the biotechnology, pharmaceuticals, medical device, automotive, solar, semi-conductor and other industries—can meet the most stringent regulatory standards due to the ISO 9001:2015 and ISO 13485:2016 certifications we discussed previously. Being ISO-certified makes us an appealing partner in highly regulated sectors, and our focus on quality promotes stress-free international expansion.
What’s more, clients need not worry about quality audits and managing processes when they work with us. Nexus Automation will oversee every aspect of their CNC machining.
But most importantly, our CNC machine shop will create the prototype you need to your exact specifications, complete with high dimensional accuracy and superior finishing. With us by your side, you can expect reliable, safe components from a true partner.<|endoftext|>
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Physics is the art of leaving behind common assumptions in order to see everyday events as the result of simple natural causes and of describing those events in mathematical approximations. In every physics course our overarching goal is to nurture students’ ability to observe normal, everyday phenomena with more attention and in greater detail, in order to glimpse the larger patterns of the workings of the universe.
I. Students in distribution courses (non-science majors)
Physics emerged from the original artes physicae of arithmetic, geometry, astronomy, and music of the seven liberal arts of the classical curriculum and still stands solidly within the liberal arts today. The study of physics promotes understanding of the basic workings of nature. It also extends that understanding beyond the realms of our everyday lives, from the scales of atomic nuclei to those of galaxies. It is essential for understanding the role of technology in society as well as important policy issues. In all of our distribution courses our goal is that students demonstrate an understanding of the nature of physics and scientific evidence by focusing on the habits of mind of the physicist or astronomer. With that in mind students in our distribution courses should be able to demonstrate an understanding of the following key features of physics:
- the assumption that the processes of the world can be studied with sensory data which can be used to build predictive models of the universe;
- that experimentation requires a disciplined approach to the acquisition of data, which allows physicists to refine and evaluate their own models;
- that physics is an ongoing community activity which requires public presentation of results and the critique of results by independent researchers, and that disagreements among researchers and challenges to “conventional wisdom” are constant and important aspects of the process.
II. Science majors in required introductory physics courses
A number of other degree programs recognize the importance of a solid grounding in the observations and habits of thought that we use in physics. Our department has traditionally spent the bulk of our resources providing courses that fulfill this mission. In addition to the goals we have for the distribution courses, we also ask of these students a deeper understanding of the content and methodology of the discipline. In particular they should be able to demonstrate
- a basic understanding of the core concepts of physics, including mechanics, electricity and magnetism, wave motion and optics, heat and thermodynamics, and modern physics;
- some understanding of the perspective and values of physics as it differs from other natural sciences;
- skills in problem solving techniques requiring the reduction of complex situations to essential elements and the application of physical laws in their appropriate contexts;
- an awareness of uncertainty in laboratory experiments where interpretation and analysis of the data are tied specifically to physical laws studied in the classroom; and
- skills in the understanding, use, and limitations of graphical representations of data, emphasizing that physics equations are expressions of relationships among physical quantities.
III. Physics Majors
Our curriculum is designed to allow our physics majors to succeed in graduate programs in physics and associated disciplines as well as a variety of other careers. We seek to give our majors a solid grounding in physics as a liberal art, as well as theories and laboratory practices in physics that will enable them to pursue whatever career they choose. Toward this end, our specific goals for physics majors are that they develop and be able to demonstrate
- a deep understanding of the common concepts of physics developed through a core curriculum of physics courses consisting of introductory physics, modern (20th century) physics, classical mechanics, electricity and magnetism, and quantum mechanics;
- an appreciation of the history of physics as a process of continually refining and revising our understanding of the world that has resulted in our current understanding of the universe;
- the ability to apply their knowledge of this core curriculum to solve problems;
B. Experimental techniques and methods
- the ability to design an experiment to test a model;
- the ability to quantify uncertainty in measurement and to follow the propagation of uncertainty through a calculation;
- the ability to build and refine a simple model from the analysis of data;
C. The methods of mathematical models
- an understanding of mathematical relationships and a facility with mathematical manipulations from which physics models are built and physics problems are solved;
- the ability to use the computer to collect, analyze, and display data;
D. The Culture of Science
- the ability to communicate their own work effectively to appropriate audiences including oral presentations, lab reports, and lab notebooks.
Majors’ courses demand increasing levels of mathematical rigor as the student progresses through the 4 years, while the laboratory courses require increasing levels of independence. These culminate in the required senior project (SYE), where ideally the student draws on all or most of the skills listed above.<|endoftext|>
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## Tuesday, February 19, 2013
### More on quadratic equations and parabolas
This week my studies focused on the vertex of the graph of a quadratic equations. The vertex of a parabola is the point in which the graph changes direction. Imagine the highest point of a roller coaster which you climb before plummeting downwards. That is the vertex. Now imagine a slope between two mountains, the vertex would be the point where you stop coming down one of the mountains and start coming up the other. We can find the vertex of a quadratic equation in standard form by transforming it to vertex form.
The way the vertex transformation works is the following: If you can get your quadratic equation in the form of y =a(x-h)² + k, remembering that (x-h) is equal to (x+(-h)), then (h,k) is the vertex of that equation. If a is positive, then the vertex is the minimum point the parabola reaches before going up, if a is negative then the vertex is the maximum point the parabola reaches before going down.
The vertex is an important point to know since if someone gives you the vertex and another point in the parabola, you can get the standard equation for it. For example, if I know the vertex of my parabola is (3,-4) and that point (2,-3) is in the parabola we can use algebra to get the equation of it.
y =a(x-h)² + k
y =a(x-3)² + (-4) substituting the given vertex into the equation.
-3 =a(2-3)² + (-4) substituting the (x,y) point given we the solve for a.
1=a or a=1
Therefore the equation of the parabola is (since a=1, we omit it):
y =(x-3)² - 4
Or (x-3)(x-3) -4
which gives us the standard equation x²-6x+5.
Now that is cool.
It seems most of my time while going through pre-calculus in order to get to calculus has been spent on quadratic equations. I did not remember there was so much to learn from them. Whenever I am sure I am done with parabolas something else comes up. Ad that something else is amazing to understand.
Right now when I look at x2-6x+5, I can:
• Look at the standard form of that quadratic equation and tell at what point the graph passes the y axis (y intercept) and know whether the graph is concave up (like a cup) or concave down (like and umbrella)l
• Solve quadratic equations using the quadratic formula and thus getting the points were the graph crosses the x axis (x intercepts).
• Use the discriminant in the quadratic formula to know wether the equation has rational, irrational or imaginary solutions as well as the number of x intercepts.
• Use factoring (when possible) to get the same results as with the quadratic equation.
• Transform that standard form to the vertex form (y = a(x-h)² + k) to find the vertex or the point where the parabola changing direction (its maximum or minimum). I will talk about this form later in the post.
And I can graph it since I know these points:
Y intercept is 5 so (0,5)
2 X intercepts 1 and 5 so (1,0) and (5,0)
The vertex is (3,-4)
Let's graph it:
Notice I added a new point at the top right corner, (6,5). I knew that point should be there since a parabola is simetrical across its vertex. In other words, one side is a mirror image of the other. And since point (0,5) was on the left side, then point (6,5) must exist in the right. If you want to be sure, just plug 6 in the formula substituting it for x. I'll wait until you do so.
Going back to the vertex transformation, that transformation got me to think that someone had to discover this relationship between quadratic equations and their graphs through serious and hard study. I must remind myself that all these formulas, equivalencies, and transformations I am now given, where not always available. It is easy to forget that someone put a lot of effort coming up with these "shortcuts" we now use is class. I thank immensely all those mathematical scholars who made this learning possible.
If you liked this quadratic stuff as much as I did let me know.<|endoftext|>
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C is a popular programming language which is commonly used by scientists and engineers to write programs for any specific application. C is also a widely accepted programming language in the software industries. This beginner’s guide to computer programming is for student programmers to effectively write programs for solving numerical problems. All that is required of a beginner programmer is not experience in computing but interest in computing. The programs illustrated in the book have been accumulated, experimented and tested by the author during his teaching of the subject to a few thousand students in over a decade. In addition, numerous problems are adapted form university question papers. Short questions and answers and objective questions are an added feature. All these would build confidence of the students and those appearing for interview/viva voce in a practical lab. The special topic of the book is C graphics and animation which helps students develop simple programs to generate geometrical and graphical objects.<|endoftext|>
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A neuron (also known as a neurone or nerve cell) is an electrically excitable cell that processes and transmits information through electrical and chemical signals. These signals between neurons occur via synapses, specialized connections with other cells.
Neurons can connect to each other to form neural networks. Neurons are the core components of the nervous system, which includes the brain, spinal cord–which together comprise the central nervous system (CNS)–and the ganglia of the peripheral nervous system (PNS) .
Specialized types of neurons include: sensory neurons which respond to touch, sound, light and all other stimuli affecting the cells of the sensory organs that then send signals to the spinal cord and brain, motor neurons that receive signals from the brain and spinal cord to cause muscle contractions and affect glandular outputs, and interneurons which connect neurons to other neurons within the same region of the brain, or spinal cord in neural networks.
A typical neuron possesses a cell body (soma), dendrites, and an axon. The term neurite is used to describe either a dendrite or an axon, particularly in its undifferentiated stage. Dendrites are thin structures that arise from the cell body, often extending for hundreds of micrometres and branching multiple times, giving rise to a complex “dendritic tree”. An axon is a special cellular extension that arises from the cell body at a site called the axon hillock and travels for a distance, as far as 1 meter in humans or even more in other species.
The cell body of a neuron frequently gives rise to multiple dendrites, but never to more than one axon, although the axon may branch hundreds of times before it terminates. At the majority of synapses, signals are sent from the axon of one neuron to a dendrite of another. There are, however, many exceptions to these rules: neurons that lack dendrites, neurons that have no axon, synapses that connect an axon to another axon or a dendrite to another dendrite, etc.
All neurons are electrically excitable, maintaining voltage gradients across their membranes by means of metabolically driven ion pumps, which combine with ion channels embedded in the membrane to generate intracellular-versus-extracellular concentration differences of ions such as sodium, potassium, chloride, and calcium.
Changes in the cross-membrane voltage can alter the function of voltage-dependent ion channels. If the voltage changes by a large enough amount, an all-or-none electrochemical pulse called an action potential is generated, which travels rapidly along the cell’s axon, and activates synaptic connections with other cells when it arrives.
Neurons do not undergo cell division. In most cases, neurons are generated by special types of stem cells. A type of glial cell, called astrocytes (named for being somewhat star-shaped), have also been observed to turn into neurons by virtue of the stem cell characteristic pluripotency. In humans, neurogenesis largely ceases during adulthood; but in two brain areas, the hippocampus and olfactory bulb, there is strong evidence for generation of substantial numbers of new neurons.
The features that define a neuron are electrical excitability and the presence of synapses, which are complex membrane junctions that transmit signals to other cells. The body’s neurons, plus the glial cells that give them structural and metabolic support, together constitute the nervous system. In vertebrates, the majority of neurons belong to the central nervous system, but some reside in peripheral ganglia, and many sensory neurons are situated in sensory organs such as the retina and cochlea.
Although neurons are very diverse and there are exceptions to nearly every rule, it is convenient to begin with a schematic description of the structure and function of a “typical” neuron. A typical neuron is divided into three parts: the soma or cell body, dendrites, and axon.
The soma is usually compact; the axon and dendrites are filaments that extrude from it. Dendrites typically branch profusely, getting thinner with each branching, and extending their farthest branches a few hundred micrometers from the soma. The axon leaves the soma at a swelling called the axon hillock, and can extend for great distances, giving rise to hundreds of branches.
Unlike dendrites, an axon usually maintains the same diameter as it extends. The soma may give rise to numerous dendrites, but never to more than one axon. Synaptic signals from other neurons are received by the soma and dendrites; signals to other neurons are transmitted by the axon. A typical synapse, then, is a contact between the axon of one neuron and a dendrite or soma of another.
Synaptic signals may be excitatory or inhibitory. If the net excitation received by a neuron over a short period of time is large enough, the neuron generates a brief pulse called an action potential, which originates at the soma and propagates rapidly along the axon, activating synapses onto other neurons as it goes. This is called saltatory conduction.
Many neurons fit the foregoing schema in every respect, but there are also exceptions to most parts of it. There are no neurons that lack a soma, but there are neurons that lack dendrites, and others that lack an axon. Furthermore, in addition to the typical axodendritic and axosomatic synapses, there are axoaxonic (axon-to-axon) and dendrodendritic (dendrite-to-dendrite) synapses.
The key to neural function is the synaptic signaling process, which is partly electrical and partly chemical. The electrical aspect depends on properties of the neuron’s membrane. Like all animal cells, the cell body of every neuron is enclosed by a plasma membrane, a bilayer of lipid molecules with many types of protein structures embedded in it.
A lipid bilayer is a powerful electrical insulator, but in neurons, many of the protein structures embedded in the membrane are electrically active. These include ion channels that permit electrically charged ions to flow across the membrane, and ion pumps that actively transport ions from one side of the membrane to the other. Most ion channels are permeable only to specific types of ions.
Some ion channels are voltage gated, meaning that they can be switched between open and closed states by altering the voltage difference across the membrane. Others are chemically gated, meaning that they can be switched between open and closed states by interactions with chemicals that diffuse through the extracellular fluid.
The interactions between ion channels and ion pumps produce a voltage difference across the membrane, typically a bit less than 1/10 of a volt at baseline. This voltage has two functions: first, it provides a power source for an assortment of voltage-dependent protein machinery that is embedded in the membrane; second, it provides a basis for electrical signal transmission between different parts of the membrane.
Neurons communicate by chemical and electrical synapses in a process known as neurotransmission, also called synaptic transmission. The fundamental process that triggers the release of neurotransmitters is the action potential, a propagating electrical signal that is generated by exploiting the electrically excitable membrane of the neuron. This is also known as a wave of depolarization.
Neurons are highly specialized for the processing and transmission of cellular signals. Given their diversity of functions performed in different parts of the nervous system, there is, as expected, a wide variety in their shape, size, and electrochemical properties. For instance, the soma of a neuron can vary from 4 to 100 micrometers in diameter.
- The soma is the body of the neuron. As it contains the nucleus, most protein synthesis occurs here. The nucleus can range from 3 to 18 micrometers in diameter.
- The dendrites of a neuron are cellular extensions with many branches. This overall shape and structure is referred to metaphorically as a dendritic tree. This is where the majority of input to the neuron occurs via the dendritic spine.
- The axon is a finer, cable-like projection that can extend tens, hundreds, or even tens of thousands of times the diameter of the soma in length. The axon carries nerve signals away from the soma (and also carries some types of information back to it). Many neurons have only one axon, but this axon may—and usually will—undergo extensive branching, enabling communication with many target cells.
- The part of the axon where it emerges from the soma is called the axon hillock. Besides being an anatomical structure, the axon hillock is also the part of the neuron that has the greatest density of voltage-dependent sodium channels. This makes it the most easily excited part of the neuron and the spike initiation zone for the axon: in electrophysiological terms it has the most negative action potential threshold. While the axon and axon hillock are generally involved in information outflow, this region can also receive input from other neurons.
The axon terminal contains synapses, specialized structures where neurotransmitter chemicals are released to communicate with target neurons.
Although the canonical view of the neuron attributes dedicated functions to its various anatomical components, dendrites and axons often act in ways contrary to their so-called main function.
Axons and dendrites in the central nervous system are typically only about one micrometer thick, while some in the peripheral nervous system are much thicker. The soma is usually about 10–25 micrometers in diameter and often is not much larger than the cell nucleus it contains. The longest axon of a human motoneuron can be over a meter long, reaching from the base of the spine to the toes.
Sensory neurons have axons that run from the toes to the dorsal columns, over 1.5 meters in adults. Giraffes have single axons several meters in length running along the entire length of their necks. Much of what is known about axonal function comes from studying the squid giant axon, an ideal experimental preparation because of its relatively immense size (0.5–1 millimeters thick, several centimeters long).
Fully differentiated neurons are permanently postmitotic; however, recent research shows that additional neurons throughout the brain can originate from neural stem cells found throughout the brain but in particularly high concentrations in the subventricular zone and subgranular zone through the process of neurogenesis.
Action on other Neurons
A neuron affects other neurons by releasing a neurotransmitter that binds to chemical receptors. The effect upon the postsynaptic neuron is determined not by the presynaptic neuron or by the neurotransmitter, but by the type of receptor that is activated.
A neurotransmitter can be thought of as a key, and a receptor as a lock: the same type of key can here be used to open many different types of locks. Receptors can be classified broadly as excitatory (causing an increase in firing rate), inhibitory (causing a decrease in firing rate), or modulatory (causing long-lasting effects not directly related to firing rate).
The two most common neurotransmitters in the brain, glutamate and GABA, have actions that are largely consistent. Glutamate acts on several different types of receptors, and have effects that are excitatory at ionotropic receptors and a modulatory effect at metabotropic receptors. Similarly GABA acts on several different types of receptors, but all of them have effects (in adult animals, at least) that are inhibitory.
Because of this consistency, it is common for neuroscientists to simplify the terminology by referring to cells that release glutamate as “excitatory neurons”, and cells that release GABA as “inhibitory neurons”. Since over 90% of the neurons in the brain release either glutamate or GABA, these labels encompass the great majority of neurons. There are also other types of neurons that have consistent effects on their targets, for example “excitatory” motor neurons in the spinal cord that release acetylcholine, and “inhibitory” spinal neurons that release glycine.
The distinction between excitatory and inhibitory neurotransmitters is not absolute, however. Rather, it depends on the class of chemical receptors present on the postsynaptic neuron. In principle, a single neuron, releasing a single neurotransmitter, can have excitatory effects on some targets, inhibitory effects on others, and modulatory effects on others still. For example, photoreceptor cells in the retina constantly release the neurotransmitter glutamate in the absence of light.
So-called OFF bipolar cells are, like most neurons, excited by the released glutamate. However, neighboring target neurons called ON bipolar cells are instead inhibited by glutamate, because they lack the typical ionotropic glutamate receptors and instead express a class of inhibitory metabotropic glutamate receptors. When light is present, the photoreceptors cease releasing glutamate, which relieves the ON bipolar cells from inhibition, activating them; this simultaneously removes the excitation from the OFF bipolar cells, silencing them.
It is possible to identify the type of inhibitory effect a presynaptic neuron will have on a postsynaptic neuron, based on the proteins the presynaptic neuron expresses. Parvalbumin-expressing neurons typically dampen the output signal of the postsynaptic neuron in the visual cortex, whereas somatostatin-expressing neurons typically block dendritic inputs to the postsynaptic neuron.
Neurons communicate with one another via synapses, where the axon terminal or en passant boutons (terminals located along the length of the axon) of one cell impinges upon another neuron’s dendrite, soma or, less commonly, axon. Neurons such as Purkinje cells in the cerebellum can have over 1000 dendritic branches, making connections with tens of thousands of other cells; other neurons, such as the magnocellular neurons of the supraoptic nucleus, have only one or two dendrites, each of which receives thousands of synapses. Synapses can be excitatory or inhibitory and either increase or decrease activity in the target neuron.
In a chemical synapse, the process of synaptic transmission is as follows: when an action potential reaches the axon terminal, it opens voltage-gated calcium channels, allowing calcium ions to enter the terminal. Calcium causes synaptic vesicles filled with neurotransmitter molecules to fuse with the membrane, releasing their contents into the synaptic cleft.
The neurotransmitters diffuse across the synaptic cleft and activate receptors on the postsynaptic neuron. High cytosolic calcium in the axon terminal also triggers mitochondrial calcium uptake, which, in turn, activates mitochondrial energy metabolism to produce ATP to support continuous neurotransmission.
The human brain has a huge number of synapses. Each of the 1011 (one hundred billion) neurons has on average 7,000 synaptic connections to other neurons. It has been estimated that the brain of a three-year-old child has about 1015 synapses (1 quadrillion). This number declines with age, stabilizing by adulthood. Estimates vary for an adult, ranging from 1014 to 5 x 1014 synapses (100 to 500 trillion).
Mechanisms for Propagating Action Potentials
In 1937, John Zachary Young suggested that the squid giant axon could be used to study neuronal electrical properties. Being larger than but similar in nature to human neurons, squid cells were easier to study. By inserting electrodes into the giant squid axons, accurate measurements were made of the membrane potential.
The cell membrane of the axon and soma contain voltage-gated ion channels that allow the neuron to generate and propagate an electrical signal (an action potential). These signals are generated and propagated by charge-carrying ions including sodium (Na+), potassium (K+), chloride (Cl−), and calcium (Ca2+).
There are several stimuli that can activate a neuron leading to electrical activity, including pressure, stretch, chemical transmitters, and changes of the electric potential across the cell membrane. Stimuli cause specific ion-channels within the cell membrane to open, leading to a flow of ions through the cell membrane, changing the membrane potential.
Thin neurons and axons require less metabolic expense to produce and carry action potentials, but thicker axons convey impulses more rapidly. To minimize metabolic expense while maintaining rapid conduction, many neurons have insulating sheaths of myelin around their axons. The sheaths are formed by glial cells: oligodendrocytes in the central nervous system and Schwann cells in the peripheral nervous system.
The sheath enables action potentials to travel faster than in unmyelinated axons of the same diameter, whilst using less energy. The myelin sheath in peripheral nerves normally runs along the axon in sections about 1 mm long, punctuated by unsheathed nodes of Ranvier, which contain a high density of voltage-gated ion channels. Multiple sclerosis is a neurological disorder that results from demyelination of axons in the central nervous system.
Some neurons do not generate action potentials, but instead generate a graded electrical signal, which in turn causes graded neurotransmitter release. Such nonspiking neurons tend to be sensory neurons or interneurons, because they cannot carry signals long distances.
History of the Neuron
The term neuron was coined by the German anatomist Heinrich Wilhelm Waldeyer. The neuron’s place as the primary functional unit of the nervous system was first recognized in the early 20th century through the work of the Spanish anatomist Santiago Ramón y Cajal.
Ramón y Cajal proposed that neurons were discrete cells that communicated with each other via specialized junctions, or spaces, between cells. This became known as the neuron doctrine, one of the central tenets of modern neuroscience. To observe the structure of individual neurons, Ramón y Cajal improved a silver staining process known as Golgi’s method, which had been developed by his rival, Camillo Golgi.
Cajal’s improvement, which involved a technique he called “double impregnation”, is still in use. The silver impregnation stains are an extremely useful method for neuroanatomical investigations because, for reasons unknown, it stains a very small percentage of cells in a tissue, so one is able to see the complete micro structure of individual neurons without much overlap from other cells in the densely packed brain.
The neuron doctrine is the now fundamental idea that neurons are the basic structural and functional units of the nervous system. The theory was put forward by Santiago Ramón y Cajal in the late 19th century. It held that neurons are discrete cells (not connected in a meshwork), acting as metabolically distinct units.
Later discoveries yielded a few refinements to the simplest form of the doctrine. For example, glial cells, which are not considered neurons, play an essential role in information processing. Also, electrical synapses are more common than previously thought, meaning that there are direct, cytoplasmic connections between neurons. In fact, there are examples of neurons forming even tighter coupling: the squid giant axon arises from the fusion of multiple axons.
Ramón y Cajal also postulated the Law of Dynamic Polarization, which states that a neuron receives signals at its dendrites and cell body and transmits them, as action potentials, along the axon in one direction: away from the cell body. The Law of Dynamic Polarization has important exceptions; dendrites can serve as synaptic output sites of neurons and axons can receive synaptic inputs.
Top Illustration: “Blausen 0657 MultipolarNeuron” by BruceBlaus – Own work. Licensed under CC BY 3.0 via Wikimedia Commons<|endoftext|>
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Given below is a description of three of the most influential Republicans in the history of the United States. These members of the Republican Party represent its political motivations and beliefs during different eras.
The first Republican and 16th President of the United States, Lincoln is one of the most famous leaders of the country. He was part of the Whig Party until the formation of the Republican Party in 1854, whose anti-slavery beliefs attracted the young man and got him involved in politics once again after a lull. He was chosen as the Presidential candidate for the 1860 elections for his more moderate views on slavery and because he was from Kentucky, one of the Western States.
He won the election with almost no support from the South, and soon afterwards a number of Southern States declared themselves as the Confederate States of America, triggering the chain of events leading to the Civil War. In 1863, he gave his famous Emancipation Proclamation, ordering the freedom of all slaves in the rebelling states. While freedom was not immediate granted, it was a huge moment for slavery abortion in the United States. The same year saw the famous Gettysburg Address being made by Lincoln, which was one of the most significant speeches of American history.
Lincoln’s views reflected that of the Republican Party during its early years; a liberal viewpoint that focuses on the abolishment of slavery, resulting in the Thirteenth Amendment of 1865 that made slavery illegal.
Theodore ‘Teddy’ Roosevelt was the 26th U.S. President and one of the most famous ones in American history. He served under the 1st U.S. Volunteer Cavalry, called Roughriders, during the Spanish-American War. He joined the Republican Party and, on becoming President, focuses his efforts on making the United States a powerful nation. He described his foreign policy with the proverb ‘Speak softly but carry a big stick’, and he played a major role in the expansion of the U.S. Navy. He sent naval envoys to various countries as a demonstration of the country’s power.
Roosevelt was extremely active in foreign policy, building the Panama Canal and ending the war between Japan and Russia in 1905. He won the Nobel Peace Prize the following year for his efforts. Within the country, he strived to give American equal opportunities for success under his ‘Square Deal’ policies. These included regulation of big businesses, support of labor laws to protect the working class, protection of the environment, and regulation of drugs and meat. His policies formed the foundation for the expansion of the government later on, a belief that the Republican Party does not share any longer.
A successful second leading man in movies, Ronald Reagan became increasingly active in politics towards the end of his acting career. He later became the eldest man to be elected President of the United States during the period from 1981 to 1989. A man of strong conservative values, Reagan was a Democrat until 1962, when he shifted to the Republican Party. This was during a time when the Democrats were becoming more liberal and the Republicans more conservative. Reagan is known for his aggressive policy against Russia, created the biggest peacetime military buildup in the history of the U.S. This buildup significantly increased the country’s foreign debts.
Towards the end of his political career, he became friendlier towards Russia as he developed a strong friendship with his Russian counterpart, Mikhail Gorbachev. He was also famous for his ‘Tear Down This Wall’ speech at the Brandenburg Gate in West Germany, expressing his desire for peace between the Soviet Union and the West.
- The Republican Party Platform
- History of the Republican Party
- Republican Presidents
- Republican Party Beliefs
- The Birth Of The Republican Party
- What is the GOP Convention?
- Donald Trump on Foreign Policy
- Democratic Views On Foreign Policy
- Republican Views On Foreign Policy
- Democratic Views on Big Business<|endoftext|>
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# Coterminal Angle
How to find the coterminal angle of the given angle: definition, formula, 5 examples, and their solutions.
## Definition
### Definition
The angle on a coordinate is formed by
the x-axis and the terminal side.
So, coterminal angles are the angles
that have the same terminal side.
This angle θ and below angles
are coterminal angles
because they have the same terminal side.
## Formula
### Formula
As you can see,
coterminal angles have 360⋅n part.
(360⋅n means the number of rotation counterclockwise.)
So, to find the coterminal angles,
to the given angle θ.
(n is an integer.)
360⋅n: 360, 720, 1080, ...
2π⋅n: 2π, 4π, 6π, ...
## Example 1
### Solution
To find the coterminal angles,
(Choose any integer n.)
Put 1 into the n.
Then the first coterminal angle of 60º is
360⋅1 + 60 degrees.
360 + 60 = 420
So 420º is the first coterminal angle.
Put 2 into the n.
Then the next coterminal angle is
360⋅2 + 60 degrees.
360⋅2 = 720
720 + 60 = 780
So 780º is the second coterminal angle.
Put -1 into the n.
Then the third coterminal angle is
360⋅(-1) + 60 degrees.
-360 + 60 = -300
So -300º is the third coterminal angle.
So 420º, 780º, and -300º
are the coterminal angles.
So 420º, 780º, -300º is the answer.
## Example 2
### Solution
To find the coterminal angles,
(Choose any integer n.)
Put 1 into the n.
Then the first coterminal angle of π/4 is
2π⋅1 + π/4.
Then 9π/4.
So 9π/4 is the first coterminal angle.
Put 2 into the n.
Then the next coterminal angle of π/4 is
2π⋅2 + π/4.
Then 17π/4.
So 17π/4 is the second coterminal angle.
Put 3 into the n.
Then the third coterminal angle of π/4 is
2π⋅3 + π/4.
Then 25π/4.
So 25π/4 is the third coterminal angle.
So 9π/4, 17π/4, and 25π/4
are the coterminal angles.
So 9π/4, 17π/4, 25π/4 is the answer.
## Example 3
### Solution
Write the 360⋅n numbers
that seems to cover 1000
on a number line:
0, 360, 720, 1080.
1000 is between 720 and 1080.
Then the number on the left side of 1000, 720,
is the 360⋅n number.
And the number between 720 and 1000
is the coterminal angle θ.
This means
720 + θ = 1000.
θ = [right angle] - [left angle]
So θ = 1000 - 720.
1000 - 720 = 280
## Example 4
### Solution
Write the 360⋅n numbers
that seems to cover -520
on a number line:
0, -360, -720, -1080.
-520 is between -720 and -360.
Then the number on the left side of -520, -720,
is the 360⋅n number.
And the number between -720 and -520
is the coterminal angle θ.
This means
-720 + θ = -520.
θ = [right angle] - [left angle]
So θ = -520 - (-720).
-520 - (-720)
= -520 + 720
= 200
## Example 5
### Solution
Write the 2π⋅n numbers
that seems to cover 13π/2
on a number line:
2π, 4π, 6π, 8π.
The denominator of 13π/2 is 2.
So change the denominators of the 2π⋅n numbers
to 2:
2π = 4π/2
4π = 8π/2
6π = 12π/2
8π = 16π/2.
13π/2 is between 12π/2 and 16π/2.
Then the number on the left side of 13π/2, 12π/2,
is the 2π⋅n number.
And the number between 12π/2 and 13π/2
is the coterminal angle θ.
This means
12π/2 + θ = 13π/2.
θ = [right angle] - [left angle]
So θ = 13π/2 - 12π/2.
13π/2 - 12π/2 = π/2<|endoftext|>
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## Class 9 (Old)
### Course: Class 9 (Old)>Unit 1
Lesson 3: Operations on real numbers
# Worked example: rational vs. irrational expressions
Sal shows how to determine whether the following expressions are rational or irrational: 9 + √(45), √(45)/ (3*√(5)), and 3*√(9). Created by Sal Khan.
## Want to join the conversation?
• How do we know that an integer plus an irrational number yields an irrational number? Is there another video on that?
• We'll do a proof by contradiction. This just means that we show that the false to our statement presents a contradiction.
First, let us assume that an irrational number plus a rational number makes a rational number and make this lead to a contradiction.
If a is rational, b is irrational, and c is rational, we will try to prove that:
``a + b = c``
is rational. If this is true, a = x/y and c = e/f for integers x, y, e, and f. So:
``a + b = c x/y + b = e/f b = e/f - x/y b = ey/(fy) - xf/(fy) b = (ey - xf)/(fy)``
Since the right hand side of the equation is rational, then so is b. But we said that b is irrational! This leads to a contradiction and so the sum must be irrational. Let me know if you need anything clarified.
• Sal cancelled out 3√5/3√5 to get 1. But the order of operations, PEMDAS states that we do the powers before division. So, what happened here? Can anyone please explain me. Thanks!
Sam D
• It doesn't matter because since the numerator and denominator are the same, even if you did use PEMDAS to approximate √5 AND THEN divided them out, you would still get 1. And besides, when we have the square root of a non-perfect square, we leave the answer in radical form (not decimal form), because the decimal form goes on forever like the digits in π. This is because it is irrational. Comment if you have any questions.
• Pi is an irrational number and is the ratio of the circumference (c) over the diameter (d), therefore c/d = Pi. Does this mean that either c or d or both must be irrational or can the quotient of two rational numbers be irrational?
• Do negative square roots exist?
• @CallaJones
By definition, the square root of a negative number does not exist. it instead is called an imaginary number or complex number.
Originally there were only positive integers but over time the concepts of fractions, zero, decimals, negative numbers, irrational numbers, and then certain transcendental numbers (pi or e) were developed to make the number system complete. Leonard Euler invented the idea that we can represent sqrt(-1) with an imaginary number called "i".
For example, the square root of -16 can be expressed as 4i.
sqrt(-16) = sqrt(16) x sqrt(-1) = 4i
• Would you consider Infinity Rational or Irrational?
• Infinity is not a number. It is the concept that there is no largest number. If you think you have found the largest number, you can add 1 and get a still larger number.
Since infinity is not a number, it is not classified as rational or irrational.
• Hmm. Could one multiply irrational numbers to get rational numbers? her it an example.
sqrt(3)*sqrt(3)
sqrt(3*3)
sqrt(9)
3
3/1
• Yes you did it!
(1 vote)
• how do you find square root of 2 ? i know it is irrational but like i want to know the method in detail.
• There is a long division method to find out square roots of numbers up to any decimal place you want. You can search it up.
• Hi guys,
Could anybody help with a question I encountered that I find quite confusing.
"Write as a single fraction . . .
[SQRT(x)] + 1/SQRT(x) . . . . "
Two seperate terms, SQRT(x) and 1/SQRT(x).
I understand many of the rules but I can't get my head around this one.
Thanks for your time guys.
• find a common denominator. You can multiply anything by 1 will be equal to the same thing. Keep in mind that 1 can be 432tvx^2/432tvx^2
or anything where the topand bottom are equal
• What about the question (√2 - √3) ^ 2 ?
Write in Racial Form?
I know that it is not very relevant but there is nowhere else to ask it!
This this the activity that I am stuck on: https://www.khanacademy.org/math/in-in-grade-9-ncert/xfd53e0255cd302f8:number-systems/xfd53e0255cd302f8:simplifying-expressions/e/multiplying-and-dividing-irrational-numbers?modal=1
(1 vote)
• Radical form means you use the radical symbol where needed rather than exponential form.
-- Radical form: √2
-- Exponential form: 2^(1/2)
How to do: (√2 - √3)^2
Did you use the hints?
You need to multiply 2 binomials, which means you use FOIL. (a-b)^2 = (a-b)(a-b) = a^2-ab-ab+b^2 = a^2-2ab+b^2
Remember to simplify the radicals. For example: √2√2 = √4 = 2
Give it a try. Comment back with questions.<|endoftext|>
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# When four dice are rolled simultaneously in how many outcomes will atleast one of the dice shows?
Contents
## How many possible outcomes are there when 4 dice are rolled in which at least one dice show 2?
R D Sharma – Mathematics 9
Total Possible outcomes when 4 dice are rolled = 6^4. Total Possible outcomes where none of the dice shows 2 = 5^4. = 671. Hope this helps!
## When 4 dice are rolled simultaneously What is the probability that the same number appears on all of them?
The chances that all the dice show same number (1,1,1,1), (2,2,2,2), (3,3,3,3), (4,4,4,4), (5,5,5,5), (6,6,6,6)} is 6. Answer: There are 6 numbers on a dice.
## How many ways can 4 dice be rolled?
We can roll 4 (all ones) in 1 way; 5 in 4 ways; 6 in 10 ways, and so on, but for sums that can be rolled in many ways, like 14, this relationship tends to go away.
## When 4 dice are rolled the probability that the total score on the 4 dice is maximum is?
When 4 dice are rolled the probability that the total score on the four dice is maximum is? 2 Answers. so the answer is 140.
IT IS INTERESTING: Question: How many casinos are close to Sacramento?
## How many possibilities are there when you roll 4 six sided dice?
Two (6-sided) dice roll probability table
Roll a… Probability
4 3/36 (8.333%)
5 4/36 (11.111%)
6 5/36 (13.889%)
7 6/36 (16.667%)
## What is the probability that the dice show the same numbers?
If you roll two fair six-sided dice, the probability that the dice show the same number is 1/6.
## What is the probability of getting a sum 9 from the throws of a dice?
The probability of getting 9 as the sum when 2 dice are thrown is 1/9.
## What are the odds of rolling the same number on 6 dice?
You have about a 1.5% chance of rolling a straight. One thing worth noting is that that probability is the same whether you roll one die at a time or all six together. We’ll use the one-die-at-a-time scenario and consider what the value of each die must be as it’s rolled so that we end up with a straight.<|endoftext|>
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# Ex.7.3 Q3 Coordinate Geometry Solution - NCERT Maths Class 10
Go back to 'Ex.7.3'
## Question
Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are $$(0, -1), (2, 1)$$ and $$(0, 3).$$ Find the ratio of this area to the area of the given triangle.
Video Solution
Coordinate Geometry
Ex 7.3 | Question 3
## Text Solution
Reasoning:
Let ABC be any triangle whose vertices are $$A(x_1, y_1),$$ $$B(x_2, y_2)$$ and $$C(x_3, y_3).$$
Area of a triangle
\begin{align}=\frac{1}{2} \begin{bmatrix} x_1 \left( y_2 - y_3 \right) \\ + x_2 \left( y_3 - y_1 \right) \\ + x_3 \left( y_1 - y_2 \right) \end{bmatrix} \end{align}
What is Known:
The $$x$$ and $$y$$ co-ordinates of the vertices of the triangle.
What is Unknown?
The ratio of this area to the area of the given triangle.
Solution:
From the given figure,
Given,
• Let A $$(x_1, y_1) = (0, -1)$$
• Let B $$(x_2, y_2) = (2 , 1)$$
• Let C $$(x_3, y_3) = (0, 3)$$
Area of a triangle
\begin{align}=\frac{1}{2} \begin{bmatrix} x_1 \left( y_2 - y_3 \right) \\ + x_2 \left( y_3 - y_1 \right) \\ + x_3 \left( y_1 - y_2 \right) \end{bmatrix}\;\; \dots (1) \end{align}
By substituting the values of vertices, $$A, B, C$$ in $$(1)$$,
Let $$P, Q, R$$ be the mid-points of the sides of this triangle.
Coordinates of $$P, Q,$$ and $$R$$ are given by
\begin{align}{P = \left[ {\frac{{0 + 2}}{2},\frac{{ - 1 + 1}}{2}} \right] = (1,0)}\\{Q = \left[ {\frac{{0 + 0}}{2},\frac{{3 - 1}}{2}} \right] = (0,1)}\\{R = \left[ {\frac{{2 + 0}}{2},\frac{{1 + 3}}{2}} \right] = (1,2)}\end{align}
By substituting the values of Points $$P, Q, R$$
Area of $$\Delta PQR$$,
\begin{align}&= \frac{1}{2}\begin{bmatrix} (2 - 1) + 1(1 - 0) \\ + 0(0 - 2)\end{bmatrix} \\ &= \frac{1}{2}(1 + 1)\\ &= 1{\text{ Square units }}\end{align}
By substituting the values of Points $$A, B, C$$
Area of $$\Delta ABC$$,
\begin{align}&= \frac{1}{2} \begin{bmatrix} 0(1 - 3) + 2( 3 - ( - 1)) \\ + 0( - 1 - 1) \end{bmatrix} \\ &= \frac{1}{2}(8) \\ &= 4{\text{ Square units }}\end{align}
Therefore, Ratio of this area $$\Delta \, PQR$$ to the area of the triangle \begin{align}\Delta {\text{ABC = 1: 4}}\end{align}
Learn from the best math teachers and top your exams
• Live one on one classroom and doubt clearing
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Sentence stress is the combination of strong words and weak words in a sentence, which creates a sort of rhythm. Strong words, such as nouns and verbs, are spoken stronger and more clearly than weak words, such as articles, pronouns or prepositions. (See Sentence Stress Part 2)
Weak words are not usually pronounced very clearly because they are spoken quickly, and they get reduced. This means that part of their sound is missing. Reducing weak words makes it easier to say them quickly.
Here are some examples of weak words that are used every day, and are normally reduced.
Prepositions: to / for / at / then / from / on / with.
Pronouns: you / your / he / she / it / them / his / him / her.
Helping verbs and BE: can / do / have / am / are / is / been / will.
Others words: and / an / the / or / than.
Notice, however, that when I say these words as part of a list, I do not say them with reductions. The reductions happen when they are in a sentence.
The most common ways that weak words get reduced are the following.
The most basic kind of reduction is using a schwa sound instead of a clearly pronounced vowel. In these examples, the word in parentheses after the sentence is the word that has a schwa rather than a full vowel sound.
I love to read. (to)
Look at this! (at)
They took it already. (it)
Do you like it? (do/you)
What is that? (is)
We need more than that. (than)
Sometimes a vowel is reduced so much, that it gets skipped completely. This is how some contractions are formed. In these sentences, the vowel of the weak word is missing.
He has to go. (to)
Thanks for your help. (for/your)
I’m almost finished. (am)
It’s getting late. (is)
We’re not ready yet. (are)
Another very common kind of reduction is skipping the “H” in these words: have / has / had / he / her / him / his. In these sentences, those words are weak and the “H” is missing.
They will call him later.
We should have waited.
Can her brother drive?
I had hoped to finish sooner.
It is important to know that H-deletion is only for weak words. If one of these words is said with strong stress, the “H” does not get deleted. For example, if I say: “That’s his book not mine” the “H” in the word “his” is said clearly because the word is strong. Also, when the word “have” is a main verb, it does not get reduced, as in “We have two days left.”
Other skipped consonants
Besides “H”, there are a few other consonant sounds that regularly get skipped.
“F” in the word “of”: We have lots of time.
“V” in “have”: He should have told us.
“D” in “and”: Let’s stop and eat.
“W” in “will”: That’ll be fine.
Sentence Stress Part 5 shows how some frequently used weak words can be confused with each other, because they can sometimes sound the same!<|endoftext|>
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Note:
———————————————————————————————————————————–
We have,
$\displaystyle \lim \limits_{x \to a} \frac{x^m-a^m}{x^n-a^n}$
$\displaystyle = \lim \limits_{x \to a} \Bigg\{ \frac{x^m-a^m}{x-a} \times \frac{x-a}{x^n-a^n} \Bigg\}$
$\displaystyle = \lim \limits_{x \to a} \Bigg\{ \frac{x^m-a^m}{x-a} \div \frac{x^n-a^n}{x-a} \Bigg\}$
$\displaystyle = \lim \limits_{x \to a} \frac{x^m-a^m}{x-a} \div \lim \limits_{x \to a} \frac{x^n-a^n}{x-a}$
$\displaystyle = ma^{m-1} \div na^{n-1}$
$\displaystyle = \frac{m}{n} a^{m-n}$
———————————————————————————————————————————–
$\displaystyle \text{Question 1: } \lim \limits_{x \to a} \frac{(x+2)^\frac{5}{2} - ( a+2)^\frac{5}{2}}{x-a }$
We have,
$\displaystyle \lim \limits_{x \to a} \frac{(x+2)^\frac{5}{2} - ( a+2)^\frac{5}{2}}{x-a }$
$\displaystyle \text{Let } y = x+2 \text{ and } b = a+2$
$\displaystyle \text{When } x \rightarrow a, \text{ then } x+2 \rightarrow a+2 \Rightarrow y \rightarrow b$
$\displaystyle = \lim \limits_{y \to b} \frac{y^\frac{5}{2} - b^\frac{5}{2}}{y-b }$
$\displaystyle = \frac{5}{2} b^{\frac{5}{2}-1}$
$\displaystyle = \frac{5}{2} b^{\frac{3}{2}}$
$\displaystyle = \frac{5}{2} (a+2)^{\frac{3}{2}}$
$\\$
$\displaystyle \text{Question 2: } \lim \limits_{x \to a} \frac{(x+2)^\frac{3}{2} - ( a+2)^\frac{3}{2}}{x-a }$
We have,
$\displaystyle \lim \limits_{x \to a} \frac{(x+2)^\frac{3}{2} - ( a+2)^\frac{3}{2}}{x-a }$
$\displaystyle \text{Let } y = x+2 \text{ and } b = a+2$
$\displaystyle \text{When } x \rightarrow a, \text{ then } x+2 \rightarrow a+2 \Rightarrow y \rightarrow b$
$\displaystyle = \lim \limits_{y \to b} \frac{y^\frac{3}{2} - b^\frac{3}{2}}{y-b }$
$\displaystyle = \frac{3}{2} b^{\frac{3}{2}-1}$
$\displaystyle = \frac{3}{2} b^{\frac{1}{2}}$
$\displaystyle = \frac{3}{2} (a+2)^{\frac{1}{2}}$
$\\$
$\displaystyle \text{Question 3: } \lim \limits_{x \to 0} \frac{(1+x)^6-1}{(1+x)^2-1 }$
We have,
$\displaystyle \lim \limits_{x \to 0} \frac{(1+x)^6-1}{(1+x)^2-1 }$
$\displaystyle = \lim \limits_{x \to 0} \Bigg\{ \frac{(1+x)^6-1}{x} \times \frac{x}{(1+x)^2-1} \Bigg\}$
$\displaystyle = \lim \limits_{x \to 0} \Bigg\{ \frac{(1+x)^6-1^6}{(1+x) - 1} \times \frac{(1+x)-1}{(1+x)^2-1^2} \Bigg\}$
$\displaystyle \text{Let } y = 1+x$
$\displaystyle \text{When } x \rightarrow 0, \text{ then } 1+x \rightarrow 1 \Rightarrow y \rightarrow 1$
$\displaystyle = \lim \limits_{x \to 0} \Bigg\{ \frac{y^6-1^6}{y - 1} \times \frac{y-1}{y^2-1^2} \Bigg\}$
$\displaystyle = \frac{6}{2} ( 1^{6-2} )$
$\displaystyle = 3$
$\\$
$\displaystyle \text{Question 4: } \lim \limits_{x \to a} \frac{x^\frac{2}{7} - a^\frac{2}{7}}{x-a }$
We have,
$\displaystyle \lim \limits_{x \to a} \frac{x^\frac{2}{7} - a^\frac{2}{7}}{x-a }$
$\displaystyle = \frac{2}{7}a^{\frac{2}{7}-1}$
$\displaystyle = \frac{2}{7} a^{\frac{-5}{7}}$
$\\$
$\displaystyle \text{Question 5: } \lim \limits_{x \to a} \frac{x^\frac{5}{7} - a^\frac{5}{7}}{x^\frac{2}{7} - a^\frac{2}{7} }$
$\displaystyle \text{We have } \lim \limits_{x \to a} \frac{x^\frac{5}{7} - a^\frac{5}{7}}{x^\frac{2}{7} - a^\frac{2}{7} }$
$\displaystyle = \Bigg\{ \lim \limits_{x \to a} \frac{x^\frac{5}{7} - a^\frac{5}{7}}{x-a} \times \frac{x-a}{x^\frac{2}{7} - a^\frac{2}{7} } \Bigg\}$
$\displaystyle = \Bigg\{ \lim \limits_{x \to a} \frac{x^\frac{5}{7} - a^\frac{5}{7}}{x-a} \div \frac{x^\frac{2}{7} - a^\frac{2}{7} }{x-a} \Bigg\}$
$\displaystyle = \frac{5}{7} a^{\frac{5}{7}-1} \div \frac{2}{7} a^{\frac{2}{7}-1}$
$\displaystyle = \frac{5}{7} a^{\frac{-2}{7}} \div \frac{2}{7} a^{\frac{-5}{7}}$
$\displaystyle = \frac{5}{2} a^{\frac{-2}{7} +\frac{-5}{7} }$
$\displaystyle = \frac{5}{2} a^\frac{3}{7}$
$\\$
$\displaystyle \text{Question 6: } \lim \limits_{x \to -\frac{1}{2}} \frac{8x^3+1}{2x+1 }$
$\displaystyle \text{We have } \lim \limits_{x \to -\frac{1}{2}} \frac{8x^3+1}{2x+1 }$
$\displaystyle = \lim \limits_{x \to -\frac{1}{2}} \frac{(2x)^3-(-1)^3}{2x-(-1) }$
$\displaystyle \text{Let } y = 2x$
$\displaystyle \text{When } x \rightarrow -\frac{1}{2}, \text{ then } 2x \rightarrow -1$
$\displaystyle = \lim \limits_{x \to -\frac{1}{2}} \frac{y^3-(-1)^3}{y-(-1) }$
$\displaystyle = 3 (-1)^{3-1}$
$\displaystyle = 3 (-1)^2$
$\displaystyle = 3$
$\\$
$\displaystyle \text{Question 7: } \lim \limits_{x \to 27} \frac{(x^\frac{1}{3}+3)(x^\frac{1}{3}-3)}{ x-27}$
$\displaystyle \text{We have, } \lim \limits_{x \to 27} \frac{(x^\frac{1}{3}+3)(x^\frac{1}{3}-3)}{ x-27}$
$\displaystyle = \lim \limits_{x \to 27} \frac{((x^\frac{1}{3})^2-3^2)}{ (x^\frac{1}{3})^3-3^3}$
$\displaystyle \text{When } x \rightarrow 27, \text{ then } x^\frac{1}{3} \rightarrow 3$
$\displaystyle \text{Let } y = x^\frac{1}{3}$
$\displaystyle = \lim \limits_{y\to 3} \frac{(y^2-3^2)}{ (y^3-3^3}$
$\displaystyle = \frac{2}{3} (3)^{2-3}$
$\displaystyle = \frac{2}{9}$
$\\$
$\displaystyle \text{Question 8: } \lim \limits_{x \to 4} \frac{x^3-64}{x^2-16 }$
$\displaystyle \text{We have, } \lim \limits_{x \to 4} \frac{x^3-64}{x^2-16 }$
$\displaystyle = \lim \limits_{x \to 4} \frac{x^3-4^3}{x^2-4^2 }$
$\displaystyle = \lim \limits_{x \to 4} \Bigg\{ \frac{x^3-4^3}{x-4} \times \frac{x-4}{x^2-4^2} \Bigg\}$
$\displaystyle = \lim \limits_{x \to 4} \Bigg\{ \frac{x^3-4^3}{x-4} \div \frac{x^2-4^2}{x-4} \Bigg\}$
$\displaystyle = 3(4)^{3-1} \div 2(4)^{2-1}$
$\displaystyle = \frac{3 \times 16}{2 \times 4}$
$\displaystyle = 6$
$\\$
$\displaystyle \text{Question 9: } \lim \limits_{x \to 1} \frac{x^{15}-1}{ x^{10}-1}$
$\displaystyle \text{We have, } \lim \limits_{x \to 1} \frac{x^{15}-1}{ x^{10}-1}$
$\displaystyle = \lim \limits_{x \to 1} \Bigg\{ \frac{x^{15}-1^{15}}{x-1} \times \frac{x-1}{x^{10}-1^{10}} \Bigg\}$
$\displaystyle = \lim \limits_{x \to 1} \Bigg\{ \frac{x^{15}-1^{15}}{x-1} \div \frac{x^{10}-1^{10}}{x-1} \Bigg\}$
$\displaystyle = 15(1)^{15-1} \div 10(1)^{10-1}$
$\displaystyle = \frac{15}{10}$
$\displaystyle = \frac{3}{2}$
$\\$
$\displaystyle \text{Question 10: } \lim \limits_{x \to -1} \frac{x^3+1}{x+1 }$
$\displaystyle \text{We have, } \lim \limits_{x \to -1} \frac{x^3+1}{x+1 }$
$\displaystyle = \lim \limits_{x \to -1} \frac{x^3-(-1)^3}{x-(-1) }$
$\displaystyle = 3(-1)^{3-1}$
$\displaystyle = 3$
$\\$
$\displaystyle \text{Question 11: } \lim \limits_{x \to a} \frac{x^\frac{2}{3}- a^\frac{2}{3}}{ x^\frac{3}{4}- a^\frac{3}{4}}$
$\displaystyle \text{We have: } \lim \limits_{x \to a} \frac{x^\frac{2}{3}- a^\frac{2}{3}}{ x^\frac{3}{4}- a^\frac{3}{4}}$
$\displaystyle= \lim \limits_{x \to a} \frac{x^\frac{2}{3}- a^\frac{2}{3}}{x-a } \times \frac{x-a} { x^\frac{3}{4}- a^\frac{3}{4}}$
$\displaystyle= \lim \limits_{x \to a} \frac{x^\frac{2}{3}- a^\frac{2}{3}}{x-a } \div \frac{ x^\frac{3}{4}- a^\frac{3}{4}}{x-a}$
$\displaystyle= = \frac{2}{3} a^{\frac{2}{3}-1} \div \frac{3}{4} a^{\frac{3}{4}-1}$
$\displaystyle= = \frac{8}{9} a^{(-\frac{1}{3}+\frac{1}{4})}$
$\displaystyle= = \frac{8}{9} a^{-\frac{1}{12}}$
$\\$
$\displaystyle \text{Question 12: If } \lim \limits_{x \to 3} \frac{x^n-3^n}{x-3 } = 108, \text{ find the value of } n.$
$\displaystyle \text{Given } \lim \limits_{x \to 3} \frac{x^n-3^n}{x-3 } = 108$
$\displaystyle\Rightarrow n(3)^{n-1} = 108$
$\displaystyle\Rightarrow n(3)^{n-1} = 4(3)^{3}$
$\displaystyle\Rightarrow n(3)^{n-1} = 4(3)^{4-1}$
$\displaystyle\Rightarrow n=4$
$\\$
$\displaystyle \text{Question 13: If } \lim \limits_{x \to a} \frac{x^9-a^9}{x-a } = 9, \text{ find the value of } a.$
$\displaystyle \text{Given } \lim \limits_{x \to a} \frac{x^9-a^9}{x-a } = 9, \text{ find the value of } a.$
$\displaystyle\Rightarrow 9(a)^{9-1} = 9$
$\displaystyle\Rightarrow 9a^8 = 9$
$\displaystyle\Rightarrow a^8 = 1$
$\displaystyle\Rightarrow a = \pm 1$
$\\$
$\displaystyle \text{Question 14: If } \lim \limits_{x \to a} \frac{x^5-3^5}{x-a } = 405, \text{ find the value of } a.$
$\displaystyle \text{Given } \lim \limits_{x \to a} \frac{x^5-3^5}{x-a } = 405$
$\displaystyle\Rightarrow 5(a)^{5-1} = 405$
$\displaystyle\Rightarrow 5a^4 = 405$
$\displaystyle\Rightarrow a^4 = 81$
$\displaystyle\Rightarrow a = \pm 3$
$\\$
$\displaystyle \text{Question 15: If } \lim \limits_{x \to a} \frac{x^9-a^9}{x-a } = \lim \limits_{x \to 5} (4+x), \text{ find the value of } a.$
$\displaystyle \text{Given } \lim \limits_{x \to a} \frac{x^9-a^9}{x-a } = \lim \limits_{x \to 5} (4+x)$
$\displaystyle\Rightarrow 9(a)^{9-1} = 9$
$\displaystyle\Rightarrow a^8 = 1$
$\displaystyle\Rightarrow a = \pm 1$
$\\$
$\displaystyle \text{Question 16: If } \lim \limits_{x \to a} \frac{x^3-a^3}{x-a } = \lim \limits_{x \to 1} \frac{x^4-1}{x-1} , \text{ find the value of } a.$
$\displaystyle \text{Given } \lim \limits_{x \to a} \frac{x^3-a^3}{x-a } = \lim \limits_{x \to 1} \frac{x^4-1}{x-1}$
$\displaystyle\Rightarrow 3(a)^{3-1} = 4(1)^{4-1}$
$\displaystyle\Rightarrow 3a^2 = 4$
$\displaystyle\Rightarrow a^2 = \frac{4}{3}$
$\displaystyle\Rightarrow a = \pm \frac{2}{\sqrt{3}}$<|endoftext|>
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PUMPA - THE SMART LEARNING APP
Take a 10 minutes test to understand your learning levels and get personalised training plan!
### Theory:
Let us recall the union and intersection of two sets.
Union of two sets:
The union of two sets $$A$$ and $$B$$ is the set of all elements which are either in $$A$$ or in $$B$$ or in both. It is denoted by $$A\cup B$$ and read as '$$A$$ union $$B$$'.
Intersection of two sets:
The intersection of two sets $$A$$ and $$B$$ is the set of all elements common to both $$A$$ and $$B$$. It is denoted by $$A\cap B$$ and read as '$$A$$ intersection $$B$$'.
Commutative property for union of two sets:
The union of two sets will not change if you interchange the order of the two sets.
$$A\cup B$$ $$=$$ $$B \cup A$$
Let $$A$$ $$=$$ $$\{$$$$2$$, $$3$$, $$5$$, $$7$$$$\}$$, and $$B$$ $$=$$ $$\{$$$$4$$, $$6$$, $$8$$, $$10$$$$\}$$
$$A \cup B$$ $$=$$ $$\{$$$$2$$, $$3$$, $$5$$, $$7$$$$\}$$ $$\cup$$ $$\{$$$$4$$, $$6$$, $$8$$, $$10$$$$\}$$
$$A \cup B$$ $$=$$ $$\{$$$$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$10$$$$\}$$
$$B \cup A$$ $$=$$ $$\{$$$$4$$, $$6$$, $$8$$, $$10$$$$\}$$ $$\cup$$ $$\{$$$$2$$, $$3$$, $$5$$, $$7$$$$\}$$
$$B \cup A$$ $$=$$ $$\{$$$$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$10$$$$\}$$
From the above results, we see that:
$$A \cup B$$ $$=$$ $$B \cup A$$
This is known as commutative property of union of two sets.
Commutative property for intersection of two sets
The intersection of two sets will not change if you interchange the order of the two sets.
$$A\cap B$$ $$=$$ $$B \cap A$$
Let $$A$$ $$=$$ $$\{$$$$3$$, $$5$$, $$8$$, $$9$$, $$10$$$$\}$$, and $$B$$ $$=$$ $$\{$$$$4$$, $$5$$, $$7$$, $$10$$$$\}$$
$$A \cap B$$ $$=$$ $$\{$$$$3$$, $$5$$, $$8$$, $$9$$, $$10$$$$\}$$ $$\cap$$ $$\{$$$$4$$, $$5$$, $$7$$, $$10$$$$\}$$
$$A \cap B$$ $$=$$ $$\{$$$$5$$, $$10$$$$\}$$
$$B \cap A$$ $$=$$ $$\{$$$$4$$, $$5$$, $$7$$, $$10$$$$\}$$ $$\cap$$ $$\{$$$$3$$, $$5$$, $$8$$, $$9$$, $$10$$$$\}$$
$$B \cap A$$ $$=$$ $$\{$$$$5$$, $$10$$$$\}$$
From the above results, we see that:
$$A \cap B$$ $$=$$ $$B \cap A$$
This is known as commutative property of intersection of two sets.
Important!
The generalized meaning of the commutative property is the result will not change if you interchange the order of the sets.<|endoftext|>
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The United Nations High Commissioner for Refugees estimated that by the end of 2017 more than 65 million individuals had been forcibly displaced from their homes. Some 22 million of these people are refugees, over half of whom are under the age of 18. The others remain displaced within the borders of the country as internally displaced persons. Three countries, Syria, Afghanistan, and South Sudan, account for more than half of all refugees.
Since the outbreak of the Syrian uprising and civil war in 2011, more than half of all Syrians have been displaced from their homes. Over 5.5 million of these have fled the country due to war crimes and crimes against humanity including persecution, torture, the besieging of communities and aerial bombardment at the hands of the Syrian regime and of extremist forces, including the self-proclaimed Islamic State (ISIS). Over six million are internally displaced within Syria.
In addition to Syria, large numbers of displaced persons have fled atrocity crises in recent years in Burma, the Central African Republic, the Democratic Republic of the Congo, Iraq, and South Sudan, among others. Since 2014, hundreds of thousands of Iraqis have fled the campaign of religious persecution and mass murder waged by ISIS in northern Iraq. In November 2015, the United States Holocaust Memorial Museum published its findings that ISIS had committed genocide against the Yezidi and widespread crimes against humanity against other religious communities, including both Christian and Muslim groups.
Refugees and International Protections
Recognizing its moral failure to help Jews and others fleeing Nazi persecution before World War II, and faced with hundreds of thousands of displaced persons at the war's end, the international community made important commitments to assist and protect refugees.
In 1948, the United Nations adopted the Universal Declaration of Human Rights, which recognizes the right of every individual to seek and enjoy asylum from persecution. In 1951, the UN established the office of the United Nations High Commissioner for Refugees and adopted the Convention Relating to the Status of Refugees. The latter established the basic international obligation not to return people to countries where their life or freedom might be threatened. The United States accepted this obligation in 1968.
The Refugee Convention defines refugees as persons who are outside their native country or country of habitual residence and who cannot return to their country or call on its protection because they fear that they will be persecuted on the basis of their “race, religion, nationality, membership in a particular social group, or political opinion.” The convention guarantees refugees a wide range of civil and human rights, including freedom of association, the right of legal redress, and protection from discrimination.
These landmark commitments by the UN established the plight of refugees as a responsibility of the international community. They continue to shape policy today.
Other Categories of Displaced Persons
Not all persons forcibly displaced from their homes are refugees under international law. “Internally displaced persons” are those who have fled their homes, perhaps for the same reasons as refugees, but have not left the country in which they have been living. Under international law, they still technically fall under the protection of their own government, even if that government is responsible for their displacement.
At the end of 2015, over 40 million people were classified as “internally displaced persons.” Individuals who have crossed an international border fleeing economic misery, civil war, or natural disasters, such as floods, earthquakes, and drought, also do not qualify for refugee status. As a result, such persons do not receive the same legal protections as refugees.
The Global Impact of the Refugee Crisis
Today's refugee crisis is the product of conflicts that involve mass atrocities and human rights violations. The vast majority of current refugees are in countries neighboring their homelands. For example, 97% of registered Syrian refugees in 2017 are in the neighboring states of Turkey, Lebanon, Jordan and Iraq. Lebanon alone, with a population of 4.3 million in 2011, has since taken in more than one million Syrian refugees.
Such large inflows of refugees put serious strains on the host countries' resources. Uprooted from their homes, communities, and cultures and often traumatized by their experiences, most refugees live in crowded conditions, unable to find work or provide for their families. Some refugees are seeking to move on to more distant countries where they hope for the opportunity to lead safe and productive lives.
In addition to straining resources, large inflows of refugees can raise national and regional tensions that may have far-reaching consequences. The refugee flow created by Rwanda's genocide contributed to the igniting of two international wars and ongoing insurgencies that have led to more than 5 million deaths in neighboring Democratic Republic of the Congo.
In Europe today, the influx of refugees and an even larger number of migrants has sparked political backlash and contributed to a rise in racism and xenophobia.
Preventing and responding to conflicts before they put populations at risk and providing adequate support and protection for refugees are key requirements for preventing future genocides.
Critical Thinking Questions
- How has international law about refugees evolved since the Holocaust?
- What pressures and motivations may affect decision makers and citizens in another country considering how to respond to a refugee crisis?
- What responsibilities do (or should) other nations have regarding refugees from oppressive regimes?<|endoftext|>
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The 35th Regiment of Foot, were one of the main British units involved in the siege and massacre at Fort William Henry, during the French and Indian War.
Following heavy bombardment and siege operations that progressively neared the fort's walls, the garrison was forced to surrender on August 8th when it became apparent that General Daniel Webb, the commander at Fort Edward, was not sending any relief.The terms of surrender were that the British and their camp followers would be allowed to withdraw, under French escort, to Fort Edward, with the full honours of war, on condition that they refrain from participation in the war for 18 months. They were allowed to keep their muskets but no ammunition, and a single symbolic cannon. In addition, British authorities were to release French prisoners within three months.The next morning, even before the British column began to form up for the march to Fort Edward, the Indians renewed attacks on the largely defenceless British. At 5 am, Indians entered huts in the fort housing wounded British who were supposed to be under the care of French doctors, and killed and scalped them. Monro complained that the terms of capitulation had in essence been violated already, but his contingent was forced to surrender some of its baggage in order to even be able to begin the march. As they marched off, they were harassed by the swarming Indians, who snatched at them, grabbing for weapons and clothing, and pulling away with force those that resisted their actions, including many of the women, children, and black servants. As the last of the men left the encampment, a war whoop sounded, and warriors seized a number of men at the rear of the column.While Montcalm and other French officers tried to stop these attacks, others did not, and explicitly refused further protection to the British. At this point, the column dissolved, as some prisoners tried to escape the Indian onslaught, while others actively tried to defend themselves. Massachusetts Colonel Joseph Frye reported that he was stripped of much of his clothing and repeatedly threatened. He fled into the woods, and did not reach Fort Edward until August 12, three days later.Estimates of the numbers captured, wounded or killed varied widely. Ian Steele has compiled estimates ranging from 200 to 1,500.His detailed reconstruction of the action and its aftermath indicates that the final tally of missing and dead ranges from 69 to 184, at most 7.5% of the 2,308 who surrendered.Atrocities described in accounts of the massacre include the killing and scalping of sick and wounded individuals, and the digging up graves to take additional trophies from those who died of wounds or disease during the siege. As a result, many Indians who participated in the action may have contracted smallpox, which they carried back to their communities<|endoftext|>
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# Geometric definitions of hyperbolic functions
I've learned in school that all the trigonometric functions can be constructed geometrically in terms of a unit circle:
Can the hyperbolic functions be constructed geometrically as well? I know that $\sinh$ and $\cosh$ can be constructed based on the area between a ray through the origin and the unit hyperbola, but what about $\tanh$, $\mathrm{csch}$, $\mathrm{sech}$ and $\coth$?
-
The easy arrangement around the circle, as you showed it, makes use of the fact that every radius of the circle is a unit length, so triangles with that unit length as one edge and a right angle at one corner are easy to find. Furthermore, right angles can be used to transfer the angle argument $\theta$ between these triangles.
In the hyperbola situation, things are a bit more complicated. The point $(1,0)$ is easily obtained as the location of one vertex, which has distance $1$ from the origin. Using e.g. a line parallel to one asymptote, you can also obtain a point $(0,1)$. The area cannot easly be transferred, so one has to make do with the single occurence of that area, and find everything else in relation to it.
+1. My answer to a similar question intentionally left out $\operatorname{sech}$ and $\operatorname{csch}$. Where you've put these values is pretty good ---the reciprocal relations with $\sinh$ and $\cosh$ are clear--- but I'm holding out for a configuration that also neatly demonstrates the corresponding Pythagorean relations. – Blue Jul 30 '13 at 22:39
@Blue: The distance between the origin and the point $(\operatorname{csch},1)$ is $\operatorname{coth}$, so there you have the equation $\operatorname{csch}^2+1^2=\operatorname{coth}^2$. And if you form a right triangle with $(0,0)$ and $(1,0)$ as endpoints of its hypothenuse, and with one leg pointing in the $(1,\sinh)$ direction (along the upper of the two cyan lines), then the lengths of the leg will be $\operatorname{sech}$ and the other leg will be $\operatorname{tanh}$, together prooving $\operatorname{sech}^2+\operatorname{tanh}^2=1$. Marking these lengths in an image is tricky, though. – MvG Jul 30 '13 at 23:06
Certainly, the relations are (and must be) implicitly there. As I mention, there are lots of ways to introduce segments whose lengths happen to be whatever we need them to be. But, eg, the only way we know that the distance from origin to $(\operatorname{csch}, 1)$ is $\operatorname{coth}$ is by already knowing the Pythagorean identity; there's no clear visual tie between that hypotenuse and the segment your figure defines to have length $\operatorname{coth}$. I want a figure that makes everything as obvious as in the circular trig diagram (which, btw, inspired my company logo). – Blue Jul 30 '13 at 23:26
For dis-satisfaction from other side: Consider the point $P(u,v)$ where the line joining the origin to $(\cosh, 1)$ meets the hyperbola in your figure. By virtue of $P$ lying on the curve, the coordinates satisfy the hyp-Pythagorean relation: $u^2-v^2=1$. Terrific! One can check we must in fact have $u = \operatorname{coth}$ and $v = \operatorname{csch}$ ... but this time we lack a clear visual connection to the proportionality relations $\operatorname{coth} = \cosh/\sinh$ and $\operatorname{csch} = 1/\sinh$. Somewhere is an ideal diagram that shows both Pythagorean-ness and proportionality. – Blue Jul 30 '13 at 23:57<|endoftext|>
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Definition - What does Carcinogen mean?
A carcinogen is a substance or agent that is directly responsible for causing cancer. These substances cause changes in a cell's DNA, either directly or indirectly. Carcinogens have varying levels of cancer-causing potential and not all who are exposed to carcinogens will get cancer.
Safeopedia explains Carcinogen
According to the World Health Organization (WHO), there are over 400 documented agents that are classified as carcinogenic. The most well known of these, tobacco, is known to cause lung cancer. Many workers come into daily contact with carcinogens. These agents include asbestos, chromium, nickel compound and arsenic. Such carcinogens may be absorbed through the skin or inhaled. It is important that all who work in industries where one may come into contact with such agents utilize appropriate safety equipment and PPE at all times.<|endoftext|>
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What is astigmatism?
Astigmatism is a vision condition in which light entering the eye is unable to be brought to a single focus, resulting in vision being blurred at all distances.
Astigmatism is not a disease, but rather, a vision condition that is quite common. It often occurs in conjunction with other refractive errors like nearsightedness and farsightedness.
Why does astigmatism occur?
Typically, astigmatism is caused by the front of your eye (the cornea) being more oval than round, and not allowing light to focus properly on the back of your eye (retina). The causes of this irregular shape vary.
In some cases, it may be hereditary or it may result from such factors as pressure of the eyelids on the cornea, incorrect posture or an increased use of the eyes for close work.
How common is astigmatism?
Most people have some degree of astigmatism. However, only individuals with moderate to highly astigmatic eyes usually need corrective lenses.
What are signs/symptoms of astigmatism?
People with significant amounts of astigmatism will usually have blurred or distorted vision. Those with mild astigmatism may experience headache eye strain, fatigue or blurred vision at certain distances.
How is astigmatism diagnosed?
A comprehensive eye examination by your doctor of optometry will include testing for astigmatism.
How is astigmatism treated?
In general, astigmatism can be optically corrected with properly prescribed and fitted eyeglasses or contact lenses. In recent years, a number of options to surgically alter the shape of the cornea, to correct low or moderate astigmatism, have been developed. These include procedures called radial keratotomy (RK) and photorefractive keratectomy (PRK).
Your doctor of optometry can help you decide if these procedures are right for you. There is also a procedure called orthokeratology which uses a series of rigid contact lenses to provide improved vision for extended periods of time for people with astigmatism.
Does astigmatism get progressively worse?
Astigmatism may change slowly. Regular optometric care can, however, help to ensure that proper vision is maintained.
How will astigmatism affect my lifestyle?
You may have to adjust to wearing eyeglasses or contact lenses if you do not wear them now. Other than that, astigmatism probably will not significantly affect your lifestyle at all.<|endoftext|>
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According to About.com, areas of low pressure within the Earth's atmosphere are caused by unequal heating across the surface and the pressure gradient force. Incoming solar radiation largely concentrates at the equator, resulting in warmer air at the lower latitudes. This warm air has a lower barometric pressure than the cooler, denser air near the poles, and the differences between these types of air create the pressure gradient force.
According to Sciences 360, both high- and low-pressure systems develop from the interactions of an unequally heated environment, such as the difference between a land mass and a nearby body of water. The low-pressure weather system forms at areas of wind divergence in the upper layer of the troposphere. This divergence creates an empty space for the warmer air to rise in to, lowering the surface air pressures as that warm air moves upward.
The warm air carries moisture up with it, creating clouds and ultimately, precipitation. Areas of low pressure are also associated with higher winds, as the air present in the system is moving toward the areas of higher pressure above and away from the lower-pressure system. This constant churning of air between areas of high and low pressure is known as the pressure gradient force.<|endoftext|>
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# Chapter 1 - Section 1.1 - Functions and Their Graphs - Exercises - Page 12: 12
$P(m)=(\frac{1}{m^2},\frac{1}{m})$
#### Work Step by Step
$\mathrm{See\: the\:figure\:below}$. Let's say $\ P(a,b)\$ is any point on the graph of $\ f(x)=\sqrt{x}\$. If we input $\ x\$ into the function $\ f(x)=\sqrt{x}\$, we get the corresponding value of $\ y\$ to be $\ \sqrt{x}\$. And, since $\ P\$ is on the graph of the function $\ \sqrt{x}\$, we have a relation for its coordinates. $P=(x,\sqrt{x})$ The slope of the line joining the point $\ P\$ and the origin is $\ m=\frac{\sqrt{x}}{x}.$ After rearranging the equation, we have: $\ m=\frac{\sqrt{x}}{x}$ $\Rightarrow\ m=\frac{1}{\sqrt{x}}$ $\Rightarrow\ \sqrt{x}=\frac{1}{m}$ $\Rightarrow\ x=\frac{1}{m^2}$ So, the coordinates of $\ P\$ as a function of the slope $\ m\$ can be written as: $P(m)=(\frac{1}{m^2},\frac{1}{m})$ where $\ m\$ represents the slope of the line.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.<|endoftext|>
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find the slope of the line passing through the points calculator
If the y-values are decreasing, it referred to as the line has a negative slope. The coefficient of x is said to be as 1/2, means the slope of the line is 1/2.
Now, you ought to calculate where the line intersects with the y-axis: You ought to enter one of the coordinates into this slope equation: y – mx = b. Slope tells us the nature of change of function. The formula becomes increasingly useful as the coordinates take on larger values or decimal values. For further assistance, please contact us. Given m, it is possible to determine the direction of the line that m describes based on its sign and value: Slope is essentially change in height over change in horizontal distance, and is often referred to as "rise over run." Yes, slope point calculator helps you in finding the slope and shows you the slope graph corresponding to the given points by using the simple slope equation. Let us the formula to calculate the slope of the line passing through the points $(2,5)$ and $(-5, 1)$; Subtract the second coordinates and first coordinates, this gives us $y_B-y_A=1-5=-4$ and $x_B-x_A=-5-2=-7$; Simplify the fraction to get the slope of $\frac 47$. If you trace the line by using your finger, means from left to right (same like the direction that you read a book), the line will go down to the right. Simply, all you have to remember is that the slope is equal to the tangent of the angle. Remember that there is not a slope for these types of lines.
$$y_2 = y_1 + m \times \frac{d}{\sqrt(1 + m^2)}$$.
The formulas to find x and y of the point to the right of the point are as: $$x_2 = x_1 + \frac{d}{\sqrt(1 + m^2)}$$ Solution for Find the slope of the line passing through the given points, when possible. Subtract the lower bound from the upper bound. For example, an angle of 30° has a tangent of 0.577.
Parallel lines have the same slope, to find the parallel line at a given point you should simply calculate the... To create your new password, just click the link in the email we sent you. Multiply c by x and add this to the new line. =8/4 If it decreases when moving from the upper left to lower right, then the gradient is negative. Just as slope can be calculated using the endpoints of a segment, the midpoint can also be calculated. You can take a look on the definition of slope that demonstrates the amount of horizontal change is in the denominator of a fraction. If we have coordinates of two points $A(x_A,y_A)$ and $B(x_B,y_B)$ in the two-dimensional Cartesian coordinate plane, then the slope $m$ of the line through $A(x_A,y_A)$ and $B(x_B,y_B)$ is fully determined by the following formula Thanks for the feedback. Just stick to the following steps to attain best! You can perform calculations with the following: Once you entered all the above parameters, then simply hit the calculate button, the slope calculator helps to find slope from two points and generate: Once done, then simply hit the calculate button, this slope intercept form calculator will provide you with: Once you added all the above-parameters, hit the calculate button, this point slope calculator will generates: Once you filled the all parameters, then simply hit the calculate button, this slope (m) calculator will generates: Once you entered the line equation, then hit the calculate button, this slope of a line calculator will generates: Thankfully, you come to know how to find the slope using the simple slope of a line formula. If any two sides of a triangle have slopes that multiply to equal -1, then the triangle is a right triangle. For Example: If angle = 90, then the slope is equal to tan (90). \text{Slope} m=\frac{y_B-y_A}{x_B-x_A} The slope work with steps shows the complete step-by-step calculation for finding the slope of line through the point A A at coordinates (2,5) (2, 5) and point B B at coordinates (−5,1) (− 5, 1). In this equation, you re-work the equation until you isolate y: y = x/2 – 5. In other words, the formula for the slope can be written as
These values must be real numbers or parameters; Slope calculator will give the slope of the line that passes through $A$ and $B$. Refer to the Triangle Calculator for more detail on the Pythagorean theorem as well as how to calculate the angle of incline θ provided in the calculator above. the slope of the line that passes through the points. From the question.
Second calculator finds the line equation in parametric form, that is,. m = 2. The line is horizontal; the slope is expressed as a 0. The formula to determine the distance (D) between 2 different points is: $$Distance (d) = \sqrt {(x₂ – x₁)^2 + (y₂ – y₁)^2 }$$. So, the equation of the line is $x=a$. The slope is an important concept in mathematics that is usually used in basic or advanced graphing like linear regression; the slope is said to be one of the primary numbers in a linear formula. The slope calculator, formula, work with steps and practice problems would be very useful for grade school students (K-12 education) to learn about the concept of line in geometry, how to find the general equation of a line and how to find relation between two lines. Practice Problem 1:Find the slope of the line through $(-1,6)$ and $(-10,15)$. Get check the result and you ought to make sure that this slope make sense by thinking about the points on the coordinate plane. The simple slope calculator is the tool that helps to find slope & distance between two points, slope & angle, x and y intercept, and slope intercept form for a given parameters. In many cases, we can find the slope for a given points by hand, especially for integers. This number is the gradient of the hill if it increases linearly. The slope intercept form calculator will find the slope of the line passing through the two given points, its y-intercept and slope-intercept form of the line, with steps shown. matter!
Objective : It will help you to find the coefficients of slope and y-intercept, as well as the x-intercept, using the slope intercept formulas. If you can only measure the change in x, multiply this value by the gradient to find the change in the y axis. Check out 23 similar coordinate geometry calculators , Input the values into the formula. (If an answer is undefined, enter UNDEFINED.) The run is the change in $x$, $\Delta x$.
Rejecting cookies may impair some of our website’s functionality. How To Find The Slope of a Line Given 2 Points? If the y-values are increasing as the x-values increase, it referred to as the line has a positive slope. The slope $m$ of a line $y=mx+b$ can be defined also as the rise divided by the run. Finding the Slope of a Line from the Graph, Finding the Slope of a Line from Two Points, Finding the Slope of a Line from the Equation, Finding the Equation of a Line Given a Point and a Slope, Finding the Equation of a Line Given Two Points. Rejecting cookies may impair some of our website’s functionality. To find the area under a slope you need to integrate the equation and subtract the lower bound of the area from the upper bound. To find the slope of a line we need two coordinates on the line.
A line is increasing, and goes upwards from left to right when m > 0, A line is decreasing, and goes downwards from left to right when m < 0, A line has a constant slope, and is horizontal when m = 0. If you want to calculate slope, all you need to divide the different of the y-coordinates of 2 points on a line by the difference of the x-coordinates of those same 2 points. Rate of change is particularly useful if you want to predict the future of previous value of something, as, by changing the x variable, the corresponding y value will be present (and vice versa). For any other coordinates of points, just supply four real numbers and click on the "GENERATE WORK" button. 1/4″ per foot pitch equals to 2% (percent), and remember that it is not expressed as 2 degrees.
What is the equation? Solution: Slopes are very important tool to determine whether two lines perpendicular or not.
As we know, the Greek letter $\Delta$, means difference or change. As the slope of a curve changes at each point, you can find the slope of a curve by differentiating the equation with respect to x and, in the resulting equation, substituting x for the point at which you’d like to find the gradient.
The formula becomes increasingly useful as the coordinates take on larger values or decimal values. The equation point slope calculator will find an equation in either slope intercept form or point slope form when given a point and a slope. Message received. Divide m by the new number of the order and put it in front of the new x. The sign in front of the gradient provided by the slope calculator indicates whether the line is increasing, decreasing, constant or undefined. Slope definition is very simple; it is said to be a measure of the difference in position between two points on a line. All Rights Reserved. The slope of a roof will change depending on the style and where you live. To find the gradient of other polynomials, you will need to differentiate the function with respect to x. The calculator also has the ability to provide step by step solutions. Remember that if the line is horizontal anytime (flat from left to the right), the slope is said to be zero. Notice that the slope of a line is easily calculated by hand using small, whole number coordinates.
Download Slope Calculator App for Your Mobile, So you can calculate your values in your hand. Answer to: Find the slope of a line passing through the points (-3, 1/2) and (2, 5). No doubt, points on a line can be readily solved given the slope of the line and the distance from another point. Enter coordinates $(x_A,y_A)$ and $(x_B,y_B)$ of two points $A$ and $B$ in the box. According to the mathematician, if the line is plotted on a 2-dimensional graph, then the slope is something that shows how much the line moves along the x-axis and the y-axis between those 2 points. We have the final answer as. The symbol Δ is used to express the delta of x and y, simply, it is the absolute value of the distance between x values or y values of 2 points. This will result in a zero in the numerator of the slope formula. The slope of these points (-10, 1) and (-4, 0) is perpendicular to this line.
In the case of a road the "rise" is the change in altitude, while the "run" is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth's curvature should be considered as a factor. if it goes up from left to right; Negative slope $m<0$, if a line $y=mx+b$ is decreasing, i.e. This new value is the length of the slope. The method for finding the slope from an equation will vary depending on the form of the equation in front of you. By using this website, you agree to our Cookie Policy. Free line equation calculator - find the equation of a line step-by-step This website uses cookies to ensure you get the best experience.
If the y-values are not changing as x increases, it is indicated as the line will have a slope of 0. Yes, slope can be determined as a percentage that is calculated in much the same as the gradient.
A 1/20 slope is equivalent to a gradient of 1/20 (strangely enough) and forms an angle of 2.86° between itself and the x-axis. where (x1 , y1) and (x2 , y2) are the points. Well, we can easily calculate ‘b’ from this equation: Now, let’s plug-in the values into the above equation: Very next, we plug-in the value of ‘b’ and the slope into the given equation: Also, you can use the above point slope calculator to perform instant calculations instead of sticking to these manual calculation steps! \text{Slope} m=\frac{10-2}{7-3} An online point slope calculator allows you to find the slope or gradient between two points in the Cartesian coordinate system. Write a new line where you add 1 to the order of the x (e.g., x becomes x^2, x^2.5 becomes x^3.5). If the product of slopes of two lines in the plane is $-1$, then the lines are perpendicular and vice-versa. © 2019 Coolmath.com LLC. Also, our slope intercept calculator will also show you the same answer for these given parameters.
It doesn’t make any legitimate sense to divide by 0 so it is said that the slope of a vertical line is undefined.
.
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Jews as Refugees – Biblical Connection
The Exodus from Egypt, one of the central stories in Jewish tradition, is a foundation document of Jews as refugees. Having been held slaves under the pharoahs for several hundred years, the Israelites were desperate for their freedom. Their dramatic departure reached its climax at the Red Sea, when the Israelites were finally able to escape Pharoah’s soldiers and Egypt.
Jewish law - Spiritual Connection
Jewish texts and laws provide a constant reminder about the Exodus and about the experience of slavery in Egypt. The Exodus is the central experience recounted in the Torah. It is mentioned as part of our weekly Friday night blessings, and is told in far greater detail at our annual Passover seders. At the seder, surrounded by comfort and good food, we are encouraged to remember the story as if we were the ones who had been slaves and refugees.
The experience in Egypt is repeated often in the Torah, with the explicit admonition to treat others with compassion and justice. Our Biblical laws – requiring us to provide for the widow and orphan, to treat workers fairly, and to help the foreigners in our midst - explain that we must do so because we ourselves were foreigners in Egypt. This idea is echoed more than 30 times in the Torah. Here too, we are asked to put ourselves in the position of our ancestors: slaves and foreigners in Egypt.
Jews as Refugees - Historical Connections
The Shoah (Holocaust) was another defining moment for the Jewish people. The genocide of six million Jews was a tragedy, heightened by the indifference of those free countries who refused to admit Jews to safety on their shores. Canada is one of several countries who bear the shame of having refused to receive 900 Jewish refugees who had managed to escape Germany aboard the ocean liner St. Louis. Many Canadian Jews remember the treatment of these desperate Jewish refugees.
Jews and Roma – Parallel Histories of Persecution
The persecution of Jews in Europe for many centuries, bears certain similarities to that experienced by the Roma people, who also faced restrictions on their employment and permitted living areas, violence, expulsions, and other forms of oppression. The Roma people, like the Jews, were the only other group legally targeted for extermination by the Nazis, and experienced their own genocide known as the Porajmos. Unfortunately, the Roma in Europe continue to suffer many forms of violence, discrimination, at the hands of bigots and fascists with the collusion of some governments.<|endoftext|>
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# Congruence and similarity
### 2) Similar shapes
When a shape is enlarged, the image is similar to the original shape. It is the same shape but a different size.
These two shapes are similar as they are both rectangles but one is an enlargement of the other.
##### Similar triangles
Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio. Either of these conditions will prove two triangles are similar.
Triangle B is an enlargement of triangle A by a scale factor of 2. Each length in triangle B is twice as long as in triangle A.
The two triangles are similar.
##### Example 1
State whether the two triangles are similar. Give a reason to support your answer.
Yes, they are similar. The two lengths have been increased by a scale factor of 2. The corresponding angle is the same.
##### Example 2
State whether the two triangles are similar. Give a reason to support your answer.
To decide whether the two triangles are similar, calculate the missing angles.
Remember angles in a triangle add up to 180°.
Angle $yzx = 180-85-40=55$°
Angle $YZX = 180-85-55=40$°
Yes, they are similar. The three angles are the same.
##### Example 3
State whether the two triangles are similar. Give a reason to support your answer.
No. The two sides of the triangle are increased by a scale factor of 1.5. The other side has been increased by a scale factor of 2.
##### Calculating lengths and angles in similar shapes
In similar shapes, the corresponding lengths are in the same ratio. This fact can be used to calculate lengths.
##### Example
Calculate the length PS.
The scale factor of enlargement is 2.
Length PS is twice as long as length ps.
PS = 9 x 2 = 18 cm
Similar shapes may be inside one another.
Question 1
Show that triangles ABC and DBE are similar and calculate the length DE.
Answer: Angle BCA = angle BED because of corresponding angles in parallel lines.
Angle BAC = angle BDE because of corresponding angles in parallel lines.
Angle DBE = angle ABC because both triangles share the same angle.
All three angles are the same in both triangles so they are similar.
To calculate a missing length, draw the two triangles separately and label the lengths.
To calculate the scale factor, divide the two corresponding lengths.
The scale factor of enlargement is $\frac{6}{4} = 1.5$.
Therefore, DE = 7.5 : 1.5 = 5 cm
Question 2
Calculate the length TR.
Answer: To calculate a missing length, draw the two triangles separately and label the lengths.
The scale factor is $\frac{6}{3} = 2$.
To calculate TR, first find QR.
QR = 6 x 2 = 12 cm
QR = QT + TR
TR = QR – QT
TR = 12 – 6 = 6 cm
< Previous | 1 | 2 | 3 | 4 | Next ><|endoftext|>
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# Algorithms
Within these castle walls be forged Mavens of Computer Science ...
— Merlin, The Coder
Put simply, an algorithm is a set of instructions to perform a computation. For example, to determine if a number is prime or composite, finding the factors of a number, generating passwords using a given set of characters, etc.
## Functions Crash Course
We'll talk more about functions later on, but for now just know that a function in computer science works similarly to functions in math: the function takes input, does something with it, and returns something. These are useful when writing algorithms because they allow us to write an algorithm once in a program and use it as many times as needed. Here's an example function that returns the given number multiplied by two:
```func multiplyByTwo(n: Int) -> Int {
// 'n' is now defined as the argument passed to the function
return n * 2
}
let x = multiplyByTwo(n: 8) // x is 16
```
Put simply, a function in Swift is defined by the keyword func, followed by the function's name, list of parameters, then its return type and lastly the actual code the function will run.
For the following exercises, the function boilerplate will be provided.
## Exercises
Since most of the characters are reserved by Swift, the characters '[?]' will be used to show a fill in the blank.
### Parity
The parity of an integer is whether it is even or odd. In math class, you have most likely already learned that a number is even if it is divisible by two and odd otherwise.
For this exercise, you will complete an algorithm to determine if a number is even. To do this, you can begin by using the below function as a starting point, and replacing the blank with the correct arithmetic operator. The comparison operator == checks for equality between the terms on both sides. So, you'll need an operator that returns the integer 0 if the number is evenly divisible by 2, or any other number otherwise.
```func isEven(n: Int) -> Bool {
return n [?] 2 == 0
}
```
```func isOdd(n: Int) -> Bool {<|endoftext|>
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(a) Name a process by which hydrogen gas is manufactured.
(b) Give equations for the reactions.
(c) How is hydrogen separated from carbon dioxide and carbon monoxide?
(a) Commercially, hydrogen is prepared by Bosch process.
(b) (i) Steam is passed over hot coke at 1000°C in a furnace called converters. As a result water gas is produced which is a mixture of carbon monoxide and hydrogen gases.
This reaction is endothermic in nature.
(ii) Water gas is mixed with excess of steam and passed over a catalyst ferric oxide (Fe2O3) and a promotor chromium trioxide (Cr2O3).
This reaction is exothermic in nature
(c) (i) The products are hydrogen, carbon dioxide and some
unreacted carbon monoxide. Hydrogen is separated from carbon dioxide by passing the mixture through water under pressure, in which carbon dioxide gets dissolved leaving behind hydrogen. Carbon dioxide can also be separated by passing it through caustic potash (KOH) solution.
2KOH + CO2 → K2CO3 + H2O
(ii) To separate carbon monoxide the gaseous mixture is passed through ammoniacal cuprous chloride in which carbon monoxide dissolves leaving behind hydrogen.
Thus hydrogen gas is obtained.<|endoftext|>
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# How do solve the following linear system?: x - 3y = 8 , 3y = 15 -6x ?
Dec 5, 2015
The solution for the system of equations is :
color(blue)(x=23/7, y=(-11)/7
#### Explanation:
$x - 3 y = 8$
x=color(blue)(8 +3y....................equation $1$
$3 y = 15 - 6 x$................equation $2$
Solving by substitution.
Substituting equation $1$ in $2$
$3 y = 15 - 6 x$
$3 y = 15 - 6 \cdot \left(\textcolor{b l u e}{8 + 3 y}\right)$
$3 y = 15 - 48 - 18 y$
$3 y + 18 y = 15 - 48$
$21 y = - 33$
$y = \frac{- \cancel{33}}{\cancel{21}}$
color(blue)(y=(-11)/7
Now, we substitute this value of $y$ in equation $1$ to find $x$
$x = 8 + 3 y$
$x = 8 + 3 \cdot \left(- \frac{11}{7}\right)$
$x = 8 - \frac{33}{7}$
$x = \frac{56}{7} - \frac{33}{7}$
color(blue)(x=23/7<|endoftext|>
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Henry V is a history play by William Shakespeare, believed to be written in 1599. It is based on the life of King Henry V of England, and focuses on events immediately before and after the Battle of Agincourt (1415) during the Hundred Years' War. The play is the final part of a tetralogy, preceded by Richard II, Henry IV, part 1 and Henry IV, part 2. The original audiences would thus have already been familiar with the title character, who was depicted in the Henry IV plays as a wild, undisciplined lad known as 'Prince Hal.' In Henry V, the young prince has become a mature man and embarks on an attempted conquest of France.
About the Author
William Shakespeare (baptised 26 April 1564 23 April 1616) was an English poet and playwright, widely regarded as the greatest writer in the English language and the world's preeminent dramatist.He is often called England's national poet and the Bard of Avon (or simply The Bard ). His surviving works consist of 38 plays,154 sonnets, two long narrative poems, and several other poems. His plays have been translated into every major living language and are performed more often than those of any other playwright. Shakespeare was born and raised in Stratford-upon-Avon. At the age of 18, he married Anne Hathaway, who bore him three children: Susanna, and twins Hamnet and Judith. Between 1585 and 1592, he began a successful career in London as an actor, writer, and part owner of a playing company called the Lord Chamberlain's Men, later known as the King's Men. He appears to have retired to Stratford around 1613, where he died three years later. Few records of Shakespeare's private life survive, and there has been considerable speculation about such matters as his physical appearance, sexuality, religious beliefs, and whether the works attributed to him were written by others. Shakespeare produced most of his known work between 1590 and 1613. His early plays were mainly comedies and histories, genres he raised to the peak of sophistication and artistry by the end of the sixteenth century. He then wrote mainly tragedies until about 1608, including Hamlet, King Lear, and Macbeth, considered some of the finest works in the English language. In his last phase, he wrote tragicomedies, also known as romances, and collaborated with other playwrights. Many of his plays were published in editions of varying quality and accuracy during his lifetime. In 1623, two of his former theatrical colleagues published the First Folio, a collected edition of his dramatic works that included all but two of the plays now recognised as Shakespeare's. Shakespeare was a respected poet and playwright in his own day, but his reputation did not rise to its present heights until the nineteenth century. The Romantics, in particular, acclaimed Shakespeare's genius, and the Victorians hero-worshipped Shakespeare with a reverence that George Bernard Shaw called bardolatry .In the twentieth century, his work was repeatedly adopted and rediscovered by new movements in scholarship and performance. His plays remain highly popular today and are constantly studied,<|endoftext|>
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# ALGEBRA
Mae open her coin purse and found pennies,nickels and dimes with a total value of \$2.85.If there are twice as many pennies as there are nickels and dimes combined.How many pennies,dimes and nickels are there if she has a collection of 90 coins
given:
penny= 1 cent =\$0.01
nickel= 5 cents =\$0.05
dine= 10 cents =\$0.10
quarter= 25 cents =\$0.25
half= 50 cents =\$0.50
dollar= 100 cents =\$1
1. 👍
2. 👎
3. 👁
1. number of nickels --- n
number of dimes ---- d
number of pennies --- n+d
n + d + n+d = 90
2n+2d = 90
n+d = 45 or d = 45-n
value equation:
5n + 10d + 1(n+d) = 285
6n + 11d = 285
using substitution:
6n + 11(45-n) = 285
6n + 495 - 11n = 285
-5n = -210
n = 42
d = 45-42 = 3
so we have 42 nickels, 3 dimes and 45 pennies
check:
do they add up to 90 ? YEAH
what is their value?
42(5) + 3(10) + 45(1) = 285. YEAHHH!!!
1. 👍
2. 👎
2. by Reiny's formation of equations,
eqn.1. should look like n + d + 2(n + d)=90
3n + 3d=90
3(n + d)=90
n + d=30
n=30 - d
eqn.2. 5n + 10d + 2(n + d)=285
7n + 12d=285
7(30 - d) + 12 d=285
5d= 70
[d=15]
then lets find out n:
n=30-15
[n=15]
lets check if it all adds to 90 coins:
n + d + 2(n + d)
15 + 15 + 2(15 + 15)
30 + 60
90 coins checks out
1. 👍
2. 👎
## Similar Questions
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In Vanessa's coin bank there are 4 times as many quarters as nickels, 2 fewer dimes than nickels, and 15 pennies. The total amount in the bank is \$5.80. How many of each coin are in the bank? A. Assign a variable to the unknown
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1. ### algebra
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Kevin has \$6.45 in coins in his cash box. The numbers of quarters is one less than twice the number of dimes. The number of nickels is one less than twice the number of quarters. The value of the pennies is the same as the value
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This is a word problem I am supposed to solve with systems. I am having trouble setting it up if anyone could help me. A coin bank contains only dimes and nickels. The bank contains 46 coins. If 5 dimes and 2 nickels were removed<|endoftext|>
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1. ## solving for substitution
having trouble understanding substitution this is the problem :
Solve for the system of equations by substitution
5x-7=-y
2x-y=0
This is another one..
6x+7y=1
x=55-9y
2. $\displaystyle 5x-7=-y$
$\displaystyle 2x-y=0$.
Substituting $\displaystyle -y$ into the second equation gives
$\displaystyle 2x+5x-7=0$
$\displaystyle 7x-7=0$
$\displaystyle 7x = 7$
$\displaystyle x = 1$.
You know $\displaystyle -y = 5x-7$ so
$\displaystyle -y = 5\cdot 1 - 7$
$\displaystyle -y = 5 - 7$
$\displaystyle -y = -2$
$\displaystyle y = 2$.
Therefore $\displaystyle (x, y) = (1, 2)$ is the solution.
3. 5x-7=-y
2x-y=0
First, solve one of the equations for one of the variables. It really doesn't matter which equation or which variable. You probably will want to choose the way that's the easiest. I'm going to start by solving the 2nd equation for y:
\displaystyle \begin{aligned} 2x - y &= 0 \\ 2x &= y \\ \end{aligned}
Now, substitute this into the 1st equation wherever you see a y, and then solve for x:
\displaystyle \begin{aligned} 5x - 7 &= -y \\ 5x - 7 &= -2x \\ -7 &= -7x \\ x &= 1 \end{aligned}
Now, plug this into the altered version of the 2nd equation:
\displaystyle \begin{aligned} 2x &= y \\ 2(1) &= y \\ y &= 2 \end{aligned}
The answer is (1, 2). You want to try your other example now?
EDIT: Too slow!
4. thank you! does anyone know solving for elimination?
4x=15+3y
-6/5x+y=-17/5
5. I can't really read that...
Is it
$\displaystyle 4x=15+3y$
$\displaystyle -\frac{6}{5x}+y = -\frac{17}{5}$?
6. yes the 6/5x , the x is for both the 6 and 5 not just the 5.. not even sure if that matters but you have it right.
7. Obviously it's not right if it's meant to be
$\displaystyle 4x=15+3y$
$\displaystyle -\frac{6}{5}x+y = -\frac{17}{5}$
since $\displaystyle \frac{6}{5}x = \frac{6x}{5}$, not $\displaystyle \frac{6}{5x}$...
Anyway...
$\displaystyle 4x=15+3y$
$\displaystyle -\frac{6}{5}x+y = -\frac{17}{5}$
Multiply the second equation by $\displaystyle 3$...
$\displaystyle 4x=15+3y$
$\displaystyle -\frac{18}{5}x+3y=-\frac{51}{5}$
$\displaystyle 4x-3y=15$
$\displaystyle -\frac{18}{5}x+3y=-\frac{51}{5}$.
$\displaystyle (4x-3y) + \left(-\frac{18}{5}x+3y\right) = 15-\frac{51}{5}$
$\displaystyle \frac{2}{5}x=\frac{24}{5}$
$\displaystyle 2x = 24$
$\displaystyle x = 12$.
Substituting into the first equation
$\displaystyle 4x=15+3y$
$\displaystyle 4\cdot 12 = 15 + 3y$
$\displaystyle 48 = 15 + 3y$
$\displaystyle 33 = 3y$
$\displaystyle y = 11$.
So $\displaystyle (x, y) = (12, 11)$ is the solution.
8. Originally Posted by eumyang
5x-7=-y
2x-y=0
First, solve one of the equations for one of the variables. It really doesn't matter which equation or which variable. You probably will want to choose the way that's the easiest. I'm going to start by solving the 2nd equation for y:
\displaystyle \begin{aligned} 2x - y &= 0 \\ 2x &= y \\ \end{aligned}
Now, substitute this into the 1st equation wherever you see a y, and then solve for x:
\displaystyle \begin{aligned} 5x - 7 &= -y \\ 5x - 7 &= -2x \\ -7 &= -7x \\ x &= 1 \end{aligned}
Now, plug this into the altered version of the 2nd equation:
\displaystyle \begin{aligned} 2x &= y \\ 2(1) &= y \\ y &= 2 \end{aligned}
The answer is (1, 2). You want to try your other example now?
EDIT: Too slow!
yeah if you can help me out with the other example that would be great.. this is one part of math that i do not like haha
9. double posted by accident sorry
10. OP: Prove It's point is that you have to be careful with notation. Ideally, you should learn LaTeX so that it's not ambiguous when you type $\displaystyle \frac{6}{5}x$. 6/5x is really read as $\displaystyle \frac{6}{5x}$. If you are not using LaTeX and you want to indicate the fraction 6/5 times x, use parentheses: (6/5)x.<|endoftext|>
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# Problem Solving (Part 2)
In this lesson, we will learn how to use different strategies to solve questions involving money.
Quiz:
# Intro quiz - Recap from previous lesson
Before we start this lesson, let’s see what you can remember from this topic. Here’s a quick quiz!
Q1.Emily bought ONE of each sweet. How much did the sweets cost altogether?
1/3
Q2.Emily bought two different sweets. She paid 8p and did not need any change. What sweets did she buy?
2/3
Q3.Emily bought some sweets. The total cost of her sweets was 20p. She paid with a 50p coin. How much change did she receive?
3/3
Quiz:
# Intro quiz - Recap from previous lesson
Before we start this lesson, let’s see what you can remember from this topic. Here’s a quick quiz!
Q1.Emily bought ONE of each sweet. How much did the sweets cost altogether?
1/3
Q2.Emily bought two different sweets. She paid 8p and did not need any change. What sweets did she buy?
2/3
Q3.Emily bought some sweets. The total cost of her sweets was 20p. She paid with a 50p coin. How much change did she receive?
3/3
# Video
Click on the play button to start the video. If your teacher asks you to pause the video and look at the worksheet you should:
• Click "Close Video"
• Click "Next" to view the activity
Your video will re-appear on the next page, and will stay paused in the right place.
# Worksheet
These slides will take you through some tasks for the lesson. If you need to re-play the video, click the ‘Resume Video’ icon. If you are asked to add answers to the slides, first download or print out the worksheet. Once you have finished all the tasks, click ‘Next’ below.
Quiz:
# Further Problem Solving
This quiz will help you asses what you have learned in this lesson.
Q1.How many pennies are equal to £1
1/3
Q2.George wants to buy one of each type of button. What is the cost of the buttons altogether?
2/3
Q3.George's dad bought two different types of buttons. He only spent 30p. What two buttons did he buy?
3/3
Quiz:
# Further Problem Solving
This quiz will help you asses what you have learned in this lesson.
Q1.How many pennies are equal to £1
1/3
Q2.George wants to buy one of each type of button. What is the cost of the buttons altogether?
2/3
Q3.George's dad bought two different types of buttons. He only spent 30p. What two buttons did he buy?
3/3
# Lesson summary: Problem Solving (Part 2)
## Time to move!
Did you know that exercise helps your concentration and ability to learn?
For 5 mins...
Move around:
Jog
On the spot:
Dance<|endoftext|>
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Quadratic Formula and Quadratic Equations
More Lessons for Algebra II
A series of free, online Intermediate Algebra Lessons or Algebra II lessons.
Videos, worksheets, and activities to help Algebra students.
Deriving the Quadratic Formula
The quadratic formula is one of the most important formulas that you will learn in Algebra and chances are that you have probably memorized it. But where does it come from? When deriving the quadratic formula, we first start with a generic quadratic formula using coefficients a, b and c and then derive the formula by completing the square.
This video shows the proof of the quadratic formula by solving ax2 + bx + c by completing the square.
Overview of the Different Methods of Solving a Quadratic Equation
Solving quadratic equations can be difficult, but luckily there are several different methods that we can use depending on what type of quadratic that we are trying to solve. The four methods of solving a quadratic equation are factoring, using the square roots, completing the square and the quadratic formula.
More details about the methods used to solve a quadratic equation
How to know which method to use when solving a quadratic equation.
The Discriminant of a Quadratic Equation
The discriminant is part of the quadratic formula which lies underneath the square root. The quadratic equation discriminant is important because it tells us the number and type of solutions. This information is helpful because it serves as a double check when solving quadratic equations by any of the four methods (factoring, completing the square, using square roots, and using the quadratic formula).
A discussion on what the discriminant is, how it's used, and it's connection to graphs of quadratic functions.
Solving Quadratic Equations in Disguise
There are four methods to solving quadratic equations: factoring, completing the square, using square roots, and using the quadratic formula. Sometimes there are more complex quadratic equations including equations that have fractional exponents and negative exponents. To solve these types of problems, we either make a substitution for a term or factor out negative exponents.
Using quadratic methods to solve quadratic equations and exponential equations that can be modified to look like quadratics.
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.<|endoftext|>
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Meet the Stillmans
All vaccines against N. meningitidis work by the induction of antibodies in serum. Transudation from serum may block adherence of the bacteria to epithelial cells in the nasopharynx to prevent contagion spread and provide herd immunity. Serum bactericidal antibody levels of greater than or equal to 1:4 correlate with protection. However, a specific level of antibody is not an absolute correlate of protection for every person because there is genetic variation in susceptibility to disease, there can be differences in virulence among strains, there can be differences in innate immune responses among individuals, there can be variation in the inoculum of the pathogen, and there may be effects from concurrent illness or coinfection. Therefore, a specified protective level of antibody should be considered as a close estimate applicable in the majority of potentially susceptible persons.
Antibody levels usually decline over time after vaccination if boosters are not given, although there are exceptions. Persistence of vaccine-induced antibodies usually goes well beyond the time when antibodies should have disappeared according to the mathematics of their half-life. This may be caused by ongoing “natural” boosting or other immunologic mechanisms. Natural boosting can occur by asymptomatic colonization by the pathogen or by a nonpathogen expressing a cross-reactive antigen. Natural boosting can decrease over time as a pathogen circulates less widely in a population because of increasing use of a vaccine and/or the establishment of herd immunity. This is an ongoing issue relative to several vaccines because the absence of natural boosting among vaccinees may lead to a return to disease susceptibility.
While many of the components of the immune response triggered by vaccination are known, the immune components that are needed to sustain a protective immune response are not fully understood. Vaccines that activate multiple antigen-presenting cell subsets through a diverse array of Toll-like receptors produce immunity that is more durable. The combination of receptor activation induces a set of proinflammatory cytokines that affect the Th1/Th2 helper cell balance and provides a strong stimulus for B-cell and T-cell memory. This is especially true in the case of the conjugate vaccines in which the naked polysaccharide of the bacteria is poorly immunogenic in very young children. However, conjugation of the polysaccharide with a protein activates T cells to help the B-cells in producing antibodies to the polysaccharide.
The production of memory B cells and T cells is a complex developmental process. The likelihood that a B-cell or T-cell memory response will be fast enough in the absence of a protective circulating antibody level likely depends in large part on the pace of pathogenesis of the infection caused by a specific organism.
The speed of production of measurable antibody responses following memory B-cell stimulation has been measured for N. meningitidis. A detectable response occurs 2-7 days following exposure. It takes that amount of time for antibody production because the bacteria must be taken up and processed by antigen presenting cells, then these cells must interact with B cells and T cells, then the B cells must proliferate and mature to plasma cells, and finally the plasma cells release the antibodies into circulation. Even if the antigen directly activates memory B cells by interaction with the B-cell receptor and no T-cell help is necessary, the memory B cell still takes time to process the antigen and mature to an antibody-secreting plasma cell.
Credit: © 2018 National Meningitis Association. All rights reserved.
In the United States, the Centers for Disease Control and Prevention’s Advisory Committee on Immunization Practices has recommended since 2005 the routine vaccination against N. meningitidis serogroups A, C, W, and Y for individuals aged 11-18 years. There were no N. meningitidis serotype B (MenB) vaccines available until 2014. There are now two vaccines (Bexero and Trumenba) that were licensed under the accelerated approval of biological products regulations for use in individuals aged 10-25 years. This approval was granted with the caveat that postmarketing clinical studies would be performed in order to further investigate the immunogenicity and tolerability of the products and to assess effectiveness in the targeted population in the United States. In 2015, the CDC’s Advisory Committee on Immunization Practices issued a category B recommendation to vaccinate against MenB in individuals aged 16-23 years to provide short-term protection against most strains of serogroup B meningococcal disease.
The human serum bactericidal assay for measuring functional antibody levels was used as the immunological correlate of protection and acceptance for licensure. At the time of licensure, limited information regarding the length of protection against disease was available. However, data on antibody persistence after Bexsero vaccination continue to accumulate.
For example, in a study of adolescent vaccinees, antibody levels were measured 18-23 months after a two-dose series: When measured for three of the four test strains, as many as 94%, 82%, and 77% of the subjects had protective titers to each of the three strains (Santolaya et al.Hum Vaccin Immunother. 2013 Nov 1; 9:2304-10).
The immunogenicity of Bexsero among adult laboratory staff using the recommended schedule of two doses at a 5-week interval was assessed. Immunogenicity was evaluated 6 weeks and 1 year after the second dose. All participants showed an increase in their bactericidal titers against the components of Bexsero 6 weeks after the second dose; however, titers declined significantly 1 year later (Hong et al. Hum Vaccin Immunother. 2017 Mar;13:645-8).
The Trumenba vaccine (Pfizer) consists of two variants of factor H–binding protein, one from subfamily A and one from subfamily B. It was licensed in the United States in October 2014 for use in individuals aged 10-25 years, with a three-dose schedule administered over a 6-month period. Recently, a two-dose schedule with the two doses administered 6 months apart has been licensed.
In a study of adult health care workers, three doses of Trumenba elicited short-term protective SBA responses to diverse disease-causing serogroup B strains. For some strains, serum titers declined to 1:4 by 9-11 months, which raised concerns about the duration of broad, long-term protection (Lujan et al. Clin Vaccine Immunol. 2017 Aug 4. doi: 10.1128/CVI.00121-17).
Studies investigating the persistence of the immune response elicited by Trumenba found that there is a decline in antibodies over the 6-12 months following the last vaccine dose, after which the decline levels off (Shirley and Taha. Drugs. 2018 Feb;78:257-68). Substantial increases in antibody responses against all primary test strains were observed following a booster dose of Trumenba administered 4 years after the primary series, increases that were consistent with induction of immunological memory. There was no clear benefit of a three-dose series over a two-dose series in terms of the persistence of antibodies or the response to a booster dose.
Given the persistence data and given that circulating serum antibodies are considered necessary to convey protection against invasive meningococcal disease (with immunological memory alone not likely to be sufficient), some authorities recommended that, following a primary vaccination series, a booster dose of N. meningitidis serogroup B vaccine should be considered for individuals at continued risk of invasive meningococcal disease. The appropriate timing of a booster dose would be dependent on several factors, including the declining antibody titers, the local epidemiology and the antigen expression of circulating strains, and the possible effects of herd immunity. There are currently no data I am aware of on the persistence of the immune response following a booster dose of N. meningitidis serogroup B vaccine, but this will be examined in ongoing trials.
The antibody waning measured in the studies of Bexsero and Trumenba make it clear that the effective duration of protection following vaccination may be relatively short lived. It generally requires about 2-7 days for B memory cells and T memory cells to expand and give rise to increases in antibody levels on exposure to N. meningitidis. The innate immune system and preexisting circulating antibody levels must prevent progression of disease until memory responses occur. For N. meningitidis serogroup B in which the pace of pathogenesis is rapid, some individuals will contract infection before the memory response is fully activated and implemented. This is an area of active research for the future.<|endoftext|>
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Tiny pieces of plastic, now ubiquitous in the marine environment, have long been a cause of concern for their ability to absorb toxic substances and potentially penetrate the food chain. Now scientists are beginning to understand the level of threat posed to life, by gauging the extent of marine accumulation and tracking the movement of these contaminants.
So-called microplastics are described as particles sized about 5 mm or smaller. Originating from various sources, some, called microbeads, are intentionally included as exfoliating components in cosmetics. Others emerge from normal wear and tear on the products. The majority of microplastics, however, originate from the disintegration of larger pieces of plastic waste such as packaging material on land, at coastal sites or in the sea.
These particles are considered the most common form of marine litter. However, the European Food Safety Authority says that many questions remain regarding the human health effects of microplastics and nanoplastics – particles with a diameter smaller than one-thousandth of a millimetre.
Dr Ana Catarino, a postdoctoral research associate at the UK-based Natural Environment Research Council, says there is considerable data indicating that organisms ingest microplastics. However, studies demonstrate that the concentration of microplastics in the environment is several orders of magnitude lower than most tested concentrations in the laboratory, indicating that the harmful effects could be minimal, she said.
‘Microplastics may accumulate in the gut and potentially interfere with processes like nutrient uptake or the passage of waste – but studies also showed they may just be expelled without any negative effects.’
Dr Catarino served as project researcher for the MARMICROTOX project, which was conducted between 2014 and 2016 to assess the abundance and type of microplastics in wild mussels collected from a remote coastal location in Scotland. The researchers carried out tests to check whether toxic substances associated with particles are transferred into fish such as trout, and how microplastics affect mussels.
Preliminary results suggested that toxic substances associated with the surface of microplastics might be absorbed by mussels and fish when they ingest particles. However, research into understanding how this exposure to toxicity compares to plastic concentrations in different environments – such as contaminated food – needs to be conducted, Dr Catarino said.
The researchers also observed microplastics, mostly fibres, in their mussel samples. Following this, the team went on to investigate the risk of humans ingesting particles via mussels to the ingestion of plastic fibres from household dust.
After cooking ‘in our kitchens, we left open petri dishes with sticky tape to collect dust fallout in the surrounding air. We compared the amounts of plastic fibres in this dust with the quantities we found in mussels,’ Dr Catarino said.
In an unexpected turn of events, data from the study indicated that while a regular UK consumer may ingest 100 plastic particles a year from eating mussels, their average exposure to plastic particles during meals from household dust is well over 10,000 per year. However, even the risk of such a level of exposure to human health is unknown, she added.
'You could say 99% of the plastic is missing.'
Dr Erik Van Sebille, Utrecht University, the Netherlands
In order to gauge the level of health risk, it is imperative to target areas where plastic is most prevalent to understand how animals actually encounter plastic, said Dr Erik Van Sebille, associate professor of oceanography at Utrecht University in the Netherlands.
‘We just don’t know that yet, because we don’t know where the plastic is.’
To support research into the impact of microplastics on aquatic life, biodiversity and human health, scientists including Dr Van Sebille are looking into where plastic ends up in the ocean.
‘The best estimates we have are from the surface of the ocean, in terms of floating plastic, and that is probably just 1% or so of all the plastic we think has ever gone into the ocean. So you could say 99% of the plastic is missing,’ he said.
‘It’s bit like accounting, so much is going in, so much is going out. Where’s the rest?’
Dr Van Sebille is involved in the TOPIOS.org project, which is in the process of developing a 3D map of all the plastic in the ocean, combining a circulation model with various observations of its whereabouts across the earth's oceans.
Three decades ago, scientists created a virtual computer simulation of how carbon dioxide is carried by the wind, Dr Van Sebille said. One year into the ambitious five-year TOPIOS.org project, he said ‘I’m proposing to do exactly the same within the ocean for plastic.’
Because the oceans are are massive, there may not be enough observations taken so far by scientists across the world to understand which areas are at high risk of pollution. Even so, TOPIOS could provide valuable insight into which regions require more observation, said Dr Van Sebille.
It is estimated that more than 150 million tonnes of plastics have accumulated in the world’s oceans, and research shows that between 4.8 and 12.7 million tonnes was added in 2010. A handful of European nations including the UK and the Netherlands, in addition to North America, are either considering or have enforced bans on plastic microbeads typically found in cosmetics and personal care products.
The research in this article was funded by the EU. If you liked this article, please consider sharing it on social media.
Each of us harbours hundreds of man-made chemicals inside our bodies because we are exposed to them in our daily lives. While individual chemicals may not be of immediate concern to public health, scientists now worry that certain mixtures of them may pose previously underestimated risks to health.
Teenagers rarely have a say in the public health policies that concern them, but we can’t halt the childhood obesity problem without working with them, says Professor Knut-Inge Klepp, executive director of the mental and physical health division at the Norwegian Institute of Public Health.
Staring at rows of numbers or formulas on a page can be off-putting for many children studying mathematics or science in school. But music, drawing and even body movement are providing promising new ways of teaching complex subjects to youngsters.
Dedicated policies and guidelines aim to reduce everyday exposure.
Mental health and free wifi in fast food joints have been raised as pertinent issues, says public health expert.
Notre Dame restoration is a learning opportunity, says historian.<|endoftext|>
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# ABX01 - EDITORIAL
Simple/Easy
## Prerequisites
Fast modular exponentiation, Observation skills
## Problem
You are given two integers A and N. The task is to compute F(A^N) where F(X) is a function which results in a non-negative single digit integer obtained by a process of summing digits,and on each iteration using the result from the previous iteration to compute a digit sum untill single digit number is reached.
## Explanation
We can simply find A^N as it doesn’t exceed 10^{15} and then evaluate F(A^N) by the recursive process mentioned in the definition of this function.Note that we can evaluate F(A^N) in less than 5 iterations.
### For Subtask 2 and 3
Let us first observe the function F first.What will be F(X) for a single digit integer X ? It will simply be the number itself.Or we can say for a single digit integer X ,F(X)=X\%9 for X!=9 and F(X)=9 for X=9.Combining these two we can write F(X)=X\%9 + 9*(X\%9==0) for a single digit integer X where X\%9==0 will return 1 only when X is 9. Now what will be F(X) for a two digit integer X. For a two digit integer X which is a multiple of 9 observe that F(X)=9 and for X not a multiple of 9 , F(X)= X\%9. Generalising this we can define F(X) as follows : F(X)= X\%9 + 9*(X\%9==0) for any non negative integer X where (X\%9==0) will return 1 only when X is a multiple of 9.
Now lets try to find out F(X*X). Consider two Cases :
• When X is a multiple of 9 i.e F(X)=9 then X*X is also a multiple of 9 and consequently F(X*X)=9. Also see here that F(F(X)*F(X))= F(9*9)=F(81)=9.Hence when X is a multiple of 9 then F(X*X)=F(F(X)*F(X)).
• When X is not a multiple of 9 then F(X)=X\%9 . Hence, we have F(F(X)*F(X))=F((X\%9)*(X\%9))=((X\%9)*(X\%9))\%9 = (X*X)\%9 = F(X*X) .Hence in this case also F(X*X)=F(F(X)*F(X)).
So lets generalise this. F(X^2)=F(F(X)*F(X)). Similarly, F(X^4)=F( (F(F(X)*F(X)))^2 ).
Now for SUBTASK #2 where N<=100 we can evaluate F(A^N) by iterating from 1 to N and taking F at each step while multiplying.The following code evaluates this in N steps :
ll Res=1;
for(int i=1;i<=N;i++)
{
Res=Res*A;
Res=F(Res);
}
For Subtask #3 we have N<=10^{18} and hence we cannot take N steps as it will timeout. we can write a function similar to a fast modular exponentiation to evaluate F(A^N) which evaluates this in logN steps.The following code evaluates this :
int solve(long long A,long long N)
{
long long res=1;
while(N)
{
if(N%2==1)
{
res=res*F(A);
res=F(res);
}
A=F(F(A)*F(A));
N/=2;
}
return res;
}
The function F(A) for an integer A can be computed in O(1) easily as we have already defined this function.
## Time Complexity
O(logN) per testcase
## Space Complexity
O(1)
### AUTHOR’S AND TESTER’S SOLUTIONS:
Author’s solution can be found here.
Tester’s solution can be found here.
4 Likes
i have done this question in O(1)
1 Like
When you realize that F(A)\equiv A \mod 9 all you need to do is compute A^N \mod 9.
Solution.
And that’s also what the tester’s solution does.
4 Likes
Can anyone tell me the mistake in this solution? It passed for the last test case but failed in the first two.
I am not getting the mistake.
https://www.codechef.com/viewsolution/16722563
My this code is passing for 70 pts and getting WA for first two subtasks. I don’t know why it is happening atleast in first subtask because i have matched my output of all the possible test cases for first subtask with ouput of an AC code. If anyone could please help. @abx_2109
I have solved this in O(no_of_digits_in_N) but not sure whether my approach is fully correct or not. I have observed that after certain number of steps there will be a cycle and thus i just sum the digits of N untill it becomes single digit and then just print the answer for that single digit for that power.
my solution : https://www.codechef.com/viewsolution/16708827
I don’t know whether it fully correct or not and if it is and someone has some mathematical stuff regarding it’s proof then please share
This is my solution.
I did it in a different way.
I took the sum of digits according to the question given in a query. And then found out the pattern in which result is repeating.
Finally, printed the result according to the condition satisfied by the exponential term given in the query.
Here is my solution.
2 Likes
DONE THIS USING OBSERVATION https://www.codechef.com/viewsolution/16726993
And here is O(1) solution.
1 Like
@abx_2109 u can do this question more efficiently when each time for updating res u should go for checking
if(res%9==0) res=9;else res%=9;
rather than again making an recursive function as given in definition
And similarly updating A with
A = ((A%M)*(A%M))%M;
if(A%9==0) A=9;else A%=9;
Contest
Practise
Editorialist : Ashwany Aggarwal
Difficulty:
Easy
Prerequisites:
Mathematics,Observation skills
Problem
You are given two integers A and N. The task is to compute F(A^N) where F(X) is a function which results in a non-negative single digit integer obtained by a process of summing digitsuntil single digit number is reached.
Explanation:
STEP 1: First of all we find the sum of integers of the given integer A until it reaches to a single integer (say y), using function :
int digsum(unsigned long long p)
{ unsigned long long x,sum=0;
while(p>0)
{
while(p != 0)
{
x = p%10;
sum = sum+x;
p=p/10;
}
if(sum > 9)
{
p = sum;
sum = 0;
}
}
return sum;
}
Now, by the property of exponents that sum of digits repeat after every 6th exponent.i.e, digsum(A^N)= digsum(A^(N%6)), (except for A=3 or A=6 , where A^N= A for N=1 and A^N=9 for N>1).
For eg. 2^1=2, 2^7=128=1+2+8=11=1+1=2
For eg: 3^1=3; 3^4=81=8+1=9, 6^3=216=2+1+6=9 (you can check for any N).
STEP 2:
Now we have y (the single digit sum for A). Now We have 2 cases either y=(3 or 6) or y= else than 3 or 6.
1st case: If y= 3 or 6 or 9
1: If N>1 output 9
2: If N=1 output Y
2nd case: Y is not 3 or 6 or 9.(I have taken 9 also because for 9 answer is always 9)
Then take remainder of the given large N when divided by 6 (say z).(The main LOGIC)
Now single digit raised to power maximum 5 will be 3 digit in worst case this reduces the complexity of problem.
Step 3:
Find y^z and apply the same function. You will get the required answer.
Here is my solution: https://www.codechef.com/viewsolution/16728889
As some wonder why there is a magic modulo 9, here is a short explanation:
Any number can be written as a sum of power of 10 and you can write it like this:
1543 = 1 * (999+1) + 5 * (99+1) + 4 * (9+1) + 3
If you rearrange:
1543 = (1 * 999 + 5 * 99 + 4 * 9) + (1 + 5 + 4 + 3)
And then you understand why mod 9 is the magic trick.
8 Likes
We can use "set stl " also
I was trying to use recursive version of fast exponentiation algorithm but getting TLE on 2nd and 3rd subtask. Can anyone help me regarding same? Please upvote this answer so that I can ask queries
https://www.codechef.com/viewsolution/16740009
1 Like
Superfast - 0s
And it works with arbitrarily large integers!
@eugalt u really squeezed up !!!
1 Like
Are you sure it’s O(1)?
3 Likes
Consider this TC:
2
18 0
18 1
Expected o/p:
1
9
18
18
Hope this helps!
For A=18, N=0 and A=18, N=1 your code gives 9 and 0 respectively which should be 1 and 9.
1 Like<|endoftext|>
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Hypoxic respiratory failure is the general term used to describe organs of breathing not being capable of maintaining effective gas exchange in the lungs. Blood gas tests are used to ascertain whether respiratory failure is present.
In infants with hypoxic respiratory failure the blood vessels in the lungs are constricted. This limits the blood flow from the heart to the lungs and means that when the blood low in oxygen returns to the heart, it is conveyed past the lungs through ducts that remain from the foetal period and flows back out into the body.
Many infants with hypoxic respiratory failure have a sporadic lung disease which means that the air flow (and therefore oxygenation) is good in some areas of the lungs and less good or non-existent in other areas.
When nitric oxide is inhaled, the constricted blood vessels in the lungs relax, so that the blood flow from the heart to the lungs increases, and the quantity of blood that goes outside the lungs decreases. The blood flow increases in those areas of the lungs where the air flow is greatest and where the best gas exchange can take place.<|endoftext|>
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# 1995 AHSME Problems/Problem 22
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
## Problem
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13, 19, 20, 25$ and $31$, although this is not necessarily their order around the pentagon. The area of the pentagon is
$\mathrm{(A) \ 459 } \qquad \mathrm{(B) \ 600 } \qquad \mathrm{(C) \ 680 } \qquad \mathrm{(D) \ 720 } \qquad \mathrm{(E) \ 745 }$
## Solution
$[asy] defaultpen(linewidth(0.7)); draw((0,0)--(31,0)--(31,25)--(12,25)--(0,20)--cycle); draw((0,20)--(0,25)--(12,25)--cycle,linetype("4 4")); [/asy]$
Since the pentagon is cut from a rectangle, the cut-off triangle must be right. Since all of the lengths given are integers, it follows that this triangle is a Pythagorean Triple. We know that $31$ and either $25,\, 20$ must be the dimensions of the rectangle, since they are the largest lengths. With some trial-and-error, if we assign the shortest side, $13$, to be the hypotenuse of the triangle, we see the $5-12-13$ triple. Indeed this works, by placing the $31$ side opposite from the $19$ side and the $25$ side opposite from the $20$ side, leaving the cutaway side to be, as before, $13$.
To find the area of the pentagon, we subtract the area of the triangle from that of the big rectangle: $31\cdot 25-\frac{12\cdot5}{2}=775-30=745\Longrightarrow \boxed{\mathrm{(E)}745}$.<|endoftext|>
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Factoring a Sum of Cubes Without Prior Knowledge of the Formula
Many solutions related to factoring a sum of cubes suppose knowledge of the sum of cubes formula, $$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$
If you did not possess prior knowledge of this formula, how would you go about factoring a sum of cubes? I understand that some experimentation with distributing polynomials could lead to the formula, which could then be used. I think there must be a way to factor a sum of cubes without invoking it directly. For instance, how would you find some polynomial $$p(x)$$ such that $$p(x) \cdot \, (x+c) = x^3 + c^3$$
Without first resorting to applying the sum of cubes formula?
• We can show $a^3\equiv(-b)^3= -b^3 \mod (a+b)$ , so we can derive from this that $a+b$ must be a factor. Aug 26, 2023 at 19:28
• Consider $a^n-b^n$. Denote $a = x$. We have the polynomial $p(x) = x^n - b^n$, which has $x = b$ as a root, because $p(b) = 0$. So $x^n-b^n$ is divisible by $(x-b)$ . Now set $n=3$, $x = a$, and replace $b$ with $-b$. We get $a^3 - (-b)^3$ is divisible by $a - (-b)$. Aug 26, 2023 at 19:30
• To answer the last question on the latest edit, polynomial long division. Aug 26, 2023 at 19:34
• Your idea with the polynomial is fine. You just should write $p(x)\cdot (x-c)=x^3-c^3.$ Then you know that both sides are zero at $x=c.$ You can therefore perform a long division bei $(x^3-c^3):(x-c)=p(x).$ It works like a division of numbers by hand. Here is an example with polynomials: physicsforums.com/threads/… Aug 26, 2023 at 19:36
• Somewhat alternative approach (not worth posting as an answer). Notice that $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = (a^3 + b^3) + 3ab(a+b).$$ Therefore, $$~a^3 + b^3 = (a+b)^3 - 3ab(a+b).$$ Therefore, since $~(a+b)~$ is a factor of both of the RHS terms immediately above, it must be a factor of the LHS term. Aug 26, 2023 at 19:55
If you are allowed to apply the binomial theorem, we can proceed as follows: \begin{align*} (a + b)^{3} = & {3\choose 0}a^{3}b^{0} + {3\choose 1}a^{2}b^{1} + {3\choose 2}a^{1}b^{2} + {3\choose 3}a^{0}b^{3}\\\\ & = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}\\\\ & = a^{3} + b^{3} + 3ab(a + b) \end{align*}
Consequently, one arrives at the desired result according to the rearrangement: \begin{align*} a^{3} + b^{3} & = (a + b)^{3} - 3ab(a + b)\\\\ & = (a + b)[(a + b)^{2} - 3ab]\\\\ & = (a + b)(a^{2} - ab + b^{2}) \end{align*}
Hopefully this helps!
As other comments and answers have suggested (perhaps indirectly), long division of polynomials—which is an algorithm of repeatedly putting in a term in the quotient that you want then dealing with the consequences—does the trick.
Instead, here's a pretty natural way that you might discover factoring a sum of cubes on your own. (It generalizes to sum of odd powers naturally.)
Perhaps you're messing around with Pascal's triangle and binomial powers (as one does), and you write down $$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.$$ Notice the $$a^3 + b^3$$. Let's isolate it: $$a^3 + b^3 = (a+b)^3 - (3a^2b + 3ab^2).$$ Now, it's pretty clear that $$3ab$$ is a common factor of the last two terms. Let's pull it out:
$$a^3 + b^3 = (a+b)^3 - 3ab\, (a + b).$$ Now, $$(a+b)$$ is a common factor, which is fantastic news. Pull it out and clean up the contents of the quantity on the right: \begin{align} a^3 + b^3 &= (a+b) \, \bigl((a+b)^2 - 3ab \bigr) \\ &= (a+b) \, \bigl(a^2 + 2ab + b^2 - 3ab \bigr) \\ &= (a+b) \,(a^2 - ab + b^2). \end{align} Voilà, sum of cubes!
Incidentally, you can produce this identity in one chain of equations, but the adding and subtracting the same quantity (in blue) seems to come out of nowhere. It is, of course, motivated by knowing what a binomial cube looks like and the previous messing around.
\begin{align} a^3 + b^3 &= (a^3 \color{blue}{{}+ 3a^2b + 3ab^2} + b^3) \color{blue}{{}- (3a^2b + 3ab^2)} \\ &= (a+b)^3 - 3ab\,(a+b) \\ &= (a+b) \, \bigl((a+b)^2 - 3ab \bigr) \\ &= (a+b) \,(a^2 - ab + b^2) \\ \end{align}
Many helpful commenters have suggested polynomial long division to solve the question I posed, and I have carried it out here:
$$p(x) \cdot (x+c) = x^3+c^3 \\$$
$$p(x) = \frac{x^3+c^3}{x+c} \\$$
$$\begin{array}{r} x^2-cx+c^2\\ x+c{\overline{\smash{\big)}\,x^3+c^3\phantom{))))))}}}\\ \underline{-(x^3+cx^2)\phantom{))))}}\\ -cx^2+c^3\phantom{))}\\ \underline{-(-cx^2-c^2x)}\\ c^2x+c^3\phantom{)}\\ \underline{-(c^2x+c^3)}\\ 0 \end{array} \\ \\$$
$$p(x) = x^2-cx+c^2$$
Therefore,
$$p(x) \cdot (x+c) = x^3+c^3 \\$$
$$(x^2-cx+c^2)(x+c) = x^3+c^3$$
You can build up $$P(x)$$ sequentially. To get the term $$x^3$$ on the right side, first guess $$P_1(x) = x^2$$. Then $$P_1(x) (x + c) = x^3 + x^2 c$$. To get rid of this second term on the right side, try $$P_2 (x) = P_1(x) - x c$$. Then $$P_2 (x) (x + c) = x^3 - x c^2$$. To get rid of this second term on the right side, take $$P_3(x) = P_2(x) + c^2$$. We get $$P_3(x) (x-c) = x^3 + c^3$$ as desired, and we have $$P_3(x) = x^2 - xc + c^2$$.
• By the way, this is just polynomial long division in disguise! Aug 26, 2023 at 19:40<|endoftext|>
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The Great Barrier Reef is 1,800 miles long and home to a quarter of the world's ocean species. So it's no wonder that marine biologists, fearing its pollution-driven demise, started freezing corals so they can preserve them for later.
That's right. They're freezing coral sperm and embryonic cells and storing them in a frozen repository. (Does the GBR get paid for sperm donations?) Researchers believe they can make these cells get it on—even 1,000 years from now—in other ecosystems, thereby restoring and repopulating currently endangered coral habitats:
Done properly over time, samples of frozen material can be reared and placed back into ecosystems to infuse new genes into natural populations.
And what's causing this marine mayhem? Researches from the Smithsonian and others believe it's because of "increasing acidity of the ocean and the warming temperatures." Now, why would that be? [Smithsonian via Tree Hugger; Image credit: Sarah_Ackerman/FlickrCC]<|endoftext|>
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By John Barrat
If you think the American Revolution was an isolated conflict between Great Britain and the American colonies, you know only part of the story.
A new exhibition at the Smithsonian’s National Museum of American History reveals the war that earned America its independence for what it was: a world war involving multiple nations fighting battles on land and sea around the globe.
“The American colonies had no hope of winning their independence alone,” says David Allison, senior curator of the exhibition. “They had to gain support from other European powers, most importantly from France and Spain and the involvement of these nations would affect not only the history of the new United States of America, but their own histories as well.”
These pieces from “The American Revolution: A World War” put the war in a global context, which ultimately determined the outcome in the U.S. and the future of the world.
This 1782 cartoon by J. Barrow shows the colonies’ success at gaining allies who would join them against Great Britain in their fight for political freedom and control of their global trade. It depicts a British lion facing off against four different powers: France as a cockerel; Spain and Holland as two dogs, and the American colonies as a snake.
More than an uprising of discontented colonists against the British king, the American Revolution involved multiple nations fighting battles on land and sea around the globe. Britain had no major allies in this war.
The stage for the American Revolution was set during The Seven Years’ War (1754-1763), a global conflict which saw fighting in North America, when a defeated France was forced to give up its territorial claims in North America to Britain and Spain.
France sought revenge through an alliance with the colonists, who had fought with the British against them. Tensions were fueled by the fact that both France and Britain realized the potential of the growing economy of the American colonies. Impressive imports of tobacco, furs, rum, wheat, fish, sugar, rice, indigo, iron powered prosperity in the colonies, permitting the purchase of manufactured goods made elsewhere in the world.
Thousands of enslaved African Americans fled to the British forces during the Revolution, in part because the British promised them freedom.
One enslaved man went to the British as a spy. James Armistead in Virginia convinced his owner to allow him to volunteer under the Marquis de Lafayette, a French general and ally of the colonists. In 1781, Armistead infiltrated the British military and reported the plans and movements of British Gen. Charles Cornwallis back to Lafayette.
After America’s victory, enslaved people were recaptured and returned to their American owners, including two to George Washington. James Armistead was also returned to his owner. Lafayette helped him gain his freedom in 1787.
Following six long years of back-and-forth combat with the British, everything fell into place for the American Gen. George Washington and his allies during the siege at Yorktown, Virginia in 1781.
French soldiers, cannons, ammunition and warships all augmented Washington’s troops at that battle. Spain and the Dutch Republic also provided money and logistical assistance.
Older and more experienced than Washington, French Gen. Comte de Rochambeau played a central role in the outcome at Yorktown, which ended with the surrender of British Gen. Lord Charles Cornwallis.
This victory was a major turning point of the Revolution and the last major Revolutionary battle in North America. Cornwallis might have escaped by sea during the siege had not French Adm. Francois Joseph Paul de Grasse defeated a British fleet one month earlier nearby during the Battle of the Chesapeake.
More than a century after the American Revolution, as World War I erupted in Europe in 1914, American men and women crossed the Atlantic to volunteer in France as soldiers and civilians. Many were motivated by the memory of the Marquis de Lafayette, the French aristocrat and military officer who volunteered and fought with the colonists in the American Revolution.
The U.S. joined the Allies in 1917, playing an essential role in their victory, and the principle of volunteering became part of World War II as well.
“The American Revolution: A World War” will be on view at the National Museum of American History through July 9, 2019.”<|endoftext|>
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1. ## Trig Identities
Prove the given identity
$\frac{Sin\;x}{Csc\;x}\;+\;\frac{Cos\;x}{Sec\;x}\;= \;1$
For some strange reason I get $2\;=\;1$
2. Originally Posted by OzzMan
Prove the given identity
$\frac{Sin\;x}{Csc\;x}\;+\;\frac{Cos\;x}{Sec\;x}\;= \;1$
For some strange reason I get $2\;=\;1$
$
\frac{{\sin (x)}}
{{\csc (x)}} + \frac{{\cos (x)}}
{{\sec (x)}} = 1
$
$
\frac{{\sin (x)}}
{{(\frac{1}
{{\sin (x)}})}} + \frac{{\cos (x)}}
{{(\frac{1}
{{\cos (x)}})}} = 1
$
$
\sin (x) \times \sin (x) + \cos (x) \times \cos (x) = 1
$
$
\sin ^2 (x) + \cos ^2 (x) = 1
$
$1 = 1$
3. If you need to show the last step, that $sin^2(x)+cos^2(x)=1$, then draw a circle, we'll choose it to have a radius of 1, so it is a unit circle. (we can do this since trig functions are proportions, we can choose one side to be a certain length and the other sides will change accordingly.) Then the sine is the opposite over the hypotenuse. The hypotenuse is 1 since it is the unit circle, so the sine is just the opposite side, or the height. And the cosine is the adjacent over the hypotenuse which is 1, so it is just the adjacent, or the length. Then since sine is height, and cosine is length, $sin^2(x) +cos^2(x)$ can be a and b in the pythagorean theorem, which says $a^2+b^2=c^2$. Since c is the hypotenuse, c=1 and $c^2=1$. So $sin^2+cos^2=1$<|endoftext|>
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# Thread: Logarithm - Solve for X
1. ## Logarithm - Solve for X
How would you go by solving these problems to find the value of $\displaystyle x$?
1. $\displaystyle (log_3 x)(log_4 3) = 2$
2. $\displaystyle log x^3 - log y^2 =8$
$\displaystyle log x^4 + log y^6 = 2$
2. Originally Posted by chrozer
How would you go by solving these problems to find the value of $\displaystyle x$?
1. $\displaystyle (log_3 x)(log_4 3) = 2$
2. $\displaystyle log x^3 - log y^2 =8$
$\displaystyle log x^4 + log y^6 = 2$
for one rewrite it as $\displaystyle log_3(x)+\frac{1}{log_3(4)}$...for the second one rewrite it as $\displaystyle log(x^4y^6)=2$
3. 1. Know this identity: $\displaystyle \log_{a} b = \frac{\log_{c} b}{\log_{c} a}$ for any workable $\displaystyle c$
So:
$\displaystyle \log_{3}x \cdot \log_{4}{3} = 2$
$\displaystyle \frac{\log x}{\log 3} \cdot \frac{\log 3}{\log 4} = 2$
You should be able to simplify it form here.
2. Not sure where to go with MathStud's idea but I would just directly solve the system of equations:
$\displaystyle \begin{array}{ccccc} \log x^{3} & - & \log y^{2} & = & 8 \\ \log x^{4} & + & \log y^{6} & = & 2 \end{array}$ $\displaystyle . \quad \Rightarrow \quad .$$\displaystyle \begin{array}{ccccc} 3\log x & - & 2 \log y & = & 8 \\ 4\log x & + & 6\log y & = & 2 \end{array} \displaystyle \Rightarrow \begin{array}{cccccc} {\color{blue}9}\log x & - & {\color{blue}6}\log y & = & {\color{blue}24} & \mbox{(Multiplied by 3)} \\ 4\log x & + & 6\log y & = & 2 & \mbox{ } \end{array} Add them and solve for x. 4. ## here is where I was going Originally Posted by o_O 1. Know this identity: \displaystyle \log_{a} b = \frac{\log_{c} b}{\log_{c} a} for any workable \displaystyle c So: \displaystyle \log_{3}x \cdot \log_{4}{3} = 2 \displaystyle \frac{\log x}{\log 3} \cdot \frac{\log 3}{\log 4} = 2 You should be able to simplify it form here. 2. Not sure where to go with MathStud's idea but I would just directly solve the system of equations: \displaystyle \begin{array}{ccccc} \log x^{3} & - & \log y^{2} & = & 8 \\ \log x^{4} & + & \log y^{6} & = & 2 \end{array} \displaystyle . \quad \Rightarrow \quad .$$\displaystyle \begin{array}{ccccc} 3\log x & - & 2 \log y & = & 8 \\ 4\log x & + & 6\log y & = & 2 \end{array}$
$\displaystyle \Rightarrow \begin{array}{cccccc} {\color{blue}9}\log x & - & {\color{blue}6}\log y & = & {\color{blue}24} & \mbox{(Multiplied by 3)} \\ 4\log x & + & 6\log y & = & 2 & \mbox{ } \end{array}$
Add them and solve for x.
$\displaystyle \ln(x^4y^6)=2\Rightarrow{x^4y^6=e^2}$ and $\displaystyle \frac{y^2}{x^3}=e^8$ then solve...is that flawed logic?
5. Ah I see now, going with a direct comparison.
And are we dealing with base 10 log or natural log?
6. ## Haha
Originally Posted by o_O
Ah I see now, going with a direct comparison.
And are we dealing with base 10 log or natural log?
I never know anymore in math...I always taught that $\displaystyle log(x)\equiv{log_{10}(x)}$ but in most mathematical text $\displaystyle log(x)\equiv{ln(x)}$ so I just assume that I was taught wrong and use the latter...if he means $\displaystyle log_{10}(x)$ then just replace the e with 10
7. Originally Posted by o_O
1. Know this identity: $\displaystyle \log_{a} b = \frac{\log_{c} b}{\log_{c} a}$ for any workable $\displaystyle c$
So:
$\displaystyle \log_{3}x \cdot \log_{4}{3} = 2$
$\displaystyle \frac{\log x}{\log 3} \cdot \frac{\log 3}{\log 4} = 2$
You should be able to simplify it form here.
2. Not sure where to go with MathStud's idea but I would just directly solve the system of equations:
$\displaystyle \begin{array}{ccccc} \log x^{3} & - & \log y^{2} & = & 8 \\ \log x^{4} & + & \log y^{6} & = & 2 \end{array}$ $\displaystyle . \quad \Rightarrow \quad .$$\displaystyle \begin{array}{ccccc} 3\log x & - & 2 \log y & = & 8 \\ 4\log x & + & 6\log y & = & 2 \end{array} \displaystyle \Rightarrow \begin{array}{cccccc} {\color{blue}9}\log x & - & {\color{blue}6}\log y & = & {\color{blue}24} & \mbox{(Multiplied by 3)} \\ 4\log x & + & 6\log y & = & 2 & \mbox{ } \end{array} Add them and solve for x. I solved for "x" and got 100, but when I plugged it back into the equation it does not equal the answer. EDIT - NVM....I'm an idiot. Plugged the "x" value for the "y" value also. But how would you solve for "y"? I can't get the same "y" value for both equation. MathStud - Thanx for the help...but I never learned that \displaystyle log(x) = ln(x) 8. ## Ok Originally Posted by chrozer I solved for "x" and got 100, but when I plugged it back into the equation it does not equal the answer. MathStud - Thanx for the help...but I never learned that \displaystyle log(x) = ln(x) well listing the adaptation I said we have \displaystyle \frac{x^3}{y^2}=10^8\Rightarrow{x^3=10^8y^2}...therefore using substitution we get \displaystyle x=100,y=\frac{1}{10} 9. Originally Posted by Mathstud28 well listing the adaptation I said we have \displaystyle \frac{x^3}{y^2}=10^8\Rightarrow{x^3=10^8y^2}...therefore using substitution we get \displaystyle x=100,y=\frac{1}{10} I got that while trying to solve for both "x" and "y" but when I plug it back into the equation on my calculator...it does not equal to the answer of the equation. 10. \displaystyle \log (x^{3}) - \log (y^{2}) = \log\left(100^{3}\right) - \log \left(.1^{3}\right) = 6 - (-2) = 8 \displaystyle \log (x^{4}) - \log (y^{6}) = \log\left(100^{4}\right) + \log \left(.1^{6}\right) = 8 + (-6) = 2 11. ## You sure about that Originally Posted by chrozer I got that while trying to solve for both "x" and "y" but when I plug it back into the equation on my calculator...it does not equal to the answer of the equation. we have \displaystyle log_{10}((10^2)^4)+log_{10}(10^{-6})=2\Rightarrow{8log_{10}(10)-6log_{10}(10)=2}$$\displaystyle \Rightarrow{2log_{10}(10)=2}\Rightarrow{2=2}$ and $\displaystyle log_{10}((10^2)^3)-log_{10}(10^{-2})=8\Rightarrow{6log_{10}(10)+2log_{10}(10)=8}$$\displaystyle \Rightarrow{8log_{10}(10)=8}\Rightarrow{8=8}$
12. Originally Posted by o_O
$\displaystyle \log (x^{3}) - \log (y^{2}) = \log\left(100^{3}\right) - \log \left(.1^{3}\right) = 6 - (-2) = 8$
$\displaystyle \log (x^{4}) - \log (y^{6}) = \log\left(100^{4}\right) + \log \left(.1^{6}\right) = 8 + (-6) = 2$
Ok thnx alot. I see what I did wrong. I put the exponent outside the parentheses. Thnx all for your help.<|endoftext|>
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In everyday speech, the term “exponential growth” is used rather loosely to describe any situation involving rapid growth. Many life science systems have an exponential relationship with time. For example: the number of bacteria in a culture may double every hour, an example of exponential growth. In this section we will be learning more about exponential growth.
## Exponential Growth Definition
When a quantity is increased by multiplier that is greater than $1$ over fixed periods of time then we have exponential growth.
## Exponential Growth Formula
The simple formula that describes exponential growth is
$y = a^{x}$
Where $a$ is a constant that depends on the system being studied, and $x$ is the exponent also called the index.
Taking the logarithm of both sides of the general equation gives
$\log y$ = $x \log a$
So, if we plot $\log y$ against $x$, an exponential relationship will plot as a straight line with a gradient of $\log a$.
The more general formula for exponential growth can be written in the form
$y = ae^{bx}$
where $a$ and $b$ are constants that depends on the system.
Taking the natural logarithm of both sides of the this equation gives:
$\ln y = \ln(ae^{bx})$
using logarithm rules for multiplication, this is the same as
$\ln y = ln a + bx ln e$
however we know that $\ln e =1$,
so $ln y = bx + ln a$
## Exponential Growth Function
If the rate of change of a quantity, P, is proportional to P, then
$\frac{dP}{dt}$ = $bP$.....(1)
The solution to equation (1) is
$P = P_{0}e^{bt}$
$P_{0}$ is the initial value of P (i.e., when $t = 0$, $P = P_{0}$),
$b$ is the constant of proportionality.
$P$ is the amount of the quantity present at time $t$.
## Exponential Growth Graph
Graphs of exponential functions of the form
$y = ae^{bx}$
where $a$ and $b$ are constants, and $x \geq 0$
Every exponential function has the same shape as this function. The graph of an exponential decay function will be the reflection of an exponential growth function across the y-axis. Translating, reflecting, stretching, and contracting an exponential function will only change the scale or position of the function. These transformations will not change the shape of the function.
Exponential growth function can be any of the two below:
## Exponential Growth Examples
The following are the example of exponential growth.
### Solved Examples
Question 1: Solve for $x$, giving answer corrected to two decimal places.
$e^{x} = 5.47$
Solution:
Given $e^{x} = 5.47$
$x = \ln 5.47$
$e^{x} = a$; $x = log_{e}a$
$x= 1.70$
Question 2: Solve for $x$, giving answer corrected to two decimal places.
$2e^{3x} = 3.72$
Solution:
Given $2e^{3x} = 3.72$
divide both sides by $2$
$e^{3x}$ = $\frac{3.72}{2}$
$3x$ = $\ln($$\frac{3.72}{2}$$)$
$x$ = $\ln$ $\frac{\frac{3.72}{2}}{3}$
$x = 0.21$
Question 3: The current population in certain country is 8 million. what will be the population in 13 years, if the population grows at annual rate of 6%?
Solution:
Lets measure population in millions and time in years.
Given that,
Time (t) = 13 years
initial population $P_{0} = 8$ million
We know that, $P = P_{0}e^{bt}$
$P = 8 e^{0.06\times 13}$
$P$ = $17.451$ million
The population in 13 years is $P$ ≈ $17.451$ million<|endoftext|>
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One of the most famous and classical structure that everyone learns in music school is the Sonata form. This is because it was a form used so prominently by the masters such as Bach, Haydn, Brahms, Weber, and many others.
Just because they teach it in music school doesn’t mean it’s complicated; in fact, it’s quite simple. The sonata form consists of three parts: the exposition, the development, and the recapitulation.
The exposition is the first part of the music where thematic ideas and motifs are introduced. This is where you first heard the melodies that you leave concerts humming.
The development is just like what it sounds: it develops the melodic and rhythmic elements established in the exposition into many different variations. This is the part of the piece where composers can explore the many different possibilities presented by their thematic motifs. They can change keys, use chromaticism, flip the timbre upside down, etc. Sometimes the development doesn’t even sound related to the exposition at all, if the composer introduces entirely new ideas to the section!
The recapitulation is exactly what you’d expect after the development to finish off the piece. It’s where the piece returns to its original melodic material as heard in the exposition. The recapitulation also brings the piece back to its original key.
And that’s mostly it. Countless pieces that you’ve most likely heard, including famous symphonies by Mozart, are in sonata form, but you probably just didn’t notice.
- Hill, Andrew W. Music Theory for Dummies. Hoboken: John Wiley & Sons, 2006. Print.
- “Sonata Form.” Wikipedia. Wikimedia Foundation, 03 Apr. 2013. Web. 13 Mar. 2013.
- Bernstein, Leonard. “What Is Sonata Form?” Leonard Bernstein: Young People’s Concerts. Philharmonic Hall, Lincoln Center, New York. 6 Nov. 1964. Lecture.<|endoftext|>
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June 8, 2023
# What is convolution theorem?
Convolution theorem is a fundamental concept in signal processing, mathematics, and related fields. It provides a powerful mathematical tool for analyzing signals and systems in the frequency domain. In this article, we will discuss convolution theorem, its definition, mathematical properties, and applications in various fields.
## Understanding Convolution
Convolution is a mathematical operation that describes the combination of two functions to produce a third function. In simple terms, it is the process of overlapping two functions and computing the integral of their product. This operation is widely used in various fields, including physics, engineering, and computer science.
Convolution is a fundamental concept in mathematics, and it has many applications in different areas. It is a powerful tool for solving problems that involve the combination of two functions. Convolution is a complex operation, and it requires a deep understanding of mathematical concepts to use it effectively.
### Definition of Convolution
Mathematically, convolution is defined as the integral of the product of two functions over a certain range. Suppose we have two continuous functions f(x) and g(x). The convolution of f(x) and g(x) is denoted by (f*g)(x) and given by:
(f * g)(x) = ∫f(τ)g(x-τ)dτ
The integral is taken over the entire range of values of Ï„, which is the independent variable of function f(x). In general, the convolution of two functions produces a third function that has characteristics of both input functions. The resulting function represents the combined effect of the two input functions.
Convolution is a powerful operation that can be used to solve a wide range of problems. It is used in many different areas, including physics, engineering, and computer science. Convolution is an essential tool for understanding complex systems and analyzing data.
### Convolution in Mathematics
Convolution is a widely used operation in mathematics, particularly in the areas of integral and differential equations. It is used to describe the relationship between two functions and their transformations. Convolution is used in various branches of mathematics, including calculus, probability theory, and geometry.
Convolution is a powerful tool for solving mathematical problems. It allows us to combine two functions and analyze the resulting function. Convolution is used in many different areas of mathematics, including geometry, probability theory, and calculus.
### Convolution in Signal Processing
Convolution is an essential concept in signal processing. It is used to analyze and manipulate signals in the time domain and frequency domain. In signal processing, a signal is often represented as a function of time. Convolution allows us to merge two signals to produce a new signal that exhibits the behavior of both.
Convolution is a powerful tool for analyzing and manipulating signals. It is used in many different areas of signal processing, including image processing, audio processing, and speech recognition. Convolution is an essential concept for understanding the behavior of signals and processing them effectively.
## The Convolution Theorem
The convolution theorem is a mathematical tool that relates convolution in the time domain to multiplication in the frequency domain. It is a fundamental concept in signal processing and is used widely in digital signal processing.
### Fourier Transform and Convolution
The Fourier transform is an important mathematical technique that is used to represent a signal in the frequency domain. It is a way of decomposing a signal into its constituent frequencies. The convolution theorem states that the Fourier transform of a convolution of two functions is the product of their individual Fourier transforms. In other words, convolution in the time domain is equivalent to multiplication in the frequency domain.
For example, let's say we have two signals, f(x) and g(x), and we want to convolve them. We can first take the Fourier transform of both signals, which gives us F(ω) and G(ω), respectively. We can then multiply these two Fourier transforms together, which gives us F(ω)G(ω). Finally, we can take the inverse Fourier transform of F(ω)G(ω), which gives us the convolution of f(x) and g(x).
### The Convolution Theorem Statement
The convolution theorem can be stated as follows: "The Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms." This means that if we have two signals, f(x) and g(x), and we convolve them, we can find their Fourier transforms individually, multiply them together, and then take the inverse Fourier transform to get the convolution of f(x) and g(x).
This theorem is extremely useful in signal processing, as it allows us to analyze signals in the frequency domain, where certain operations are easier to perform. For example, if we want to filter out certain frequencies from a signal, we can simply multiply its Fourier transform by a filter function in the frequency domain, and then take the inverse Fourier transform to get the filtered signal in the time domain.
### Proof of the Convolution Theorem
The proof of the convolution theorem is based on the properties of the Fourier transform. We start by taking the Fourier transform of both functions f(x) and g(x), which gives us F(ω) and G(ω), respectively. We can then express the convolution of f(x) and g(x) as an integral:
f(x) * g(x) = ∫ f(t)g(x-t) dt
We can then take the Fourier transform of both sides of this equation:
F(ω)G(ω) = ∫ f(t) ∫ g(x-t) e^(-iωx) dx dt
Using a change of variables, we can rewrite the integral on the right-hand side as:
F(ω)G(ω) = ∫ f(x-u) g(u) e^(-iωu) du dx
Now, we can swap the order of integration:
F(ω)G(ω) = ∫ g(u) e^(-iωu) ∫ f(x-u) e^(-iωx) dx du
The integral on the right-hand side is just the Fourier transform of f(x-u), evaluated at ω. We can substitute this in:
F(ω)G(ω) = ∫ g(u) e^(-iωu) F(ω) e^(-iωu) du
Now, we can simplify:
F(ω)G(ω) = F(ω) ∫ g(u) e^(-2iωu) du
The integral on the right-hand side is just the Fourier transform of g(-u), evaluated at -2ω. We can substitute this in:
F(ω)G(ω) = F(ω) G(ω)'
where G(ω)' denotes the Fourier transform of g(-x). Thus, we have shown that the Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms.
## Applications of Convolution Theorem
The convolution theorem is a powerful tool that has numerous applications in various fields. It helps in simplifying complex mathematical operations and provides a better understanding of the behavior of systems in the frequency domain. Some of the major applications of the convolution theorem are:
### Image Processing
Image processing is a field that deals with the manipulation of digital images. Convolution is one of the fundamental operations used in image processing. It is used to detect edges, blur an image, and perform other operations. The Fourier transform and convolution theorem are used to analyze and manipulate images in the frequency domain. The convolution theorem helps in understanding the behavior of image processing systems and how they affect the images.
For example, in edge detection, convolution is used to detect the edges of an image. The convolution theorem is used to analyze the frequency response of the filters used for edge detection. This helps in understanding how the filters affect the image and how to choose the right filter for a particular image.
### Audio Signal Processing
Audio signal processing deals with the manipulation of sound signals. Convolution is used in audio signal processing to create reverb effects, filter out unwanted noise, and perform other operations on audio signals. The convolution theorem is essential in understanding the behavior of audio systems in the frequency domain.
For example, in creating reverb effects, convolution is used to simulate the effect of sound reflecting off surfaces in a room. The convolution theorem is used to analyze the frequency response of the system and how it affects the sound signal. This helps in creating realistic reverb effects that sound like they were recorded in a particular room.
### Communication Systems
Communication systems deal with the transmission and reception of signals. Convolution is used in communication systems to model the behavior of channels and filter signals. The convolution theorem is used to analyze the characteristics of communication channels in the frequency domain.
For example, in wireless communication, convolution is used to model the behavior of the wireless channel. The convolution theorem is used to analyze the frequency response of the channel and how it affects the transmitted signal. This helps in understanding the behavior of the wireless channel and how to design systems that can operate in such channels.
The convolution theorem is a powerful tool that has revolutionized various fields. It has made complex mathematical operations simpler and has provided a better understanding of the behavior of systems in the frequency domain. The applications of the convolution theorem are vast and varied, and it continues to be an essential tool in various fields.
## Properties of Convolution
Convolution is a mathematical operation that is used in signal processing and related fields to analyze and manipulate signals and systems. It involves combining two functions to produce a third function that expresses how the shape of one is modified by the other. Convolution has several mathematical properties that are important to understand in order to use it effectively. These properties include:
### Commutative Property
The commutative property of convolution is an important property that states that the order of a convolution operation does not matter. In other words, the convolution between two functions f(x) and g(x) is the same as the convolution between g(x) and f(x). This property is often used in simplifying convolution operations. For example, if we have two functions f(x) and g(x), we can compute their convolution as f(x) * g(x) or as g(x) * f(x), and the result will be the same. This property is essential in many applications of convolution, such as in digital signal processing, where it is used to filter signals and remove noise.
### Associative Property
The associative property of convolution is another important property that states that the grouping of convolution operations does not matter. In other words, the convolution of three functions f(x), g(x), and h(x) is the same, regardless of which functions are convolved first. This property is also used in simplifying convolution operations. For example, if we have three functions f(x), g(x), and h(x), we can compute their convolution as (f(x) * g(x)) * h(x) or as f(x) * (g(x) * h(x)), and the result will be the same. This property is also essential in many applications of convolution, such as in image processing, where it is used to blur or sharpen images.
### Distributive Property
The distributive property of convolution is yet another important property that states that the convolution of two functions with a third function is the same as the sum of the convolutions of each function with the third function. In other words, it allows us to simplify complicated convolution operations by breaking them down into smaller operations. For example, if we have three functions f(x), g(x), and h(x), we can compute their convolution as (f(x) + g(x)) * h(x) or as f(x) * h(x) + g(x) * h(x), and the result will be the same. This property is also essential in many applications of convolution, such as in audio signal processing, where it is used to create sound effects.
Understanding the properties and applications of convolution is essential in many fields, including digital signal processing, image processing, audio signal processing, communication systems, and more. Convolution is a powerful tool that allows us to analyze and manipulate signals and systems in the time domain and frequency domain. By using the commutative, associative, and distributive properties of convolution, we can simplify complicated convolution operations and make them more manageable. In conclusion, the study of convolution is a crucial concept in signal processing and related fields, and it is an area that continues to evolve and grow in importance.<|endoftext|>
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The Medford School District aligns its math curriculum with the state-adopted Common Core Standards. The Common Core State Standards are a set of shared K-12 learning expectations for students in English-language arts and mathematics intended to prepare students for college and career. Our goal is to connect every student to math strategies so they can become critical thinkers and efficient mathematicians.
Mathematics provides a foundation for the learning of science and technology as well as for the interpretation of quantitative information in other subjects. It teaches how to reason logically and develops skills that students can carry into other disciplines and many situations in real life. Mathematics helps students understand how the world works while exposing them to some of its unanswered mysteries. Ensuring all students develop a solid foundation in mathematics is our challenge.
For questions about your child's work in math, please contact your school or the Office of Elementary Student Achievement.<|endoftext|>
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# How to Solve Age Related Questions? Tips & Tricks
By Shubham Verma|Updated : September 15th, 2020
In the miscellaneous section of Quantitative Aptitude, "Ages" is one of the important topics.
To make the chapter easy for you all, we are providing you with all some Basic Concept and Tricks on Age-Related Questions which will surely make the chapter easy for you all.
Have a look at the following questions:-
Question. 1: The age of the father 3 years ago was 7 times the age of his son. At present, the father’s age is five times that of his son. What are the present ages of the father and the son?
Solution: Let the present age of son = x yrs
Then, the present age of father = 5xyr
3 years ago,
7(x – 3)= 5x – 3
Or, 7x – 21 =5x – 3
Or, 2x =18
x = 9 yrs
Therefore, the son’s age = 9 years
Father’s age = 45 years
Question. 2: At present, the age of the father is five times the age of his son. Three years hence, the father’s age would be four times that of his son. Find the present ages of the father and the son.
Solution: Let the present age of son = x yrs
Then, the present age of father = 5x yrs
3 yrs hence,
4(x+3)= 5x+3
Or, 4x + 12=5x +3
x= 9yrs.
Therefore, son’s age = 9 yrs and father’s age = 45 yrs
Question. 3: Three years earlier, the father was 7 times as old as his son. Three years hence, the father’s age would be four times of his son. What are the present ages of the father and the son?
Solution: Let the present age of son = x yrs and the present age of father = y yrs
3 yrs earlier, 7(x – 3) = y – 3
7x – y =18………….(i)
3 yes hence, 4(x+3) = y +3
4x +12 = y + 3
4x – y = – 9 …………(ii)
Solving (1) & (2) we get, x = 9 yrs & y =45 yrs
Question. 4: The sum of the ages of a mother and her daughter is 50 yrs. Also 5 yrs ago, the mother’s age was 7 times the age of the daughter. What are the present ages of the mother and the daughter?
Solution: Let the age of the daughter be x yrs.
Then, the age of the mother is (50x – x)yrs
5 yrs ago, 7(x – 5) = 50 – x – 5
Or, 8x = 50 – 5 +35 = 80
x =10
Therefore, the daughter’s age = 10 yrs and mother’s age = 40yrs
Question. 5: The sum of the ages of a son and father is 56 yrs. After 4 yrs, the age of the father will be three times that of the son. What is the age of the son?
Solution: Let the age of the son be x yrs.
Then, the age of the father is (56 – x) yrs.
After 4 yrs, 3(x+4) = 56 – x +4
Or, 4x =56 +4 – 12 = 48
x = 12 yrs
Thus, the son’s age = 12 yrs
Note: Just try to make two equations and then solve them to get your answer.
Question 6: A man’s age is 133(1/3)% of what it was 8 years ago, but 80% of what it will be after 8 years. What is his present age?
Solution: Let the present age be X years.
Then 133(1/3)% of (X-8) = X and 80%(X+8) = X
So, 133(1/3)% of (X-8) = 80%(X+8)
4(X-8)/3 = 4(X+8)/5
5(X-8) = 3(X+8)
2X = 64
X = 32
Shortcut: You don’t need to solve both equations. Solve any equation you will get the answer.
133(1/3)% of (X-8) = X
4(X-8)/3 = X
4X-32 = 3X
X = 32 years.
Question 7: The present age of Romila is one-fourth that of her father. After 6 years the father’s age will be twice the age of Kapil. If Kapil celebrated his fifth birthday 8 years ago. What is Romila’s present age?
Solution: Let the present age of Romila is X, then Father’s age = 4X
6 years hence,
father’s age = 4X+6
2 (Age of Kapil) = 4X+6
Age of Kapil = 2X+3
Present age of Kapil = 2X+3-6 = 2X-3
Kapil celebrated his 5th birthday 8 years ago
So, Present age of Kapil is 5+8 = 13 years
2X-3 = 13
2X = 16
X = 8years.
Shortcut approach: Kapil celebrated his 5th birthday 8 years ago.
The present age of Kapil = 13
After 6years, father’s age will be twice of the Kapil.
2x(13+6) = 4X+6
X= 8 years
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# Circle Center by Paperfolding
### Solution 1
Choose a corner of the sheet and a point on the circle away from the corner. Fold the paper as to place the chosen corner on top of the chosen point $(C'$ and $C$ in the diagram below):
Let the edges of the paper adjacent to $C'$ cross the circle at points $J$ and $K.$ Then, since $\angle JC'K=90^{\circ},$ $JK$ is a diameter of the circle and passes through its center. Make a fold $JK.$
Choose another point on the circle and, perhaps, another corner of the sheet, and repeat the procedure. The two so constructed folds pass both through the center of the circle and, therefore, this is where they cross.
### Solution 2
To make sure, I assume that
1. You can fold parallel to the sides of the sheet
2. Given a point you can fold parallel to one side and passing through that point
3. Given two points you can fold through them two
Then:
1. Fold parallel to one side in a way that the circle is crossed in 2 points
2. From each point fold parallel to the other side. Now you have 2 more points crossing the circle. The 4 points form a rectangle
3. Fold the diagonals of the rectangle. They cross at the center
### Solution 3
If folding to a tangent is a legitimate operation, fold on three (non parallel) tangents of the circle and then folding on the angel bisectors of the triangle created by the three tangents.
### Acknowledgment
This problem was proposed and solved by Thanos Kalogerakis in 2014. Solution 2 is by Luis García; Solution 3 is by Elia Noris. Elia also supplied the illustration.
Paper Folding Geometry<|endoftext|>
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Converting Money Values
Aligned To Common Core Standard:
Grade 2 Measurement - 2.MD.C.8
Printable Worksheets And Lessons
- Dollars to Donuts
Step-by-Step Lesson- It is mostly going to cents from dollars
and then back again.
- Guided Lesson
- Bills to cents and in some cases total value to cents.
- Guided Lesson Explanation
- I provided a good key on the worksheet itself, so I just referred
to that over and over.
- Practice Worksheet
- These will require some reading to get the hand of what is going
- Matching Worksheet
- Match the word problems to the value in cents or dollars.
View Answer Keys- All the answer keys in one file.
Cents can make dollars and dollars do make cents. A riddle must start like that somewhere?
Read all of these carefully and make sure you know what they are asking for.<|endoftext|>
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### Rectangular Coordinate System
After studying this section, you will be able to:
1. Plot a point, given the coordinates.
2. Name the coordinates of a plotted point.
3. Find ordered pairs for a given linear equation.
Plotting a Point
Many things in everyday life are clearer if we see a picture. Similarly, in mathematics, we can picture algebraic relationships by drawing a graph. To draw a graph, we need a frame of reference.
In earlier tutorial we showed that any real number can be represented on a number line. Look at the number line below. The arrow indicates the positive direction.
To form a rectangular coordinate system, we draw a second number line vertically. We construct it so that the 0 point on each number line is exactly at the same place. We refer to this location as the origin. The horizontal number line is often called the x-axis. The vertical number line is often called the y-axis. Arrows show the positive direction for each axis.
We can represent a point in this rectangular coordinate system by using an ordered pair of numbers. The first number of the pair represents the distance from the origin measured along the horizontal or x-axis. The second number of the pair represents the distance measured on the y-axis or on a line parallel to the y-axis. For example (5, 2) is an ordered pair that represents a point in the rectangular coordinate system.
The ordered pair of numbers that represent a point are often referred to as the
coordinates of a point. The first value is called the x-coordinate. The second value is called the y-coordinate. If the x-coordinate is positive, we count the proper number of squares to the right (in the positive direction). If the x coordinate is negative, we count to the left. If the y-coordinate is positive, we count the proper number of squares upward (in the positive direction). If the y coordinate is negative, we count downward.
EXAMPLE 1 Plot the point (5, 2) on a rectangular coordinate system. Label this as point A.
Since the x-coordinate is 5, we first count 5 units to the right on the x axis. Then, because the y-coordinate is 2, we count 2 units up from the point where we stopped on the x-axis. This locates the point corresponding to (5, 2). We label this point A.
It is important to remember that the first number in an ordered pair is the x-coordinate, and the second number is the y-coordinate. The points represented by (5, 2) and (2,5) are totally different, as we shall see in the next example.
EXAMPLE 2 Plot the point (2,5) on a rectangular coordinate system.
We count 2 units to the right for the x value of the point. Then we count 5 units upward for the y value of the point. We draw a dot and label it C at the location of the ordered pair (2, 5).
We observe that point C (2,5) is in a very different location from point A (5,2) in Example 1.
EXAMPLE 3 Plot the following points.
F : (0,5) G : (3, 3/2) H : (-6, 4) : (-3, -4) J : (-4, 0) K : (2, -3) L : (6.5, -7.2)
These points are plotted in the figure.
Note: When you are plotting decimal values like (6.5, -7.2), put the location of the plotted point halfway between 6 and 7 for the 6.5 and at your best approximation in the y-direction for the -7.2.
Determining the Coordinates of a Plotted Point
Sometimes, we need to find the coordinates of a point that has been plotted. First, we count the units we need on the x-axis to get as close as possible to that point. Next we count the units up or down we need to go from the x-axis to finally reach that point.
EXAMPLE 4 What ordered pair of numbers represents the point A in the figure below?
We see that the point A is represented by the ordered pair (5,4). We first counted 5 units to the right of the origin on the x-axis. Thus we obtain 5 as the first number of the ordered pair. Then we counted 4 units upward on a line parallel to the y-axis. Thus we obtain 4 as the second number on the ordered pair.
EXAMPLE 5 Write the coordinates of each point plotted below.
The coordinates of each point are:
E = (5, 1) F=(3,-4) G=(0,-6) H=(-2,-2)
I = (-5,0) J=(-2,2) K=(1,5)
Be very careful that you put the x-coordinate first and the y-coordinate second. Be careful that each sign is correct.
Finding Ordered Pairs for a Given Linear Equation
Equations such as 3x + 2y = 5 and 6x + y = 3 are called linear equations in two variables. A linear equation in two variables is an equation that can be written in the form Ax + By = C where A, B, and C are real numbers but A and B cannot both be zero. Replacement values for x and y that make true mathematical statements of the equation are called truth values and an ordered pair of these truth values is called a solution.
Consider the equation 3x + 2y = 5. The ordered pair (1, 1) is a solution to the equation. When we replace x by 1 and y by 1 we obtain a true statement.
3(1) +2(1) =5 or 3+2=5
There are an infinite number of solutions, for any given linear equation in two variables.
We obtain ordered pairs from examining such a linear equation. If one value of an ordered pair is known, the other can be quickly obtained. To do so, we replace one variable in the linear equation by the known value. Then using the methods learned in previous lesson we solve the resulting equation for the other variable.
EXAMPLE 6 Find the missing coordinate to complete the following ordered pairs for
the equation 2x + 3y = 15.
(a) (0, ?) (b) (?, 1)
(a) In the ordered pair (0, ?) we know that x = 0. Replace x by 0 in the equation.
2x + 3y = 15
2(0)+3y=15
0+ 3y = 15
y=5
Thus we have the ordered pair (0, 5).
(b) In the ordered pair ( ? , 1), we do not know the value of x. However, we do know that y = 1. So we start by replacing the variable y by 1. We will end up with an equation with one variable x. We can then solve for x.
2x + 3y = 15
2x + 3(1) = 15
2x+3= 15
2x = 12
x=6
Thus we have the ordered pair (6, 1).
The linear equations that we work with are not always written in the form Ax + By = C but are sometimes solved for y as in y = -6x + 3. Consider the equation y = -6x + 3. The ordered pair (2, -9) is a solution to the equation. When we replace x by 2 and y by -9 we obtain a true mathematical statement:
(-9) = -6(2) + 3 or -9 = -12 + 3.
EXAMPLE 7 Find the missing coordinate to complete the following ordered pairs for the equation y = -3x + 4
(a) (2, ?) (b) (-3, ? )
(a) For the ordered pair (2, ?) we know that x is 2, so we replace x by 2 in the equation and solve for y.
y= -3x+4
y=-3(2)+4
y=-6+4
y=-2
y = -3x + 4
(-2)=-3(2)+4
-2=-6+4 ✔
Thus, the solution is the complete ordered pair (2, -2).
(b) For the ordered pair (-3, ?) we replace x by -3 in the equation.
y=-3x+4
y=-3(-3)+4)
y=9+4
y= 13
Check.
y= -3x+ 4
13=-3(-3)+4
13=9+4 ✔
Thus, the solution is the ordered pair (-3, 13).
### Graphing Linear Equations
After studying this section, you will be able to:
1. Graph a straight line by finding three ordered pairs that are solutions to the linear equation.
2. Graph a straight line by finding its x- and y-intercepts.
3. Graph horizontal and vertical lines.
Graphing a Linear Equation by Plotting Three Ordered Pairs
We have seen that the graph of an ordered pair is a point. An ordered pair can also be a solution to a linear equation in two variables. Since there are an infinite number of solutions, there are an infinite number of ordered pairs and an infinite number of points. If we can plot these points, we will be graphing the equation. What will this graph look like? Let’s look at the equation y = -3x + 4. To look for a solution to the equation, we can choose any value for x. For convenience we will choose x = 0. That is, the first coordinate of the ordered pair will be 0. To complete the ordered pair (0, ), we substitute 0 for x in the equation:
y==-3x+4==-3(0) + 4==0+4==4
Thus the ordered pair is (0, 4).
To find another ordered pair that is a solution to the equation y = -3x + 4, let x= 1.
y==-3x+4==-3(1) +4==-3+4==1
Thus the ordered pair (1, 1) is another solution to the equation.
The graph of these two ordered pairs or solutions is shown below. From geometry we know that two points determine a line. Thus we can say that the line that contains the two points is the graph of the equation y = -3x + 4. In fact, the graph of any linear equation in two variables is a straight line.
While you only need two points to determine a line, we recommend that you use three points to graph an equation. Two points to determine the line and a third point to verify. For ease in plotting the points it is better that the ordered pairs contain integers.
EXAMPLE 1 Find three ordered pairs that satisfy 2x+y=4. Then graph the resulting straight line.
Since we can choose any value for x, choose those numbers that are convenient. To organize the results, we will make a table of values. We will let x = 0, x = 1, and x = 3, respectively. We write these numbers under x in our table of values. For each of these x values, we find the corresponding y value in the equation 2x + y = 4
We record these results by placing each y value in the table next to its corresponding x value. Keep in mind that these values represent ordered pairs each of which is a solution to the equation. Since there are an infinite number of ordered pairs, we will prefer those with integers whenever possible.
If we plot these ordered pairs and connect the three points, we get a straight line that is the graph of the equation 2x + y = 4. The graph of the equation is shown in the figure below.
EXAMPLE 2 Graph 5x-4y + 2 =2.
First, we simplify the equation by adding -2 to each side.
5x -4y+2=2
5x-4y+2-2=2-2
5x-4y =0
Since we are free to choose any value of x, x = 0 is a natural choice. Calculate the value of y when x = 0.
5x-4y =0
5(0)-4y =0
-4y =0
y=0 Remember: Any number times 0 is 0. Since -4y=0, y must equal 0.
Now let’s see what happens when x = 1.
5x-4y =0
5(1)-4y = 0
5-4y=0
-4y = -5
y=-5/-4 or 5/4 This is not an easy number to graph.
A better choice for a replacement of x is a number that is divisible by 4. Let’s see why.
Let x = 4 and let x = -4.
5x-4y=0 5x-4y=0
5(4)-4y=0 5(-4)-4y=0
20-4y=0 -20-4y=0
-4y=-20 -4y=20
y=-20/-4 or 5 y=20/-4 or -5
To Think About In Example 2, we picked values of x and found the corresponding values for y. An alternative approach is to first solve the equation for the variable y. Thus
5x-4y = 0
-4y = -5x Add -5x to each side.
(-4y)/-4= (-5x)/-4 Divide each side by -4.
y=5/4x
Now let x =-4, x = 0, and x = 4, and find the corresponding values of y. Graph the equation.
In the previous two examples we began by picking values for x. We could just as easily have chosen values for y.
EXAMPLE 3 Graph 3x-4y = 12.
We first find three ordered pairs by arbitrarily picking three values for y and in each case solving for x. We will choose y=0, y=3, and y=-3. Can you see why?
Graphing a Straight Line by Plotting Its Intercepts
What values should we pick for x and y? Which points should we use for plotting? For many straight lines it is easiest to pick the two intercepts . A few lines have only one intercept. We will discuss these separately.
EXAMPLE 4. Graph 5y-3x = 15 by the intercept method.
Let y = 0.
5(0)-3x = 15 Replace y by 0.
-3x = 15 Simplify.
x=-5 Divide both sides by -3.
The x-intercept is -5.
The ordered pair is (-5, 0), the x-intercept point.
Let x = 0.
5y-3(0) = 15 Replace x by 0.
5y = 15 Simplify.
y=3 Divide both sides by 5.
The y-intercept is 3.
The ordered pair is (0, 3), the y-intercept point.
We find another pair to have a third point.
Let y = 6.
5(6)-3x = 15 Replace y by 6.
30-3x = 15 Simplify.
-3x = -15 Subtract 30 from both sides.
x=-15/-3 or 5
The ordered pair is (5, 6).
Graphing Horizontal and Vertical Lines
You will notice that the x-axis is a horizontal line. It is the line y = 0, since for any x, y is 0. Try a few points. (1, 0), (3, 0), (-2, 0) all lie on the x-axis. Any horizontal line will be parallel to the x-axis. Lines such as y = 5 and y = 2 are horizontal lines. What does y = 5 mean? It means that, for any x, y is 5. Likewise y = -2 means that, for any x, y= -2.
How can we recognize the equation of a line that is horizontal, that is, parallel to the x-axis.
EXAMPLE 5 Graph y = -3.
You could write the equation as 0x + y = -3. Then it is clear that for any value of x that you substitute you will always obtain y=-3. Thus, as shown in the figure, (4, -3), (0, -3), and (-3, -3) are all ordered pairs that satisfy the equation y = -3. Since the y-coordinate of every point on this line is -3, it is easy to see that the horizontal line will be 3 units below the x-axis.
Notice that the y-axis is a vertical line. This is the line x = 0, since, for any y, x is 0. Try a few points. (0, 2), (0, -3), (0, 4) all lie on the y-axis. Any vertical line will be parallel to the y-axis. Lines such as x = 2 and x =-3 are vertical lines. Think of what x = 2 means. It means that, for any value of y, x is 2. x = 2 is a vertical line two units to the right of the y-axis.
How can we recognize the equation of a line that is vertical, that is, parallel to the y-axis?
EXAMPLE 6 Graph x=5
This can be done immediately by drawing a vertical line 5 units to the right of the origin. The x-coordinate of every point on this line is 5.
The equation x-5 = 0 can be rewritten as x=5 and graphed as shown.
### Slope of a Line
After studying this section, you will be able to:
1. Find the slope of a straight line given two points on the line.
2. Find the slope and y-intercept of a straight line, given the equation of the line.
3. Write the equation of a line, given the slope and the y-intercept.
4. Graph a line using the slope and y-intercept.
5. Find the slopes of lines that are parallel or perpendicular.
Find the Slope of a Straight Line, Given Two Points on the Line
We often use the word slope to describe the incline (the steepness) of a hill. A carpenter or a builder will refer to the pitch or slope of a roof. The slope is the change in the vertical distance compared to the change in the horizontal distance as you go from one point to another point along the roof. If the change in the vertical distance is greater than the change in the horizontal distance, the slope will be steep. If the change in the horizontal distance is greater than the change in the vertical distance, the slope will be gentle.
In a coordinate plane, the slope of a straight line is defined by the change in y divided by the change in x.
Consider the line drawn through points A and B in the figure. If we measure the change from point A to point B in the x-direction and the y-direction, we will have an idea of the steepness (or the slope) of the line. From point A to point B the change in y values is from 2 to 4, a change of 2. From point A to point B the change in x values is from 1 to 5, a change of 4.
Thus,
Informally, we can describe this move as the rise over the run: slope=rise/run.
We now state a more formal (and more frequently used) definition.
The use of subscripted terms such as x_1, x_2 and so on, is just a way of indicating that the first x value is x_1, and the second x value is x_2. Thus (x_1, y_1) are the coordinates of the first point and (x_2, y_2) are the coordinates of the second point. The letter m is commonly used for the slope.
EXAMPLE 1 Find the slope of the line that passes through (2, 0) and (4, 2).
Let (2, 0) be the first point (x_1, y_1) and (4, 2) be the second point (x_2, y_2)
By the formula,
slope = m= (y_2-y_1)/(x_2-x_1)=(2-0)/(4-2)=2/2=1
The sketch of the line is shown in the figure above.
Note that the slope of the line will be the same if we let (4, 2) be the first point (x_1, y_1) and (2, 0) be the second point (x_2, y_2).
m=(y_2-y_1)/(x_2-x_1)=(0-2)/(2-4)=-2/-2=1
Thus, given two points, it does not matter which you call (x_1, y_1) and which you call (x_2, y_2).
WARNING Be careful, however, not to put the x’s in one order and the y’s in another order when finding the slope, given two points on a line.
EXAMPLE 2 Find the slope of the line through (-3, 2) and (2, -4).
Let (-3, 2) be (x_1, y_1) and (2, -4) be (x_2, y_2)
m=(y_2-y_1)/(x_2-x_1)=(-4-2)/(2-(-3))=(-4-2)/(2+3)=-6/5
The slope of this line is negative. We would expect this since the y value decreased from 2 to -4
EXAMPLE 3 Find the slope of the line that passes through the given points.
(a) 0, 2) and (5, 2) (b) (-4, 0) and (-4, -4)
(a) Take a moment to look at the y values. What do you notice? What does this tell you about the line? Now calculate the slope.
m=(2-2)/(5-0)=0/5=0
A horizontal line has a slope of 0.
(b) Take a moment to look at the x values. What do you notice? What does this tell you about the line? Now calculate the slope.
m=(-4-0)/(-4-(-4))=-4/0
Recall that division by 0 is undefined.
The slope of a vertical line is undefined. We say that this line has no slope.
Notice in our definition of slope that x_2 != x_1. Thus it is not appropriate to use the formula for slope for the points in (b). We do so to illustrate what would happen if x_2 = x_1. We get an impossible situation, (y_2-y_1)/0. Now you can see why we include the restriction x_2 != x_1, in our definition.
Finding the Slope and y-intercept of a Line, Given the Equation of a Line
Recall that the equation of a line is a linear equation in two variables. This equation can be written in several different ways. A very useful form of the equation of a straight line is the slope—intercept form. The form can be derived in the following way. Suppose that a straight line, with slope m, crosses the y-axis at a point (0, b). Consider any other point on the line and label the point (x, y).
This form of a linear equation immediately reveals the slope of the line, m, and where the line intercepts (crosses) the y-axis, b.
EXAMPLE 4 What is the slope and the y-intercept of the line whose equation is y= 4x - 5?
The equation is in the form y = mx + b.
y = 4x + (-5)
slope = 4 y-intercept = -5
The slope is 4 and the y-intercept is -5.
By using algebraic operations, we can write any linear equation in slope—intercept form and use this form to identify the slope and the y-intercept of the line.
EXAMPLE 5 What is the slope and the y-intercept of the line 5x + 3y = 2?
We want to solve for y and get the equation in the form y = mx + b. We need to isolate the y variable.
Writing the Equation of a Line, Given the Slope and the y-intercept
If we know the slope of a line m, and the y-intercept, b, we can write the equation of the line, y = mx + b.
EXAMPLE 6 Find the equation of the line with slope 2/5 and y-intercept -3.
(a) Write the equation in slope—intercept form, y = mx + b.
(b) Write the equation in Ax + By = C form.
(a) We are given that m = 2/5 and b = -3. Since
y=mx+b
y = 2/5x+(-3)
y=2/5x-3
(b) Recall, for the form Ax + By = C, that A, B, and C are integers. We first clear the equation of fractions. Then we move the x term to the left side.
5y = 5((2x)/5)-5(3) Multiply each term by 5.
5y = 2x -15 Simplify.
-2x+5y=-15 Subtract 2x from each side.
2x-y = 15 Multiply each term by -1. The form Ax + By = C is usually written with A as a positive integer.
Graphing a Line Using the Slope and the y-Intercept
If we know the slope of a line and the y-intercept, we can draw the graph of the line.
EXAMPLE 7 Graph a line with slope m = 2/3 and y-intercept of -3. Use the coordinate system below.
Recall that the y-intercept is the point where the line crosses the y-axis. The x-coordinate of this point is 0. Thus the coordinates of the y-intercept for this line are (0, -3). We plot the point.
Recall that slope =rise/run. Since the slope for this line is 2/3, we will go up (rise) 2 units and go over (run) to the right 3 units from the point (0, -3). Look at the figure below. This is the point (3, -1). Plot the point. Draw a line that connects the two points (0, -3) and (3, -1).
This is the graph of the line with slope 2/3 and y-intercept of -3.
Finding the Slopes of Lines That Are Parallel or Perpendicular
Parallel lines are two straight lines that never touch. Look at the parallel lines in the figure below.
Notice that the slope of line a is -3 and the slope of line b is also -3. Why do you think the slopes must be equal? What would happen if the slope of line b was -1. Try it.
Perpendicular lines are two lines that meet at a 90° angle. Look at the perpendicular lines in the figure below.
The slope of line c is -3. The slope of line d is 1/3. Notice that
(-3)(1/3)=(-3/1)(1/3)=-1
You may wish to draw several pairs of perpendicular lines to determine if this will always happen.
EXAMPLE 8 Linee has a slope of 2/3.
(a) If line f is parallel to line e, what is its slope?
(b) If line g is perpendicular to line e, what is its slope?
(a) Parallel lines have the same slope.
Line f will have a slope of 2/3.
(b) Perpendicular lines have slopes whose product is -1.
m_1m_2 = -1
2/3m_2=-1 Substitute 2/3 for m_1.
(-3/2)(-2/3)m_2=-1(-3/2) Multiply both sides by 3/2.
m_2=3/2
The slope of the line g is 3/2
### Obtaining the Equation of a Line
After studying this section, you will be able to:
1. Write the equation of a line, given a point and a slope.
2. Write the equation of a line, given two points.
3. Write the equation of a line, given a graph of the line.
Writing the Equation of a Line, Given a Point and a Slope
If we know the slope of a line and the y-intercept, we can write the equation of the line in slope—intercept form. Sometimes we are given the slope and a point on the line. We use the information to find the y-intercept. Then we can write the equation of the line.
EXAMPLE 1 Find the equation of a line with a slope of 4 that passes through the point (3, 1).
We are given the following values: m = 4, x = 3, y = 1.
y=mx+b
We are given that the slope of the line is 4.
y=4x+b
Since (3, 1) is a point on the line, it satisfies the equation. Substitute x = 3 and y = 1 into the equation.
1=4(3)+b
Solve for b.
1=12+b
-11=b
Thus the y-intercept is -11. We can now write the equation of the line.
y=4x- 11
It may be helpful to summarize our approach.
EXAMPLE 2 Find the equation of a line with slope 2/3 that passes through the point (-3, 6).
We write out the values m = 2/3, x = -3, y = 6.
y=mx+b
6= (-2/3)(-3)+b Substitute known values.
6=2+b
4=b
The equation of the line is y2/3x+ 4.
Finding the Equation of a Line, Given Two Points
Our procedure can be extended to the case for which two points are given.
EXAMPLE 3 Find the equation of a line that passes through (2, 5) and (6, 3).
We first find the slope of the line. Then we proceed as in Example 2.
m=(y_2-y_1)/(x_2-x_1)
m=(3-5)/(6-2) Substitute (x_1, y_1) = (2,5) and (x_2, y_2) = (6, 3) into the formula.
m=-2/41/2
Choose either point, say (2,5), and proceed as in Example 2.
y=mx+b
51/2(x)+b
5=-1+b
6=b
The equation of the line is y = 1/2x + 6.
Note: After finding the slope m = 1/2, we could have used the other point (6, 3) and would have arrived at the same answer. Try it.
Finding the Equation of a Line from the Graph
EXAMPLE 4 What is the equation of the line in the figure below?
First, look for the y-intercept. The line crosses the y-axis at (0,4). Thus b=4.
Second, find the slope.
Look for another point on the line. We chose (5, -2). Count the number of vertical units from 4 to -2 (rise). Count the number of horizontal units from 0 to 5 (run).
m=-6/5
Now we can write the equation of the line, m6/5 and b = 4.
y=mx+b
y6/5x+4
### Graphing Linear Inequalities
After studying this section, you will be able to:
1. Graph linear inequalities in two variables.
In earlier tutorial we discussed inequalities in one variable. Look at the inequality x < -2 (x is less than -2). Some of the solutions to the inequality are -3, -5, and -51/2. In fact all numbers to the left of -2 on the number line are solutions. The graph of the inequality is given below. Notice that the open circle at 2 indicates that 2 is not a solution.
Graphing Linear Inequalities in Two Variables
Consider the inequality y >= x. The solution of the inequality is the set of all possible ordered pairs that when substituted into the inequality will yield a true statement. Which ordered pairs will make the statement y>=x true? Let’s try some.
(0, 6) (-2, 1) (1, -2) (3, 5) (4, 4)
6 >= 0, true 1>= -2, true -2 >= 1, false 5 >= 3, true 4>= 4, true
(0, 6), (-2, 1), (3, 5), and (4, 4) are solutions to the inequality y >= x. In fact, every point at which the y-coordinate is greater than or equal to the x-coordinate is a solution to the inequality. This is shown by the shaded region in the graph below. Notice that the solution set includes the points on the line y = x.
Did we need to test so many points? The solution set for an inequality in two variables will be the region above the line or the region below the line. It is sufficient to test one point. If the point is a solution to the inequality, shade the region that includes the point. If the point is not a solution, shade the region on the other side of the line.
EXAMPLE 1. Graph 5x + 3y > 15. Use the coordinate system below.
We begin by graphing the line 5x + 3y = 15. You may use any method discussed previously to graph the line. Since there is no equal sign in the inequality, we will draw a dashed line to indicate that the line is not part of the solution set.
Look for a test point. The easiest point to test is (0, 0). Substitute (0, 0) for (x, y) in the inequality.
5x+3y>15
5(0)+3(0)>15
0>15 False.
(0,0) is not a solution. Shade the side of the line that does not include (0, 0)
EXAMPLE 2 Graph 2y <= -3x.
Step 1 Graph 2y = -3x. Since is used, the line should be a solid line.
Step 2 We see that the line passes through (0, 0).
Step 3 Choose another test point. We will choose -3,-3
2y <= -3x
2(-3)<=-3(-3)
-6<=9 True.
Shade the region that includes (-3, -3). Shade the region below the line.
If we are graphing the inequality x<-2 on the coordinate plane, the solution will be a region. Notice that this is very different from the solution x<-2 on the number line discussed earlier.
EXAMPLE 3 Graph x < -2 on the coordinate plane.
Step 1 Graph x = -2. Since < is used, the line should be dashed.
Step 2 Test (0,0) in the inequality.
x<-2
0<-2 False.
Shade the region that does not include (0, 0). Shade the region to the left of the line x=-2
### Functions
After studying this section, you will be able to:
1. Understand and use the definition of a relation and a function.
2. Graph simple nonlinear equations.
3. Determine if a graph represents a function.
4. Use function notation.
Understanding and Using the Definitions of Relation and Function
Thus far you have studied linear equations in two variables. You have seen that such an equation can be represented by a table of values, by the algebraic equation itself, and by a graph.
The solutions to the linear equation are all the ordered pairs that satisfy the equation (make the equation true). They are all the points that lie on the graph of the line. These ordered pairs can be represented in a table of values. Notice the relationship between the ordered pairs. We can choose any value for x. But once we have chosen a value for x, the value of y is determined. For example, in the equation y=-3x+4 if x is 0, then y must be 4. We say that x is the independent variable and that y is the dependent variable .
Mathematicians call such a pairing of two values a relation.
All the first coordinates in each ordered pair make up the domain of the relation. All the second coordinates in each ordered pair make up the range of the relation. Notice that the definition of a relation is very broad. Some relations cannot be described by an algebraic expression. These relations may simply be a set of discrete ordered pairs.
EXAMPLE 1 State the domain and range of the following relation.
{(5, 7), (9, 11), (10, 7), (12, 14)}
The domain consists of all the first coordinates in the ordered pairs.
The range consists of all of the second coordinates in the ordered pairs. Thus
The domain is {5, 9, 10, 12}
The range is {7, 11, 14} We only list 7 once.
Some relations have a special property and are called functions. The relation y = -3x + 4 is a function. If you look at its table of values, no two different ordered pairs have the same first coordinate. If you look at its graph, you will notice that any vertical line intersects the graph only once. Sometimes we get a better idea if we look at some relations that are not functions. Look at y^2 = x.
Notice that, if x is 1, y could be 1 since 1^2 = 1 or y could be -1 since (-1)^2 = 1. That is, the ordered pairs (1, -1) and (1, 1) both have the same first coordinate. y^2 = x is not a function. What do you notice about the graph?
EXAMPLE 2 Determine if the relation is a function or not a function.
(a) {(3, 9), (4, 16), (5, 9), (6, 36)} (b) {(7, 8), (9, 10), (12, 13), (7, 14)}
(a) Look at the ordered pairs. No two ordered pairs have the same first coordinate. Thus (a) is a function. Note that the ordered pairs (3,9) and (5,9) have the same second coordinate, but this does not keep the relation from being a function.
(b) Look at the ordered pairs. Two different ordered pairs, (7, 8) and (7, 14), have the same first coordinate. Thus this relation is not a function.
Functions are what we hope to find when we analyze two sets of data. Look at the table of values that compares Celsius temperature with Fahrenheit temperature. Is there a relationship between degrees Fahrenheit and degrees Celsius? Is the relation a function?
Since every Fahrenheit temperature produces a unique Celsius temperature, we would expect this to be a function. We can verify our assumption by looking at the formula C = 5/9(F-32) and its graph. The formula is a linear equation, and its graph is a line with a slope 5/9 and y-intercept at about -17.8. The relation is a function. Notice that the dependent variable is C, since the value of C depends on the value of F. We say that F is the independent variable . The domain can be described as the set of possible values of the independent variable. The range is the set of corresponding values of the dependent variable. Scientists believe that the coldest temperature possible is approximately -273°C. They call this temperature absolute zero. Thus,
Domain = {all possible Fahrenheit temperatures from absolute zero to infinity}
Range = {all corresponding Celsius temperatures from -273°C to infinity}
EXAMPLE 3 Determine if the relation is a function or not a function. If it is a function, identify the domain and range.
(a) Looking at the table, we see that no two different ordered pairs have the same first coordinate. The area of a circle is a function of the length of the radius.
Next we need to identify the independent variable in order to determine the domain. Sometimes it is easier to identify the dependent variable. Here we notice that the area of the circle depends on the length of the radius. Thus radius is the independent variable. Since a negative length does not make sense, the radius cannot be a negative number.
Domain = {all nonnegative real numbers}
Range = {all nonnegative real numbers}
(b) No two different ordered pairs have the same first coordinate. Interest is a function of time.
Since the amount of interest paid on a loan depends on the number of years (term of the loan), interest is the dependent variable and time is the independent variable. Negative numbers do not apply in this situation.
Domain = {all positive real numbers greater than 1}
Range = {all positive real numbers greater than \$320}
Graphing Simple Nonlinear Equations
Thus far in this lesson we have graphed linear equations in two variables. We now turn to graphing a few nonlinear equations. We will need to plot more than three points to get a good idea of what the graph will look like.
EXAMPLE 4 Graph y = x^2.
Begin by constructing a table of values. We select values for x and then determine by the equation the corresponding values of y. We will include negative values for x as well as positive values. We then plot the ordered pairs and connect the points with a smooth curve.
This type of curve is called a parabola. We will study the graph of these types of curves more extensively in later tutorials.
EXAMPLE 5 Graph x = y^2 + 2.
We will select a value of y and then substitute it into the equation to obtain x. For convenience in graphing, we will repeat the y column at the end so that it is easy to write the ordered pairs (x, y).
If the equation involves fractions with variables in the denominator, we must use extra caution. You may never divide by zero.
EXAMPLE 6 Graph y=4/x.
It is important to note that x cannot be zero because division by zero is not defined. y = 4/0 is not allowed! Observe that when we draw the graph we get two separate branches of the curve that do not touch.
Determining If a Graph Represents a Function
Can we tell from a graph if it is the graph of a function? Recall that a function must have the property that no two different ordered pairs have the same first coordinate. That is, each value of x must have a separate unique value of y. Look at the graph of y = x^2 in Example 4. Each x value has a unique y value. Look at the graph of x = y^2 + 2 in Example 5. At x = 3 there are two y values, 1 and -1. In fact, for every x value greater than 2 there are two y values. x = y^2 + 2 is not a function.
Observe that we can draw a vertical line through (6, 2) and (6, -2). Any graph that is not a function will have at least one region in which a vertical line will cross the curve more than once.
EXAMPLE 7 Determine if each of the following is the graph of a function.
(a) The graph of the straight line is a function. A vertical line can only cross this straight line in one location.
(b) and (c) Each of these graphs is not the graph of a function. In each case a vertical line can cross the curve in more than one place.
Using Function Notation
We have seen that an equation like y = 2x + 7 is a function. For each value of x, the equation assigns a unique value to y. We could say ‘‘y is a function of x.’’ This statement can be symbolized by using the function notation y = f(x). Thus you can avoid the y variable completely and describe an expression like 2x + 7 as a function of x by writing f(x) = 2x + 7.
WARNING Be careful. The notation f(x) does not mean f multiplied by x.
EXAMPLE 8 If f(x) = 2x + 7, then find f(3).
f(3) means that we want to find the value of the function f(x) when x = 3. In this example f(x) is 2x + 7. Thus we substitute 3 for x wherever x occurs in the equation and evaluate.
f(x) = 2x +7
f(3) == 2(3)+7==6+7== 13
EXAMPLE 9 If f(x) = 3x^2-4x + 5, find:
(a) f(2) (b) f(-2) (c) f(4)
(a)
f(2) == 3(2)^2-4(2) +5 == 3(4)-4(2)+5 == 12-8 +5 ==9
(b)
f(-2) == 3(-2)^2-4(-2) + 5 == 3(4)-4(-2) +5 == 12 +8 +5 ==25
(c)
f(4) == 3(4)^2-4(4) + 5 == 3(16)-4(4) + 5 == 48-16+5 == 37
When evaluating a function it is helpful to place parentheses around the value that is being substituted for x. Taking the time to do this will minimize sign errors in your work.<|endoftext|>
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This unit is dedicated to the many African Americans who have made a difference in the world. The people we include in this section worked for equality and civil rights. You can learn about the many African American women and inventors on the "Women" and "Inventors" pages that are part of this unit.
Read about the man who broke the color barrier in baseball - Baseball Hall of Fame
Follow a timeline of her life and learn about this brave woman and the Underground Railroad.
Martin Luther King, Jr.
Follow this Timeline and read about Dr. King and his words of equality.
Read more about Black History at Scholastic.com
Try the Internet African American History Challenge and learn more.
Back to PWMAD<|endoftext|>
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A Swift Decade
Gamma-ray bursts are the most energetic events we can observe today. GRBs are brief, intense flashes of high-energy radiation from space, and for years were one of astronomy's biggest mysteries. First observed in July 1969 (but first reported in 1973), GRBs remained mysterious for many years, because these brief, intense flashes of Gamma-rays are hard to localize in space. Understanding GRBs requires the capability of near-simultaneous observations at Gamma-ray, X-ray and optical wavelengths. Enter NASA's Swift space observatory, shown above. Swift includes three telescopes, one operating in the Gamma-ray range, one in the X-ray range, and one in the optical and ultraviolet. The Gamma-ray detector, the Burst Alert Telescope, provides detection and localization of Gamma-ray bursts in a wide field of view by using the Gamma-ray shadows cast by a burst. Once found by the BAT, Swift can rapidly slew to put the source in the field of the X-ray Telescope and the UV and Optical Telescope, to measure the afterglows of the burst at those energies, and to provide an accurate position for subsequent follow-up by Swift and other space- and ground-based telescopes. Swift, launched in November 2004, has revolutionized our understanding of GRBs, and demonstrated that GRBs are associated with the formation of black holes, either through the collapse of massive stars or the merging of neutron stars. In Swift's decade of operation, it has made 315,000 individual observations of 26,000 individual targets, and has detected over 900 GRBs. Swift is very versatile too. When not hunting GRBs, Swift uses its complement of telescopes to study high-energy emission from accreting supermassive black holes in active galaxies and see them swallow the occasional star, map galaxies in the ultraviolet and in X-rays, detect and study violent high-energy emission from stars, and watch the Milky Way's supermassive black hole, Sgr A*, for long-term changes.
Published: November 24, 2014
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Each week the HEASARC
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At the refinery, thousands of tons of oil are turned into useful substances every day.
What is Crude Oil?
Crude oil is a mixture of many different chemicals called hydrocarbons.
Hydrocarbons are useful because they can be used to make many different substances, such as butane gas, gasoline and diesel fuel, plastics, medicines, and paints.
Crude oil refining process
Before it can be used, crude oil has to be refined.
In the refinery, the crude oil is heated and turned into a mixture of gases and liquids.
The mixture is then passed into a huge fractionating column.
Inside the column, there are trays at different levels.
The column is hotter at the bottom and cooler at the top.
The gases pass up the column and condense into liquids on the trays at different temperatures.
The gases that have longer molecules, with more carbon atoms, condense at higher temperatures.
Those with shorter molecules and fewer carbon atoms condense at lower temperatures.
This process is another example of fractional distillation.
Different hydrocarbons have different numbers of carbon atoms.
Hydrocarbons with 1 to 5 carbon atoms remain as gas at the top of the column. The gas is used as bottled fuel.
Gasoline has 5 to 10 carbon atoms and is used as fuel for cars.
Naphtha has 8 to 12 carbon atoms and is used to make chemicals.
Kerosene has 9 to 16 carbon atoms and is used as jet fuel.
Diesel oil has 15 to 25 carbon atoms and is used as fuel for trains and ships.
Bitumen has over 20 carbon atoms and is used for sealing roofs.<|endoftext|>
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## How do you calculate error propagation?
If you have some error in your measurement (x), then the resulting error in the function output (y) is based on the slope of the line (i.e. the derivative). The general formula (using derivatives) for error propagation (from which all of the other formulas are derived) is: Where Q = Q(x) is any function of x.
## What is the formula for uncertainty?
Relative uncertainty is relative uncertainty as a percentage = δx x × 100. To find the absolute uncertainty if we know the relative uncertainty, absolute uncertainty = relative uncertainty 100 × measured value.
## What does error propagation mean?
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables’ uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. The value of a quantity and its error are then expressed as an interval x ± u.
## Why is error propagation important?
Every measurement has an air of uncertainty about it, and not all uncertainties are equal. Typically, error is given by the standard deviation (σx) of a measurement. Anytime a calculation requires more than one variable to solve, propagation of error is necessary to properly determine the uncertainty.
## What is an error?
An error (from the Latin error, meaning “wandering”) is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. In statistics, “error” refers to the difference between the value which has been computed and the correct value.
## How do you calculate division error?
(b) Multiplication and Division: z = x y or z = x/y. The same rule holds for multiplication, division, or combinations, namely add all the relative errors to get the relative error in the result. Example: w = (4.52 ± 0.02) cm, x = (2.0 ± 0.2) cm.
## What is combination error?
In this Physics video in Hindi for class 11 we explained combination of errors. In this part we discussed on what happens to the error of a resultant quantity which is the sum or difference of two quantities. In these cases the error in the resultant quantity is the sum of the errors of those two quantities.
## What is uncertainty with example?
Uncertainty is defined as doubt. When you feel as if you are not sure if you want to take a new job or not, this is an example of uncertainty. When the economy is going bad and causing everyone to worry about what will happen next, this is an example of an uncertainty.
## What does uncertainty mean?
lack of certainty
You might be interested: Terminal speed equation
## Is the uncertainty principle true?
But even if two measurements hardly influence each other: quantum physics remains “uncertain.” “The uncertainty principle is of course still true,” the researchers confirm. “But the uncertainty does not always come from the disturbing influence of the measurement, but from the quantum nature of the particle itself.”
## What propagation means?
: the act or action of propagating: such as. a : increase (as of a kind of organism) in numbers. b : the spreading of something (such as a belief) abroad or into new regions. c : enlargement or extension (as of a crack) in a solid body.
## What does propagate mean?
verb (used without object), prop·a·gat·ed, prop·a·gat·ing. to multiply by any process of natural reproduction, as organisms; breed. to increase in extent, as a structural flaw: The crack will propagate only to this joint. (of electromagnetic waves, compression waves, etc.) to travel through space or a physical medium.
### Releated
#### How to write a regression equation
What is a regression equation example? A regression equation is used in stats to find out what relationship, if any, exists between sets of data. For example, if you measure a child’s height every year you might find that they grow about 3 inches a year. That trend (growing three inches a year) can be […]
#### Solving an absolute value equation
How do you find the absolute value? 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. Can you solve problems using absolute value? Solving absolute value equations is as easy as working with regular linear equations. The […]<|endoftext|>
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Neanderthals - an extinct sub-species of humans - had a subtle genetic influence on modern humans. A study comparing Neanderthal DNA to the genes of people of European and Asian descent found that a host of illnesses, including depression and blood disorders, are expressed in humans, thanks to our ancient ancestors.
With modern technology, it is now possible to sequence the DNA of our distant Neanderthal cousins.
In a study comparing the genetic material of our closest extinct human relative to a genetic database of 28,000 people of Eurasian descent, scientists have confirmed the biological influence of modern humans’ ancient ancestors.
The DNA was passed down through the interbreeding of Neanderthals, who lived in what is today Europe and Asia, and early humans, who had migrated out of Africa.
As a result of that co-mingling, 40 to 60,000 years ago, say researchers, humans have inherited about two percent of their DNA from Neanderthals. Previously it was speculated that between one and four percent of our genetic material was derived from Neanderthals.
John Capra, the study’s lead author and a professor of biological sciences at Vanderbilt University in Nashville, Tennessee, says it appears that modern human health conditions - including disorders of the immune system, skin, neurological system as well as reproductive health - were indirectly influenced by Neanderthals.
Capra says the ancestral genes of early man also appear to have influenced the development of clinical depression, a serious psychiatric illness.
“What our results are saying is not that Neanderthals were depressed or that they are making us depressed. It’s that we find in modern environments the bits of DNA that we’ve inherited from Neanderthals are having an influence on these systems. And what that effect is remains to be seen but it’s certainly fascinating about why that might have been," said Capra.
The findings linking Neanderthal DNA to that of modern humans, by John Capra and colleagues, is published in the journal Science.
The researchers unveiled their work at the annual meeting of the American Association for the Advancement of Science.
This is the first direct comparison of Neanderthal DNA to that of modern humans. Capra says the genes probably conferred some environmental benefit to the extinct human sub-species.
He hopes that comparing it to the genetic profile of 21st century people might tell investigators how humans evolved and how the genes contributed to modern ailments.
“And while Neanderthal DNA has a significant influence on them or risk for them, it by no means dooms us to having those diseases," he said.
But Capra says that ancient DNA may have contributed to more physical traits that that are yet to be discovered.<|endoftext|>
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Although carefully planned at twilight so all animals can attend, things go terribly wrong during this walkabout. The group creates such a terrible hullabaloo that Namarrkun, the lightning man, is forced to show his strength.
Katherine Scholes begins this informative piece by describing the multi-facted nature of the word "peace" and what it can mean to different people at different times. Then she provides concrete ways that each of us can be a peacemaker.
An anchor chart is an artifact of classroom learning. Like an anchor, it holds students' and teachers' thoughts, ideas and processes in place. Anchor charts can be displayed as reminders of prior learning and built upon over multiple lessons.
This strategy provides tools to create questions that help students engage critically with Perspectives central texts and examine them for issues of power and social inequity. The activities suggested here also encourage readers to bring their knowledge and experiences to the reading of a text.
Agree/disagree statements challenge students to think critically about their knowledge of a topic, theme or text. The strategy exposes students to the major ideas in a text before reading—engaging their thinking and motivating them to learn more. It also requires them to reconsider their original thinking after reading the text and to use textual evidence to support and explain their thinking.
Students showcase artwork and nonfiction writing that addresses issues they found in the text. The result is a visual, collaborative and creative representation of student learning and ideas. An alternative to the bulletin board is a community newsletter.
Students create a community puzzle mural, a large-scale artistic depiction, usually displayed in a community space. Puzzle pieces covered in student’s artwork relating to diversity, anti-bias or social justice themes from the central text comprise the mural.
Students work in groups to role-play or tell stories about real life situations related to fairness, community, diversity or social justice themes. Students then perform their skits or stories for others as part of a class-wide “fairness fair.”
Students write to a business, school or community leader to call for action in response to a social justice issue from the central text. Alternatively, students can write open, persuasive letters to their peers or family members.
Select the parts of your Learning Plan you'd like to print. If your Tasks or Strategies have PDF handouts, they'll need to be printed separately. These are listed on the left side of each Task or Strategy page.<|endoftext|>
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# AP Board 9th Class Maths Solutions Chapter 10 Surface Areas and Volumes InText Questions
AP State Syllabus AP Board 9th Class Maths Solutions Chapter 10 Surface Areas and Volumes InText Questions and Answers.
## AP State Syllabus 9th Class Maths Solutions 10th Lesson Surface Areas and Volumes InText Questions
Try This
Take a cube of edge T cm and cut it as we did in the previous activity and find total surface area and lateral surface area of cube. [Page No. 216]
Solution:
If we cut and open a cube of edgewe obtain a figure as shown above.
In the figure, A, B, C, D, E, F are squares of side
The faces A, C, D, F forms the lateral surfaces of the cube.
∴ Lateral surface area of the cube = 4l2
And all six faces form the cube.
∴ Total surface area of the cube = 6l2
Do This
Question 1.
Find the total surface area and lateral surface area of the cube with side 4 cm. By using the formulae deduced in above. Try this. [Page No. 216]1. Find the total surface area and lateral surface area of the cube with side 4 cm. By using the formulae deduced in above. Try this. [Page No. 216]
Solution:
Total surface area of a cube = 6l2
Given l = 4crn
∴ T.S.A. = 6 × 42 = 6 × 16 = 96 cm2
∴ L.S.A. = 4l2 = 4 × 42 = 64cm2
Question 2.
Each edge of a cube is increased by 50%. Find the percentage increase in the surface area. [Page No. 216]
Solution:
Let the edge of a cube x units
Its surface area = 6l2 = 6x2 sq. units
If its edge is increased by 50%, then
new edge = x + 50% of x
Where x = increase/decrease
Try These
Question
a) Find the volume of a cube whose edge is ‘a’ units. [Page No. 217]
Solution:
V = edge3 = a3 cubic units.
b) Find the edge of a cube whose volume is 1000 cm3. [Page No. 217]
Solution:
V = edge3 = 1000 = 10 × 10 × 10 = 103
∴ Edge =10 cm
Do These
Question 1.
Find the volume of a cuboid if l = 12 cm, b = 10 cm, h = 8 cm. [Page No. 218]
Solution:
Volume V = lbh = 12 × 10 × 8 = 960cm3
Question 2.
Find the volume of cube, if its edge is 10 cm [Page No. 218]
Solution:
Volume V = l3 = 10 × 10 × 10 = 1000cm3
Question 3.
Find the volume of isosceles right angled triangular prism. [Page No. 218]
Solution:
Volume = Area of base × height
= Area of isosceles triangle × height
Activity
Question
Take the square pyramid and cube containers of same base and with equal heights. [Page No. 218]
Fill the pyramid with a liquid and pour into the cube (prism) completely. How many times it takes to fill the cube? From this, what inference can you make?
Thus volume of pyramid = $$\frac { 1 }{ 3 }$$ × of the volume of right prism
= $$\frac { 1 }{ 3 }$$ × Area of the base × height.
Note : A right prism has bases perpendicu¬lar to the lateral edges and all lateral faces are rectangles.
Do These
Question 1.
Find the volume of a pyramid whose square base is 10 cm and height is 8 cm. (Page No. 219)
Solution:
Volume of a pyramid
= $$\frac { 1 }{ 3 }$$ × Area of the base × height
= $$\frac { 1 }{ 3 }$$ × 10 x 10 × 8 = $$\frac { 800 }{ 3 }$$ cm3
Question 2.
The volume of a cube is 1200 cubic cm. Find the volume of square pyra¬mid of the same height. (Page No. 219)
Solution:
Volume of the square pyramid
= $$\frac { 1 }{ 3 }$$ × volume of the square prism
= $$\frac { 1 }{ 3 }$$ × 1200 = 400 cm3
Activity
Cut out a rectangular sheet of paper. Paste a thick string along the line as shown in the figure. Hold the string with your hands on either sides of the rectangle and rotate the rectangle sheet about the string as fast as you can.
Do you recognize the shape that the rotating rectangle is forming ?
Does it remind you the shape of a cylinder ? [Page No. 220]
Do This
Question 1.
Find C.S.A. of each of the following cylinders. [Page No. 221]
i) r = x cm; h = y cm
Solution:
CSA = 2πrh = 2πxy cm2
ii) d = 7 cm; h = 10 cm
Solution:
CSA = 2πrh
= 2 × $$\frac{22}{7} \times \frac{7}{2}$$ × 10
= 220 cm2
iii) r = 3 cm; h = 14 cm
Solution:
CSA = 2πrh
= 2 × $$\frac{22}{7} \times \frac{7}{2}$$ × 3 × 14
= 264 cm2
Question 2.
Find the total surface area of each of the following cylinder. [Page No. 222]
i)
Solution:
r = 7 cm; h = 10 cm
T.S.A. = 2πr (r + h)
=2 × $$\frac{22}{7}$$ × 10 (7 + 10)
= 2 × $$\frac{22}{7}$$ × 10 × 17
= 1068.5 cm2
ii)
h = 7cm; πr2 = 250
πr2 = 250 = 250
$$\frac{22}{7}$$ × r2 = 250 = 250
r2 = 125 x $$\frac{7}{11}$$
Try These
Question 1.
If the radius of a cylinder is doubled keeping Its lateral surface area the same, then what is its height? [Page No. 225]
Solution:
Let the initial radius and height of the
cylinder be r and h.
Then L.S.A. = 2πrh
When r is doubled and the L.S.A. remains the same, then the height be hr By problem new L.S.A. = 2πrh
= 2π (2r) (h1)
⇒ 2πrh = 4πrh1
∴ $$\frac{2 \pi \mathrm{rh}}{4 \pi \mathrm{r}}=\frac{1}{2} \mathrm{~h}$$
Height becomes its half.
Question 2.
A hot water system (Geyser) consists of a cylindrical pipe of length 14 m and diameter 5 cm. Find the total radiating surface of hot water system. [Page No. 225]
Solution:
Radius (r) = $$\frac{\text { diameter }}{2}=\frac{5}{2}$$ = 2.5 cm
Length of the pipe = height = 14 m Radiating surface = 2πrh
= 2 × $$\frac{22}{7}$$ × 2.5 × 1400
= 22000 cm3
Activity
Question
Making a cone from a sector. [Page No. 227] Follow the instructions and do as shown in the figure.
i) Draw a circle on a thick paper Fig (a).
ii) Cut a sector AOB from it Fig (b).
iii) Fold the ends A, B nearer to each other slowly and join AB. Remember A, B should not overlap on each other. After joining A, B attach them with cello tape Fig (c).
iv) What kind of shape you have obtained? Is it a right cone?
While making a cone observe what hap-pened to the edges ‘OA’ and ‘OB’ and length of arc AB of the sector?
Try This
A sector with radius r and length of its arc / is cut from a circular sheet of paper. Fold it as a cone. How can you derive the formula of its curved surface area A = πrl. [Page No. 228]
C.S.A = πrl
Solution:
When a sector of radius ‘r’ and whose length of arc l is folded to form a cone. Radius ‘r’ becomes slant height ‘l’ and arc ‘l’ becomes perimeter of the base 2πr.
∴ Area of the sector = $$\frac{l r}{2}$$ = Area of the cone
$$\frac{2 \pi \mathrm{r} l}{2}$$ = Surface area of the cone
C.S.A = πrl
Do This
Question 1.
Cut a right angled triangle. Stick a string along its perpendicular side, as shown in fig. (T) hold both the sides of a string with your hands and rotate it with constant speed. What do you observe ? [Page No. 229]
Solution:
A right circular cone is observed.
Question 2.
Find the curved surface area and total surface area of the each following right circular cones.[Page No. 229]
Solution:
OP = 2 cm; OB = 3.5 cm
OP = h = 2 cm
OP = 3.5 cm; AB = 10 cm
r = $$\frac{\mathrm{AB}}{2}$$ = 5cm; h = 3.5cm
r = OB = 3.5 cm
C.S.A. = πrl
T.S.A. = πr (r + l)
= $$\frac{22}{7}$$ × 3.5(3.5 + 4.03)
= $$\frac{22}{7}$$ × 3.5 × 7.53 = 82.83cm2
C.S.A. = πrl
$$\frac{22}{7}$$ × 5 × 6.10
= 95.90cm2
T.S.A. = πr (r + l)
= $$\frac{22}{7}$$ × 5 × (5 + 6.10)
= 174.42 cm2
Activity
Draw a circle on a thick paper and cut it neatly. Stick a string along its diam¬eter. Hold the both the ends of the string with hands and rotate with con¬stant speed and observe the figure so formed. [Paper 235]
Try This
Question 1.
Can you find the surface area of sphere in any other way ? (Page No. 235)
Solution:
Height of the pyramid is equal to r.
To derive the formula of the surface area of a sphere, we imagine a sphere with many pyramids inside of it until the base of all the pyramids cover the entire surface area of the sphere. In the figure below, only one of such pyra¬mid is shown.
Then, do a ratio of the area of the pyramid to the volume of the pyramid.
The area of the.pyramid is A.
The volume of the pyramid is V = (1/3) × A × r = (A × r)/3
So, the ratio of area to volume is A/V = A + (A × r) / 3 = (3 × A) / (A × r) = 3 / r
For a large number of pyramids, let say that n is such large number, the ratio of the surface area of the sphere to the volume of the sphere is the same as 3/r.
For n pyramids, the total area is n × A- Also for n pyramids, the total volume is n × V.
Therefore, ratio of total area to total volume is n × A/n × V = A/ V and we already saw before that A / V = 3 / r
Further more, n × Apyramid = Asphere (The total area of the bases of all pyramids or n pyramids is approximately equal to the surface area of the sphere).
n × Vpyramid = Vsphere (The total Volume of all pyramids or n pyramids is approximately equal to the volume of the sphere.
Putting observation # 1 and # 2 together, we get
Therefore, the total surface area of a sphere, call it S.A. is 4πr 2
Do These
Question 1.
A right circular cylinder just encloses a sphere of radius r (see figure). Find i) surface area of the sphere
ii) curved surface area of the cylinder
iii) ratio of the areas obtained in (i) and (ii) [Page No. 236]
[Page No. 236]
Solution:
i) Radius of the sphere = radius of the cylinder = r
∴Surface area of the sphere = 4πr2
ii) C.S.A. of cylinder = 2πr (2r) [∵ h = 2r]
= 4πr2
iii) Ratio of (i) and (ii) = 4πr2 : 4πr2 = 1:1
Question 2.
Find the surface area of each of the following figure.
i)
Surface area = 4πr2
C.S.A = 4 × $$\frac{22}{7}$$ × 7 × 7
= 616cm2
ii)
C.S.A = 2πr2 = 2 × $$\frac{22}{7}$$ × 7 × 7 = 308cm2
Total Surface area = 3πr2
= 3 × $$\frac{22}{7}$$ × 7 × 7 = = 462cm2
Do This
Question 1.
Find the volume of the sphere given below[Page No. 238]
Solution:
r = 3 cm
V = $$\frac{4}{3} \pi r^{3}=\frac{4}{3} \times \frac{22}{7}$$ × 3 × 3 × 3
= 113.14cm3
d = 5.4 cm
r = $$\frac{\mathrm{d}}{2}=\frac{5.4}{2}$$ = 2.7 cm
V = $$\frac{4}{3} \pi r^{3}=\frac{4}{3} \times \frac{22}{7}$$ × 2.7 × 2.7 × 2.7 = 82.48cm3
Question 2.
Find the volume of sphere of radius 6.3 cm. [Page No. 238]
Solution:
r = 6.3 cm
V = $$\frac{4}{3} \pi r^{3}$$
= $$\frac{4}{3} \times \frac{22}{7}$$ × 6.3 × 6.3 × 6.3 = 1047.81cm3<|endoftext|>
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This product includes two attention-holding exercises with homophones, each with a teacher key. ( Please note: The Preview includes only 3 examples from each different practice.) Each exercise offers a different approach to identifying and distinguishing between homophones. The definition of a homophone is provided on each student sheet, as well as directions and an example.
The first exercise, "Hidden Homophones" asks students to find two homophones in each of ten sentences (ex. blew and blue). Students write the sentences and underline the "hidden homophones". The second separate exercise, "Horrible Homophones", asks students to find the incorrect homophone in each sentence and correct it (ex. "The horse had a beautiful brown tale (should be tail) and white spots." The students write these sentences in the corrected form. Please see preview to determine if these practices are appropriate for your grade level.<|endoftext|>
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# How do you find the roots of x^3-6x^2+13x-10=0?
Sep 15, 2016
$x = 2$
#### Explanation:
${x}^{3} - 6 {x}^{2} + 13 x - 10 = 0$
${x}^{3} - 3 {\left(x\right)}^{2} \left(2\right) + 3 {\left(2\right)}^{2} x + x - {2}^{3} - 2 = 0$
$\left({x}^{3} - 3 {\left(x\right)}^{2} \left(2\right) + 3 x {\left(2\right)}^{2} - {2}^{3}\right) + x - 2 = 0$
We can factorize using the polynomial identity that follows:
${\left(a - b\right)}^{3} = {a}^{3} - 3 {a}^{2} b + 3 a {b}^{2} + {b}^{3}$
where in our case $a = x$ and $b = 2$
So,
${\left(x - 2\right)}^{3} + \left(x - 2\right) = 0$ taking $x - 2$ as common factor
$\left(x - 2\right) \left({\left(x - 2\right)}^{2} + 1\right) = 0$
$\left(x - 2\right) \left({x}^{2} - 4 x + 4 + 1\right) = 0$
$\left(x - 2\right) \left({x}^{2} - 4 x + 5\right) = 0$
$x - 2 = 0$ then $x = 2$
Or
${x}^{2} - 4 x + 5 = 0$
$\delta = {\left(- 4\right)}^{2} - 4 \left(1\right) \left(5\right) = 16 - 20 = - 4 < 0$
$\delta < 0 \Rightarrow$ no root in R<|endoftext|>
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In Angle 2, we learned the principles governing the stroke order for Chinese characters. Another way of defining characters involves "principles of construction." In this scheme, there are six types of characters, with each type finding its meaning based on one of the following principles.
Picture characters are simply meant to look like the things they represent. As we mentioned before, though, many pictograms evolved over time so that the resemblance is less than obvious.
Symbol characters symbolize (what else!?) an idea or concept. Below, in the character meaning above, the vertical line and small stroke are above the horizontal line. In the character meaning below, they are underneath.
Also called Sound-Loan characters, these borrow the same written form and sound of another character, but have unrelated meanings.<|endoftext|>
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Astronomy offline app include next themes:
Topic 1. The Sun – The sun is just one of about 100 billion stars in our galaxy. The average radius of the Sun is 695508 kilometers.
Topic 2. The Earth – Earth is the third planet from the Sun and is the largest of the terrestrial planets. Formed approximately 4.54 billion years ago.
Topic 3. The Moon – At a distance of 384400 kilometers from the Earth, the Moon is our closest and only natural satellite.
Topic 4. Asteroids – An asteroid is a small body orbiting the Sun that is composed primarily of rock or metal.
Topic 5. Kuiper Belt – Is similar to the asteroid belt found between the orbits of Mars and Jupiter, but it is 20 times as wide and somewhere from 20 to 200 times more massive.
Topic 6. Comets – are usually made of frozen water and supercold methane, ammonia and carbon dioxide ices.
Topic 7. Meteors and meteoroids – A meteoroid is a small fracture of rock that enters our Solar System.
Topic 8. Stars – A star is a massive, bright, sphere of very hot gas called plasma which is held together by its own gravity.
Topic 9. Milky Way – The Milky Way is one of many galaxies that lie in the universe. This galaxy is a spiral or whirl shape and home to our planet and Sun.
Topic 10. Planets in solar system – Planets are among the many worlds and smaller objects that orbit the Sun.
Topic 11. Galaxies – A galaxy is a massive, gravitationally bound system that consists of stars, stellar objects, such as brown dwarfs and neutron stars, nebulae, an interstellar medium of gas and dust, black holes, and an unknown component of dark matter.
Topic 12. Dark Matter – Dark matter is something that an astronomer or scientist cannot observe through ordinary telescopes. It does not emit or absorb light, and is considered responsible for holding all the normal matter in the universe together.
Topic 13. The interstellar medium – Also ISM is the matter that exists in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, as well as dust and cosmic rays.
Topic 14. Orbits – In our solar system, the moon orbits around Earth. Earth and the other planets orbit around the Sun. The stars also orbit around the Sun.
Topic 15. Black holes – Black holes are among the strangest things in the universe. They are massive objects collections of mass with gravity so strong that nothing can escape, not even light.
Topic 16. Pulsars – is a highly magnetized, rotating neutron star or white dwarf, that emits a beam of electromagnetic radiation.
Topic 17. Novae and Supernovae – A nova, plural novae or novas is a cataclysmic nuclear explosion on a white dwarf. Supernova is nothing but a stellar explosion. To put in other words, it is actually an explosion of a star.
Topic 18. The Andromeda Galaxy – is the closest large galaxy to the Milky Way and is one of a few galaxies that can be seen unaided from the Earth. In approximately 4.5 billion years the Andromeda Galaxy and the Milky Way are expected to collide and the result will be a giant elliptical galaxy.
Topic 19. Kepler’s laws of planetary motion and interesting facts about Kepler – In astronomy, Kepler’s laws of planetary motion are three scientific laws describing the motion of planets around the Sun.
Topic 20. Mars Facts – Mars is the fourth planet from the Sun and is the second smallest planet in the solar system. Named after the Roman god of war, Mars is also often described as the “Red Planet” due to its reddish appearance.<|endoftext|>
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The term Big Data is used to describe the huge volumes of structured and unstructured data that is very difficult to process using the traditional methods. These large sets of data are analyzed computationally to discover certain patterns and trends. Big Data comprises of huge samples of data, enough to draw good conclusions and there is no minimum amount of it.
The concept of Big Data is new, and increasing amount of different types of data is being collected. With digitization, more and more information is moving online and it is readily available for analysts.
How to access Big Data?
Big Data is increasing with every passing minute and is available in a countless number of places on the web. Just one simple Google search brings out millions of results with large volumes of data. It can’t be righty judged how much data is available online. Here is how you can access and use the data:
- Data Extraction: Data extraction is the process of gaining data. There are many ways to do it but it is generally done by making an API call to an organization’s web service.
- Data Storage: Storage is a big problem when it comes to big data. A lot depends on the budget and expertise of the person responsible for setting up the storage. Technical skills and programming knowledge is important for this purpose. A good provider is someone who allows you a safe place to store your data.
- Data Cleaning: Data is available in different sizes and may be unorganized. Therefore, before you store it, you need to run data cleaning in order to convert it into an acceptable
- Data Mining: Data mining is the process of finding insights into the database. The purpose of it is to make predictions and decisions on the data that is currently available.
- Data Analysis: After the data is collected, the next step is to do analysis and find any interesting trends or patterns. A good analyst is someone who can find something important in the ordinary or something no one else has
- Data Visualisation: The visualization of data is deemed to be very important. It is a process of using all the previously done work and putting it in words that everyone can understand.
Important terms to know
An algorithm can be defined as a set of instructions or a mathematical formula that forms the base of a software used to analyze the data.
It is a collection of programs that allow the users to store, retrieve, and analyze the large sets of data.
- Data Scientist
A data scientist is someone who has the expertise in deriving insights and analysis from the data.
- Amazon Web Services (AWS)
AWS is the name given to the collection of cloud computing services that help the businesses to do large-scale computing operations without the requirement of any storage or processing power on-premise.
- Cloud Computing
The term cloud computing is given to the process of running software on remote servers and not locally.
- Internet of Things (IoT)
The IoT is a relatively new technology in which the objects like sensors collect, analyze, and transmit their own data without any manual input. In essence, the devices’ functioning is controlled over the internet.
- Web Scraping
Web scraping is the process of automatic collection and structuring of data from websites.
- Predictive Analysis
It is about predicting the future trends or events using the available data.
- Structured and Unstructured Data
Data that is properly organized in the form of a table so that it relates to other table or chart is called structured data while all other unorganized data is called unstructured data.
Data has always been used by businesses to gain insights and strengthen their business. Big Data takes this to next level as it takes more factors into account and does this analysis on a larger scale. If used in the right way, it is a powerful tool for companies to outperform their competitors.<|endoftext|>
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# 8 times table
## The 5-step plan
• 8 x 1 = 8
• 8 x 2 = 16
• 8 x 3 = 24
• 8 x 4 = 32
• 8 x 5 = 40
• 8 x 6 = 48
• 8 x 7 = 56
• 8 x 8 = 64
• 8 x 9 = 72
• 8 x 10 = 80
• 8 x 11 = 88
• 8 x 12 = 96
### Step 1a: View, read aloud and repeat
Step 1a is to get familiar with the table, so view, read aloud and repeat. If you think you remember them it's time to test your knowledge at step 1b.
### Step 1b: In sequence
Fill in your answers. Once you have entered all the answers, click on 'Check' to see whether you have got them all right! If you got all the answers right, practice the 8 times table shuffled in random order.
### Step 3: Shuffled
Practice the 8 times table shuffled. Fill in all answers and press 'check' to see how many you got right.
### Step 4: Multiple choice
Try to answer all the 15 questions right!
### Step 5: Tables Diploma
Answer all the 24 questions right to get the diploma!
### Games
These games give the possibility to repeat the questions and improve the knowledge of the 8 multiplication table. Enjoy the 8 times table games!
## 8 times table memory
Try to find as fast a possible the matching questions and answers!
2
2 x 1
2
2 x 1
2
2 x 1
2
2
2
2
2
2
2
2
2
2
## 8 times table chart
This is where you can practice your 8 times table. You can practice the 8 times table in sequence and once you have got the hang of that you can make it a bit harder by practicing the sums up in random order. If you want to practice the 8 times table against the clock, you can of course take the speed test. If you want to practice at your leisure, we suggest you print out the 8 times table worksheet and practice with that. The eight times table is the multiplication table of 8 where we get the product of multiplying 8 with whole numbers. It is helpful to know the multiples of 8. Below you will see the 8 multiplication chart.
What is the 8 times table?
The 8 times tables are:
• 8 x 1 = 8
• 8 x 2 = 16
• 8 x 3 = 24
• 8 x 4 = 32
• 8 x 5 = 40
• 8 x 6 = 48
• 8 x 7 = 56
• 8 x 8 = 64
• 8 x 9 = 72
• 8 x 10 = 80
• 8 x 11 = 88
• 8 x 12 = 96
How do you memorize the 8 times table?
The most common way to help a child remember the table of 8 is to learn it by rote. Look at the 8 multiplication table chart, have them say it, write it down, exercise with the 5-step plan, and play multiplication games until they have the 8 multiplication table memorized.
## Print 8 times table worksheet
Click on the worksheet to view it in a larger format. For the 8 times table worksheet you can choose between three different sorts of exercise. In the first exercise you have to draw a line from the sum to the correct answer. In the second exercise you have to enter the missing number to complete the sum correctly. In the third exercise you have to answer the sums which have been shuffled.
## Description of the 8 times table
This is where you can practice the 8 times table by entering all the answers and then checking how many you got right. There are various ways the tables can be practiced. You are now on the 8 times table page where multiplication calculations can be practiced in sequence. Once you have mastered the table in sequence you can practice the sums in random order. You can learn the tables on a PC, tablet, iPad or mobile phone. You learn this table in year 4 maths.<|endoftext|>
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In BriefAfter limiting the nutrient resources they need to thrive, researchers pitted drug-resistant pathogens against drug-sensitive ones and found that a little healthy competition may be just what's needed in the race to develop new treatments to combat increasing drug resistance.
Drug resistance is making it increasingly difficult to produce medications that effectively combat diseases like HIV, tuberculosis, and malaria — but researchers have developed a method of turning pathogens against one another that could help those drugs function as intended.
Pathogens like parasites, viruses, and bacteria sometimes develop a genetic mutation that makes them less vulnerable to certain drugs, such as antibiotics. Once one pathogen becomes able to survive treatment, it can replicate itself into a population of billions that all carry the mutation giving them resistance to the drug.
However, pathogens in the body need to compete over resources and nutrients to thrive — and drug-resistant pathogens often have greater needs. The new study took advantage of this competition by manipulating the nutrients present in drinking water given to mice.
The mice were infected with malaria parasites that were drug sensitive. When the mice were given treatment, it failed 40 percent of the time, which researchers determined was caused by the presence of drug-resistant strains. When a nutrient used by the parasites was limited, the infection didn’t return in any of the mice.
Researchers then confirmed this occurred because of competition among the drug-resistant and drug-sensitive pathogens, rather than some other consequence of limiting nutrients. When the mice had only been infected with drug-resistant pathogens, then limited the nutrients available to those pathogens, the resistant bacteria survived. But when they introduced both drug-resistant and drug-sensitive pathogens, then bred competition between them by limiting nutrient resources while the mice were receiving drug treatment, the resistant parasites didn’t survive.
A Little Healthy Competition
Researchers hope that harnessing pathogen competition could help create drugs that remain potent for a longer period of time. The technique could potentially be applied to existing drugs that already face resistant strains, though it would require many steps. First, researchers would need to identify a resource or nutrient that the strains need more than their non-resistant equivalents, confirm that limiting it would have the desired effect, then figure out how — and when — to remove the critical resource.
Such steps could be taken during the process of developing a completely new drug— though that could be a long way off. For now, we can only speculate as to how expensive or time-consuming such a process might be. Senior author of the study, Andrew Read, theorized that even if it is an expensive process, the long-term savings may offset the upfront costs.
“The initial screening would likely be more expensive, since more complex assays are required,” Read told Futurism via email. He explained that at worst, following trials would be about as expensive as those for a brand new drug. But there’s a fair chance that existing compounds – which might even already be approved for use – could be utilized, which would make the process considerably cheaper.
“The main gain would be long-term,” he added. “Every year you do not need to invent/discover another new antimicrobial drug is a huge saving[s].”
At present, if a doctor observes a particular infection’s ability to resist a drug, they won’t use that treatment. That’s not too much of a problem if there are other alternatives, but it can have major consequences if they aren’t.
“Such patients recover if their immunity gets on top of the bugs, or they die,” said Read. “But I would say that what we propose would be better used to protect existing drugs, rather than just as a last ditch method to save critically ill patients.”<|endoftext|>
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In most of the engineering career, we encounter some signals in the form of noise, disturbance, interference, etc. As an engineer in fact as control or communication engineer, we need to tackle these signals so they won’t cause a problem to our original system. But how to identify such signals the answer lies in the theory of probability, we classify such signals as random processes and they normally follow some distribution like Gaussian/Normal, Exponential, Uniform etc. Before jumping into this theory I would like you to understand some basic concept of Probability.
Some definitions of probability theory are given below:
Experiment: An experiment is a procedure or hypothetical procedure that produces some result. It is often denoted as E (don’t confuse it with Expectation) as an example we can say that the rolling of dice 3 times is an experiment.
Event: Set of outcomes from an experiment is called an event as an example the experiment defined above is the rolling of dice 3 times then we can say that the event A is a (3 times a six appears), B is (2 sixes and one 5 appears) etc.
Sample Space: Set of all possible outcomes of an experiment is called a sample space. Letter S is used to denote sample space.
Probability: The possible outcome that can occur from the given sample space or what is the chance of certain event occurring as an example the probability of occurring 1 in rolling a dice single time is 1/6 as there are 6 possible outcomes {1,2,3,4,5,6} and chances of dice being 1 is 1/6 or 16.66% .
Rules in Probability or Laws of Probability
Rule-1: There is no negative probability.
Rule-2: Probability always lies between 0 to 1.
Rule-3: Probability of sample space S is 1. As sample space consist of all possible events that can occur so the probability of all possible events should be 1, as an example we can say that the probability that if we toss a coin and we get both heads and tails then probability of both heads and tails out of their sample space which is (heads & tails) should be 1.
Rule-4: If we have two events A and B and A ∩ B = ∅ then Pr(A∪B) = P(A)+P(B). This means that the events A and B are mutually exclusive. On the other hand, if they are not mutually exclusive then Pr(A∪B) = P(A) + P(B) – P(A∩B).
Now you may be wondering what is mutually exclusive means. Mutually exclusive simply means that these two events have nothing in common it doesn’t mean that they span the whole sample space as well as an example I can say that the event A ={1,2} and B={3,4} are the outcomes of rolling a dice, it is clear that these two events are mutually exclusive events.
Rule-5: From Rule-4 if we have infinite number of mutually exclusive sets, $A_i, \; i=1,2,3 \ldots \;, \; A_i \cap A_j = \phi \; \text{for all} \;\; i \neq j \;$ ,
Then we can say that: $Pr\left( \bigcup_{i=1}^{\infty}{A_i} \right) = \sum_{i=1}^{\infty} Pr(A_i)$
Rule-6: if we have $Pr(A) = 0.3$ then its complement $Pr(\overset{-}{A}) = 1-Pr(A) = 1-0.3 = 0.7$
Most of the students complain that they don’t understand a single word what just happened and it is no doubt that probability is easy to understand. Some minds don’t get it all, as one scientist said that:
God may not play dice with the Universe, but we’re not Gods so we have to keep rolling the dice until we get it right.
Everything goes to your mind through some visual or hearing senses but Math is the only subject that goes to your brain by practising through hands. So we would like you to solve the following problems, don’t worry we will provide solutions as well. These will help you understanding the probability theory more clearly
Problem-1:
An experiment consists of rolling n (six-sided) dice and recording the sum of the n rolls. How many outcomes are there to this experiment?
Solution: Notice for n=1 die, there are 6 possible outcomes of the sum (1-6).
Now for n=2 dice, there are 11 possible outcomes of the sum (2-12).
Similarly for n=3 dice, you can obtain a sum from 3-18, which is 16 possible outcomes.
This leads to the general case for n dice, there are $6n-n+1$ cases. This is equivalent to $5n+1$.
Problem-2: An experiment consists of selecting a number x from the interval [0,1) and a number y
from the interval [0,2) to form a point (x,y).
1. Describe the sample space of all points.
2. What fraction of the points in the space satisfy x > y ?
3. What fraction of points in the sample space satisfy x=y ?
Solution:
1. Consider the x-y plane. The sample space will be all possible ordered pairs
where x is in [0,1) and y is in [0,2). This will look like a rectangle with the
Problem-2 Solution
bottom left corner at the origin with a width of 1 and height of 2, as shown in
the figure . Note the lines x=1 and y=2 form the top and right side lines of
the rectangle and are not included in the sample space, so they are represented
with dashed lines.
2. On the x-y plane with the rectangle drawn from part (1), draw the line x=y.
The area below this line but still inside the rectangle will satisfy x > y. From
the graph, you will see this area is 1/4 of the total area. Thus, the fraction
of points in the space satisfying x > y is 1/4.
3. Zero, the line x=y has zero area and therefore the fraction of the area of the
line relative to the area of the rectangle is zero.
Now I would like to give some proofs related to the Rules of Probability
Problem-3: Prove Rule-5 which states that: For infinite number of mutually exclusive sets, $A_i, \; i=1,2,3 \ldots \;, \; A_i \cap A_j = \phi \; \text{for all} \;\; i \neq j \;$ ,
Following holds: $Pr\left( \bigcup_{i=1}^{\infty}{A_i} \right) = \sum_{i=1}^{\infty} Pr(A_i)$
Solution: I will prove this by using mathematical induction. For the case M = 2 we have:
$Pr\left(\bigcup_{i=1}^{2}A_{i}\right)=\sum_{i=1}^{2}Pr\left(A_{i}\right)$
This we know to be true from the axioms of the probability. Let us assume that the proposition is true for M=k.
$Pr\left(\bigcup_{i=1}^{k}A_{i}\right)=\sum_{i=1}^{k}Pr\left(A_{i}\right)$
We need to prove that this is true for M=k+1. Define an auxiliary event B :
$B=\left(\bigcup_{i=1}^{k}A_{i}\right).$
Then the event
$\left(\bigcup_{i=1}^{k+1}A_{i}\right)=\left(B\cup A_{k+1}\right)$
$Pr\left(\bigcup_{i=1}^{k+1}A_{i}\right)=Pr\left(B\cup A_{k+1}\right).$
Since the proposition is true for M=2 we can rewrite the above equation as
$Pr\left(\bigcup_{i=1}^{k+1}A_{i}\right)=Pr\left(B\right)+Pr\left(A_{k+1}\right).$
Since the proposition is true for M=k we can rewrite this as:
$\begin{eqnarray*} & = & \sum_{i=1}^{k}Pr\left(A_{i}\right)+Pr\left(A_{k+1}\right)\\ & = & \sum_{i=1}^{k+1}Pr\left(A_{i}\right). \end{eqnarray*}$
Hence the proposition is true for M=k+1. And by induction the proposition is true for all $M$
These problems are adapted from the textbook: Probability and Random Process by Scott Miller 2ed<|endoftext|>
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Decimals | Mathematics Form 1
# KCSE ONLINE
## Esoma Online Revision Resources
### Secondary
Butterfly
Chameleon
#### Decimals - Mathematics Form 1
LESSON OBJECTIVES
By the end of this lesson you should be able to:
convert recurring decimals into fractions accurately.
convert a fractions into a decimals.
Exercise2
Express the following as fractions in their simplest forms.
1. a) 3.3636
b) 0.0909
c) 2.7272
d) 0.46846..
e) 1.313313.
f) 0.522522..
CASE II
Convert 0.45.. into a fraction in its simplest form.
Let r stand for the recurring decimal given i.e.
r=0.45..
Let the above equation be equation I
r=0.45.. I
Multiply both sides of equation by 100
100r=45.4545..
Let this be equation II
100r=45.4545;. II
Now subtract equation I from equation II
110r=45.4545.. II
-r= 0.4545.. I
99r=45
99r=45
99 99
r=5/11
but r=0.4545.
Therefore 0.4545. =5/11
NB: In both case I and case II the decimal point is
just before the recurring number or numbers.
In the case I only one number is recurring hence
we multiply by 10.
In case II two numbers re recurring and hence
& we multiply by 100.
If three numbers just after the decimal pointare
recurring we need to multiply by 1000.
In general if n numbers just after the decimal point a
re recurring we multiply by 10n.
Example: convert 0.123123. to a fraction in its
simplest form.
Let r stand for the recurring decimal given that
r=0.123123.
Let the above equation be equation I
r=0.123123. I
Multiply both sides of equation I by 1000 since three numbers
are recurring after the decimal point.
1000r=123.123123.
Let this equation be equation II.
1000r=123.123123. II
Now subtract equation I from equation II
1000r=123.123123..
- r= 0.123123..
999r=123
999r/999=123/999
r=123/999
But r=0.1231223.
Therefore 0.123123. = 123/999
CASE III
Convert 2.8333. to a fraction in the simplest form.
When there are non recurring digit(s) after the decimal point,
the decimal point is moved between the last non recurring
digit and the first recurring digit(s).
Let r stand for the recurring decimal
r=2.833333333 I
Let this equation be equation I
Multiply by 10 to move the decimal point just before 3
10r=28.333333..
Let this be equation II
10r=28.333333 II
Multiply equation II by ten since one digit is recurring.
100r=283.333.
Let this be equation III
100r=283.333. III
Subtract equation II from III
100r=283.333.
10r=28.333333
90r=255.0000.
90r=255
r=255/90
r=2 75/90
r=2 5/6
But r=2.83
Therefore 2.83=2 5/6
b) Convert t3.45676767 to a fraction in it simplest form.
Let r stand for 3.4567.
r=3.456767
let this be equation I
r=3.456767I
Multiply equation I by 100 to move the decimal point to be
between the last non recurring digit and the first recurring digit.
100r=345.676767
Let this be equation II
100r=345.676767 II
Multiply equation II by 100 since digits are recurring.
10000r=34567.676767
Let this be equation III
10000r=34567.676767 III
Subtract equation II from equation III
10000r=34567.676767 III
-100r=345.676767 II
9900r=34222
r=34222/9900
r=3 4522/9900
r=3 2261/4950
But r=3.45676767
Therefore 3.45676767..=3 2261/4950
CASE III
Convert 2.8333.. to a fraction in the simplest form.
When there are non recurring digit(s) after the decimal point,
the decimal point is moved between the last non recurring
digit and the first recurring digit(s).
Let r stand for the recurring decimal
r=2.833333333...... I
Let this equation be equation I
Multiply by 10 to move the decimal point just before 3
10r=28.333333...
Let this be equation II
10r=28.333333....... II
Multiply equation II by ten since one digit is recurring.
100r=283.333.......
Let this be equation III
100r=283.333......... III
Subtract equation II from III
100r=283.333........
10r=28.333333..........
90r=255.0000...........
90r=255
r=255/90
r=2 75/90
r=2 5/6
But r=2.83
Therefore 2.83=2 5/6
b) Convert t3.45676767....... to a fraction in it simplest form.
Let r stand for 3.4567.......
r=3.456767.......
let this be equation I
r=3.456767I
Multiply equation I by 100 to move the decimal point to be
between the last non recurring digit and the first recurring digit.
100r=345.676767.......
Let this be equation II
100r=345.676767......... II
Multiply equation II by 100 since digits are recurring.
10000r=34567.676767.......
Let this be equation III
10000r=34567.676767......... III
Subtract equation II from equation III
10000r=34567.676767...... III
-100r=345.676767........... II
9900r=34222
r=34222/9900
r=3 4522/9900
r=3 2261/4950
But r=3.45676767...........
Therefore 3.45676767.........=3 2261/4950
Exercise 3
Express the following s fractions in their simplest form
1. a) 0.222222222.
b) 2.8888888888...
c) 0.2727272727.
d) 2.4545454545.
e) 0.315315315
f) 2.815815815
g) 5.26666666.
h) 1.633333333
i) 2.163636363.
j) 2.134343434.
EXERCISE 1
Using the method given in the above examples
, convert the following into fractions to their simplest form.
(1) 0.8
(2) 3.4
(3) 0.6
DECIMALS
There many different types of decimals in mathematics.
Terminating, Non Terminating and Recurring decimals
We shall deal with reccuring decimals in this lesson.
#### Order this CD Today to Experience the Full Multimedia State of the Art Technology!
For Best results INSTALL Adobe Flash Player Version 16 to play the interactive content in your computer. Test the Sample e-Content link below to find out if you have Adobe Flash in your computer.
Sample Coursework e-Content CD
##### Other Goodies for KCSE ONLINE Members!
Coursework e-Content CD covers all the topics for a particular class per year and costs 1200/- ( Per Subject per Class ).
Purchase Online and have the CD sent to your nearest Parcel Service. Pay the amount to Patrick 0721806317 by M-PESA then provide your address for delivery of the Parcel. Alternatively, you can use BUY GOODS TILL NUMBER 827208 Ask for clarification if you get stuck.
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Almost 85% of the universe is covered with dark matter. Dark matter is 5 times more abundant than ordinary matter and is responsible for bending light. This affects the gravity of nearby bodies and adjusting a galaxy’s shape. Understanding the dark matter is a little bit difficult but also essential to understand the size, shape, and future of the universe. But, it requires scientists to develop technologies never before used.
To do so, scientists at the University of Zurich created the world’s most sensitive dark matter detector. Scientists named this system as XENON1T. This is for the first time, scientists used a giant vat (3.2 tons) of liquid xenon to create this dark matter detection system. Xenon is an element that’s usually a gas at room temperature, buried deep in mine shafts or in mountains.
Xenon — the colorless, odorless, and dense atomic number 54— seems like the perfect material for the detector. It produces the lowest possible radioactive content, making those pings of particle interference a little bit easier to identify.
Laura Baudis, a professor at the Physik Institute of the University of Zurich said, “I think the most exciting thing is the fact that the detector works as we expect.”
According to scientists, sensitive dark matter particles only interacts very weakly with the nucleus of regular matter atoms. Those particles hit the liquid xenon nuclei and produce light particles or knocking off an electron. The time between the initial photon signal from the strike and another photon signal from a released electron migrating out of the experiment determines where in the chamber the dark matter would have struck. A particle interaction in liquid xenon makes light flashes.
Luca Grandi, assistant professor of physics at the University of Chicago and member of the XENON Collaboration said, “The care that we put into every single detail of the new detector is finally paying back. We have excellent discovery potential in the years to come because of the huge dimension of XENON1T and its incredibly low background. These early results already are allowing us to explore regions never explored before.”
The XENON1T central detector is not visible. It sits within a cryostat in the middle of the water tank, that keeps the xenon at a temperature of -95°C without freezing the surrounding water. The mountain above it, shields this dark matter detector, preventing perturbations from cosmic rays.
Now, scientists are studying and recording to infer the position and the energy of the interacting particle, and whether or not it might be dark matter.
Christopher Tunnell, fellow at the Kavli Institute for Cosmological Physics at the University of Chicago said, “The best news is that the experiment continues to accumulate excellent data, which will allow us to test quite soon the WIMP (weakly interacting massive particles) hypothesis in a region of mass and cross-section with normal atoms as never before.”
“A new phase in the race to detect dark matter with ultra-low background massive detectors on Earth has just begun with XENON1T. We are proud to be at the forefront of the race with this amazing detector, the first of its kind.”
Tunnell said, “It has been a large, concentrated effort and seeing XENON1T back on the front line makes me forget the never-ending days spent next to my colleagues to look at plots and distributions. There is no better thrill than leading the way in our knowledge of dark matter for the coming years.”<|endoftext|>
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# What’s the Domain, Why You Need It and How You Get It
As you teach domain and range, do you get the question, “Why are we doing this?”
No doubt the question, “When am I ever going to use this?” comes up too, right?
I’ve asked myself that question my whole teaching and tutoring life. And now’s the time for an answer.
### A function is really a question.
As I wrote in a previous post, a function is a question with only one answer to a valid question.
When I write:
y = 3x + 2 where x = 4
I mean: “What is three times a number (that number is four), plus two?”
The domain is all the possible questions:
• What is three times a number (that number is five,) plus two?
• What is three times a number (that number is six,) plus two?
• What is three times a number (that number is seven,) plus two?
• What is three times a number (that number is eight,) plus two?
• <how long will I have to do this – Egad!>
Not only do the questions go on forever, but they also have all the fractions and decimals in between. And all the negative versions of those numbers (and zero).
So, what’s the point, you might ask. Looks like the questions go on forever and you can just pick any number.
### The domain might not include all the numbers.
The two sticky points for the definition of “function” are bolded:
A function is a question with only one answer to a valid question.
The “valid question” part is where the domain comes in.
The numbers that make “valid” – meaning we actually can get some answer – are the numbers that aren’t negative.
Many functions have “all real numbers” as a domain. There are no limits on the things you can put in, other than numbers that aren’t imaginary or alligators.
For the most part, there are only two places where you have to be careful of limited domains. Those are
• Numbers that cause a zero to turn up in the denominator
• Numbers that cause negatives to turn up in square roots.
Here are two videos tackling each:
What do you think? Is this easier to teach when you consider “analyzing” the function rather than “solving” it? Share your thoughts and tips in the comments!
This post may contain affiliate links. When you use them, you support us so we can continue to provide free content!<|endoftext|>
| 4.5 |
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### The Sum of an Infinite Series
#### Definition
A finite series is given by all the terms of a finite sequence, added together.
An infinite series is given by all the terms of an infinite sequence, added together.
To find the sum of an infinite series, start by summing the first few terms, i.e. $\sum^{2}_{k=1}$, $\sum^{3}_{k=1}$, $\sum^{4}_{k=1}$, $\dotso$. These sums of the first terms of the series are called partial sums. The $n^{\text{th} }$ partial sum of a series is the sum of the first $n$ terms.
If the partial sums tend towards a single number, this is called the limit and also the sum of the series.
#### Worked Examples
###### Example 1
What is the sum of the infinite series $\displaystyle{\sum^{\infty}_{k=1}\frac{1}{2^k$? }}
###### Solution
Work out the first few partial sums:
$\sum^{1}_{k=1}\frac{1}{2^k} = \frac{1}{2}$
$\sum^{2}_{k=1}\frac{1}{2^k} = \frac{1}{2}+\frac{1}{4}=\frac{3}{4}$
$\sum^{3}_{k=1}\frac{1}{2^k} = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}=\frac{7}{8}$
$\sum^{4}_{k=1}\frac{1}{2^k} = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}=\frac{15}{16}$
$\sum^{5}_{k=1}\frac{1}{2^k} = \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}=\frac{31}{32}$
Write down the partial sums as a sequence:
$\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \frac{31}{32}, \dotso$
We can see that the partial sums form a sequence that has a limit of $1$. So this series has sum $1$. Written as:
$\sum^{\infty}_{k=1} \frac{1}{2^k} = 1$
###### Example 2
What is the sum of the infinite series $\displaystyle{\sum^{\infty}_{k=1}1}$?
###### Solution
Work out the first few partial sums:
$\sum^{1}_{k=1}1 = 1$
$\sum^{2}_{k=1}1 = 1+1 =2$
$\sum^{3}_{k=1}1 = 1+1+1 =3$
$\sum^{4}_{k=1}1 = 1+1+1 +1 =4$
Write down the partial sums as a sequence:
$1, 2, 3, 4, \dotso$
This sequence of partial sums does not tend to a real limit, it tends to infinity.
Therefore this series does not have a sum.
#### More Support
You can get one-to-one support from Maths-Aid.<|endoftext|>
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The Declaration of Independence is a brilliant piece of writing built upon the ideas of the Enlightenment. It is one of the most well-known documents in the United States. It was created by Thomas Jefferson, John Adams, Roger Sherman, Robert R. Livingston, and Benjamin Franklin. The whole purpose of The Declaration of Independence was to announce and explain the separation from Great Britain to form the United States. The Declaration of Independence was the document that the Second Continental Congress used for the meeting in the Pennsylvania State House on July 4, 1776, the day the United States gained its Independence. This essay will talk about the effectiveness of Jefferson’s organization of The Declaration of Independence and how The Declaration of Independence was influenced by the Enlightenment. The Declaration of Independence was based a lot on the ideas of the Enlightenment and its thinkers like John Locke. One of the two big things it was influenced by was the law of people having their Natural Rights. Natural rights are rights that every human being has just for being a human. These rights are also called unalienable, meaning you cannot take these rights away from any human being. The three parts of Natural Rights are Life, Liberty, and Property. However, Thomas Jefferson made some minor changes changing them to Life, Liberty, and the Pursuit of Happiness. The other Big influence to The Declaration of Independence was the Social Contract. The Social contract was created by the Parisian philosopher, Jean-Jacques Rousseau. The Social Contract stated that the Government could not make rules by divine right or intrinsic authority. All of the authors of The Declaration of Independence wanted to make a free country, which Great Britain has never been. They wanted to make a country in which the people living there can much more opportunities than in other countries. In Great Britain, it is very hard to get into the higher social class where in America it’s easier. The Enlightenment really influenced the writing of The Declaration of Independence not only to convince people from Great Britain to come to the United States but also to convince the loyalists that declaring Independence would be good for them. On June 11, 1776, the Congress created a “Committee of Five” which was a group of five people. The Committee consisted of these five people, John Adams of Massachusetts, Thomas Jefferson of Virginia, Benjamin Franklin of Pennsylvania, Roger Sherman of Connecticut, and Robert R. Livingston of New York. Before Thomas Jefferson was chosen for being the main author of The Declaration of Independence, Richard Henry Lee was actually supposed to be. The congress really wanted Richard to write because of his work, the Lee Revolution. But things changed because he was already assigned to the Committee of Confederation for to write the Articles of Confederation. So him helping with The Declaration of Independence was way to much work for Henry so he declined the offer to write it. So the Second Continental Congress selected Thomas Jefferson as the author of The Declaration of Independence. Thomas was just a young delegate from Virginia at the time. But however, he would soon become one of the most well-known people in American History. At first, Thomas Jefferson did not want to write The Declaration of Independence. Instead, he wanted John Adams to write it. John Adams had three reasons that he should not write The Declaration of Independence that he said to Thomas Jefferson. The first reason was that Thomas Jefferson was from Virginia, and many Virginians are usually head of the business kind of people. His second reason was that he was pretty unpopular, but Thomas Jefferson was pretty well-known. The third reason was that he said his writing was not really good and Thomas Jefferson wrote ten times better than he did. This convinced Thomas Jefferson that he should write The Declaration of Independence. Over the next seventeen days, Thomas Jefferson wrote The Declaration of Independence while under the advisement of the Committee of Confederation. This act put Thomas Jefferson’s name down in history forever. Thomas Jefferson’s organization of The Declaration of Independence was very successful and the Continental Congress declared the colonies independent on July 4, 1776.And in conclusion, The Declaration of Independence was signed on July 4, 1776, and the United States of America gained its independence. The day that marked a huge step in History. Still today the 4th of July is celebrated all across the country. Thomas Jefferson establishing The Declaration of Independence was very successful and he and the four others of the Committee of five will always be remembered as a huge part of United States history.<|endoftext|>
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The Parthenon: A universal symbol of Classical Greece
The Parthenon describes all the ideas of Greek thought during the Classical era through artistic means. The idealism of the Greek way of life, as well as the understanding of a mathematically explained harmony in the natural world, were notions that in every Athenian eye distinguished them from the barbarians. These ideals are represented in the perfect proportions of the building, its intricate architectural features and the anthropomorphic statues that adorned it.
Some of these details were found in other Greek temples, while some were unique to the Parthenon. The temple owes its sophisticated appeal to the subtle details that were built on the architectural elements to meet practical needs or enhance the visual appeal of the building.
In the Parthenon there are no absolute straight lines because it gives a thin organic character to an obvious geometric structure. The columns of the peristyle cone in a light arc as they reach the top of the building give the impression that they are swollen with enthusiasm as if they were burdened by the weight of the roof. A thin feature that classifies anthropomorphic transfers to other wisely detectable objects.
The peristyle columns have a height of over ten meters and are oriented slightly towards the center of the building at the top of about 7 cm. While the platform on which the arcs support in a soft arc that brings the corners about 12 cm closer to the ground
The architecture of The Parthenon
The Parthenon architects seem to be exceptional scholars of visual illusion, a feature undoubtedly accentuated by years of architectural improvement and observation of the natural world. They have designed the columns appearing at the corners of the temple to be 1 / 40o about 6 cm larger in diameter than all the other columns, while making the space around the smaller ones of the remaining columns about 25 cm.
The reason for this small adjustment of the corner columns is due to the fact that they are placed in the bright sky, which would make them appear a little thinner and a little farther from the columns placed on the darkest background of the building wall. Increasing the size and reducing the space thus compensates for the illusion that usually causes the bright background.
These distinct features place the Parthenon apart from all other Greek temples, because the overall result is a deviation from the static Doric structures of the past, towards a more dynamic form of architectural expression.
Athenian citizens were proud of their cultural identity and aware of the historical size of their ideas. They believe they were civilized among the barbarians and that their cultural and political achievements were inevitably to change the history of all civilized people. The catalyst for all their achievements was the development of a system of government that the world had never seen: democracy.
Democracy, undoubtedly was the epitome of the Athenian way of thinking, was at the center, while the Parthenon was built. This was a direct democracy where every citizen had a voice on common issues through the Convention that met on the Pnyx Hill next to the Acropolis forty times a year to decide on all internal or foreign policy issues.
The fact that citizens were depicted as individuals for the first time in the Parthenon frieze was due to the fact that for the first time in history every citizen of a city was recognized as an important entity and a major driving force in the city and observable universe.<|endoftext|>
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Early Modern Cornwall 1485 – 1688
A time of new religious and political ideas. During this period English rulers sought to establish firmer control over the furthermost parts of Britain – Scotland and Ireland especially but this trend also affected Cornwall. Here the loyalty of many gentry, particularly in East Cornwall was secured as they were increasingly Anglicised, their loyalty to England secured by representation in Parliament in great numbers and the Cornish language was relegated in significance. Cornish was retained almost exclusively in the West of the county. So there was a divide between better-off East involving itself in English affairs, and the poorer and more lawless West.
This was a time of new religious and political ideas, economic expansion, and the beginnings of European expansion and conflict over trade and possessions in the New World. Cornwall was not immune to all of this and Cornish people ‘flexed their muscles’ on many occasions.
In 1497, there was considerable unrest when Henry VII tried to raise taxation to pay for his war against the Scottish.A small poorly armed Cornish force organized by Michael Joseph An Gof and Thomas Flamank marched on London along with Lord Audley only to be met and defeated by a 10,000 strong professional army at the Battle of Deptford Bridge, Blackheath (just south of Greenwich). The leaders were all caught and executed. There is a commemorative plaque on Greenwich Park wall as pictured. Later in that year Perkin Warbeck landed in Cornwall as a Pretender to the throne and received some support from the Cornish, who still felt aggrieved from the previous incidents, although this was a short lived attempt.
One of the later results of this (as a concession from Henry) was the passing of the 1508 ‘Charter of Pardon’ This established the rights of the Stannary Parliament to veto English legislation.
However, over the period as a whole this institution became less and less effective as a defender of local interests.
During Henry VIIIs reign, international security became more important and the Cornish Coast was fortified with the castles of Pendennis, St Mawes and Fowey. Invasion from France, Spain (especially) and pirates from further afield was a recurring threat as the century progressed.
Religious houses were dissolved by 1540 here as elsewhere in Britain but this provoked little reaction, even though the Cornish were religiously conservative.
However in 1549 the introduction of English Protestant services, in the Book of Common Prayer was to have a major impact in the form of an armed rebellion through Devon and Cornwall in which some 5,000 were killed. Moreover the change to English language services played a large part in the later demise of the Cornish language. Estimates suggest that by 1700 only a small core of some 5,000 Cornish speakers existed within the county.
One of many recently erected memorials in Cornwall and Devon to the 1549 Prayer Book rebellion It was during the long reign of Elizabeth that much Anglicisation took place. Many towns were granted Charters and the right to return MPs. In addition to the 2 members returned for the county, 21 other constituencies each returned 2 MPs.
During the late 16th century there were many conflicts with Spain with some Spanish vessels landing on the Cornish coast. Cornish privateers, such as the Killigrews worked out of the Helford and Spanish ships in Saltash and Fowey were seized while peace existed between England and Spain.
When war was declared in 1585, Cornwall was even more involved.The main Spanish Armada battle took place just off neighbouring Plymouth, but there are incidents of the Spanish attacking Fowey and in particular landing at Mousehole in 1595. Incidently, one of the surnames found in that area is Jose pronounced as Jose in English but Hose in its native Spanish) .
The tin industry also expanded considerably, especially in the central and western areas. A pattern was set at this time of the predominance of the tin industry in terms of Cornwall’s overall economy and this was to be the case for the next two centuries and more.
Things settled down after the Civil War, with winners and losers across the county. One of the main beneficiaries were the Killigrews of Falmouth – the town being granted a charter in return for the Killigrew’s notable support of the King towards the end of the war. This is the point from which Falmouth becomes the main port on the river, taking over from the ports of Truro and Penryn. Falmouth parish church may look similar to many Cornish churches but inside can instantly be seen as a Restoration building dedicated as it is to Charles the Martyr. One of the main ‘losers’ were the Basset family who had to sell St Michael’s Mount to the St Aubyns to fund the Royalist cause
Although in 1688 Bishop Trelawny (of Cornish origin but Bishop of Bristol) was imprisoned in the Tower of London for opposing the King’s attempts towards religious tolerance, at the time this event made little impact on the county. Later he was made famous by Rev. Hawker’s 19th century song ‘Trelawney’ – the Cornish Anthem. 20,000 men did not march on London to support Trelawney – Hawker seems to have confused this with the AnGof rebellion of 1497 – at the beginning of the Early Modern period!
History of Cornwall
Bronze Age cultures began to appear in Cornwall around 2200 BC with new ideas spreading from the Continent to the existing population, but the changes were gradual not sudden and stone tools continued to be used...
New, stronger iron ploughs and axes meant that farming improved. Cornwall contains many archaeological remains from this time: small villages with field systems around them, hillforts and cliff castles that...
After the Roman withdrawal from Great Britain, Saxons and other peoples were able to conquer and settle most of the east of the island. But Cornwall remained under the rule of local Romano-British and Celtic elites...
With the arrival of the Normans to the British mainland in 1066, the River Tamar became the agreed border. There was also a general acceptance that Cornwall had a separate identity to the rest of England...
During this period English rulers sought to establish firmer control over the furthermost parts of Britain - Scotland and Ireland especially but this trend also affected Cornwall. Here the loyalty of many gentry, particularly in East Cornwall...
The Civil Wars of the mid seventeenth century were a result of political, constitutional, religious and social changes and disagreements, which culminated in a struggle for control of the country between King and Parliament...<|endoftext|>
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#### Find the area of the triangle whose vertices are (–8, 4), (–6, 6) and (–3, 9).
Solution.
$Area \, of \, triangle = \frac{1}{2}\left [ x_{1} \left ( y_{2} -y_{3}\right )+x_{2} \left ( y_{3} -y_{1}\right )+x_{3} \left ( y_{1} -y_{2}\right )\right ]$
Given vertices are (–8, 4), (–6, 6), (–3, 9)
x1 =-8 x2 =-6 x3 = -3
y1=4 y2= 6 y3= 9
We know that area of triangle is
$= \frac{1}{2}\left [ x_{1} \left ( y_{2} -y_{3}\right )+x_{2} \left ( y_{3} -y_{1}\right )+x_{3} \left ( y_{1} -y_{2}\right )\right ]$
$= \frac{1}{2}\left [ \left ( -8 \right )\left ( 6-9 \right )+\left ( -6 \right )\left ( 9-4 \right )+\left ( -3 \right ) \left ( 4-6 \right )\right ]$
$= \frac{1}{2} \left [ 24+30+6 \right ]$
$= \frac{1}{2} \left [ 60 \right ]$
$= 30 sq.units$<|endoftext|>
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# Lesson 9: Applying Area of Circles
Let’s find the areas of shapes made up of circles.
## 9.1: Still Irrigating the Field
The area of this field is about 500,000 m2. What is the field’s area to the nearest square meter? Assume that the side lengths of the square are exactly 800 m.
1. 502,400 m2
2. 502,640 m2
3. 502,655 m2
4. 502,656 m2
5. 502,857 m2
## 9.2: Comparing Areas Made of Circles
1. Each square has a side length of 12 units. Compare the areas of the shaded regions in the 3 figures. Which figure has the largest shaded region? Explain or show your reasoning.
2. Each square in Figures D and E has a side length of 1 unit. Compare the area of the two figures. Which figure has more area? How much more? Explain or show your reasoning.
## 9.3: The Running Track Revisited
The field inside a running track is made up of a rectangle 84.39 m long and 73 m wide, together with a half-circle at each end. The running lanes are 9.76 m wide all the way around.
What is the area of the running track that goes around the field? Explain or show your reasoning.
## Summary
The relationship between $A$, the area of a circle, and $r$, its radius, is $A=\pi r^2$. We can use this to find the area of a circle if we know the radius. For example, if a circle has a radius of 10 cm, then the area is $\pi \boldcdot 10^2$ or $100\pi$ cm2. We can also use the formula to find the radius of a circle if we know the area. For example, if a circle has an area of $49 \pi$ m2 then its radius is 7 m and its diameter is 14 m.
Sometimes instead of leaving $\pi$ in expressions for the area, a numerical approximation can be helpful. For the examples above, a circle of radius 10 cm has area about 314 cm2. In a similar way, a circle with area 154 m2 has radius about 7 m.
We can also figure out the area of a fraction of a circle. For example, the figure shows a circle divided into 3 pieces of equal area. The shaded part has an area of $\frac13 \pi r^2$.<|endoftext|>
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Having laid out the long course to Northern ascendancy at the South’s expense, Calhoun turned from Southern grievances to Northern politics. Specifically, why did Northern dominance so threaten the South? Calhoun maintained that the North and South entered the Constitution with the tacit agreement for sectional equality, but he took care to unconvincingly extricate their sectional identities from the issue of slavery. In so doing he asked too much of himself, as time and time again he returned to the centrality of slavery to South identity.
But if there was no question of vital importance to the South, in reference to which there was a diversity of views between the two sections, this state of things might be endured, without the hoard of destruction to the South. But such is not the fact. There is a question of vital importance to the Southern section, in reference to which the views and feelings of the two sections are as opposite and hostile as they can possibly be.
I refer to the relation between the races in the Southern Section, which constitutes a vital portion of her social organization. Every portion of the North entertains views and feelings more or less hostile to it. Those most opposed and hostile, regard it as a sin, and consider themselves under the most sacred obligation to use every effort to destroy it. Indeed, to the extent that they conceive they have power; they regard themselves as implicated in the sin, and responsible for not suppressing it by the use of all and every means. Those less opposed and hostile, regard it as a crime – an offence against humanity, as they call it; and, although not so fanatical, feel themselves bound to use all efforts to effect the same object; while those who are least opposed and hostile, regard it as a blot and a stain on the character of what they call the Nation, and feel themselves accordingly bound to give it no countenance or support.
At last Calhoun drops the pretense entirely. The same man who began saying slavery did not form the heart of sectional discontent, but rather a growing inequality between the sections did, now comes around to say that in fact the South could suffer inequality as long as the North did not campaign against slavery. Clearly preserving slavery, not sectional equality, comes first.
And how did it come to this?
With the success of their first movement, this small fanatical party [antislavery] began to acquire strength; and with that, to become an object of courtship to both the great parties. The necessary consequence was, a further increase of power, and a gradual tainting of the opinions of both of the other parties with their doctrines, until the infection has extended over both; and the great mass of the population of the North, who, whatever may be their opinion of the original abolition party, which still preserves its distinctive organization, hardly ever fail, when it comes to acting, to co-operate in carrying out their measures.
Had proslavery forces achieved that, Calhoun would have crowed victory from the roof of every plantation house in the South. His own project to unify the South amounted to much the same as he accused the abolitionists of achieving. I can’t say with any confidence, but I imagine Calhoun must have bitterly envied the antislavery success on some level.
Calhoun continued with a list of Northern offenses to prove his point: personal liberty laws, Northerners aiding fugitive slaves, abolitionist petitions, the Wilmot Proviso. All of it aimed for a single thing: complete abolition. Abolitionists never had greater strength. The South never had greater vulnerability.<|endoftext|>
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Home > Differentiation > Differentiating Polynomials from first principles
## Differentiating Polynomials from first principles
The gradient of a curve is always changing. To calculate the gradient at a point we can consider the gradient of a chord going through that point and gradually make the length of the chord shorter. As the length gets closer to zero the gradient of the chord should get closer to the gradient of the tangent at the point.
When we are finding the gradient function we are differentiating
Consider the diagram
You can see from the graph that as $\delta x$ gets smaller so does $\delta y$ and that the gradient of the chord gets closer to that of the tangent.
So as $\delta x \rightarrow 0$, $\frac{\delta y}{\delta x} \rightarrow$ the gradient at x
We refer to the gradient at x as $\frac{dy}{dx}$
### Finding the gradient function of $x^2$
Let $y=x^2$
The point that is $\delta x$ across from $x$ has the coordinates $(x+\delta x, y+\delta y)$
Substituting into $y=x^2$ gives $y+\delta y=(x+\delta x)^2 \Rightarrow y+\delta y=x^2+2x(\delta x)+(\delta x)^2$
Since $y=x^2$ we get $x^2+\delta y=x^2+2x(\delta x)+(\delta x)^2 \Rightarrow \delta y=2x(\delta x)+(\delta x)^2$
If we now divide through by $\delta x$ we get $\frac{\delta y}{\delta x} = 2x+\delta x$
Now if we let $\delta x \rightarrow 0$ we get $\frac{dy}{dx}=2x$
### Finding the gradient function of $x^n$
Note that $(x+a)^n = x^n + nax^{n-1} + \frac{n(n-1)}{2!}a^2x^{n-2} + \frac{n(n-1)(n-2)}{3!}a^3x^{n-3} + ...$
Let $y = x^n$
The point that is $\delta x$ across from $x$ has the coordinates $(x+\delta x, y+\delta y)$
Substituting into $y=x^n$ gives $y+\delta y=(x+\delta x)^n$
So $y+\delta y = x^n + n\delta x x^{n-1} + \frac{n(n-1)}{2!}(\delta x)^2x^{n-2} + \frac{n(n-1)(n-2)}{3!}(\delta x)^3x^{n-3} + ...$
Since $y=x^n$
we get $x^n+\delta y = x^n + n\delta x x^{n-1} + \frac{n(n-1)}{2!}(\delta x)^2x^{n-2} + \frac{n(n-1)(n-2)}{3!}(\delta x)^3x^{n-3} + ...$
If we now divide through by $\delta x$
we get $\frac{\delta y}{\delta x} = n x^{n-1} + \frac{n(n-1)}{2!}\delta xx^{n-2} + \frac{n(n-1)(n-2)}{3!}(\delta x)^2x^{n-3} + ...$
Now if we let $\delta x \rightarrow 0$ we get $\frac{dy}{dx}=n x^{n-1}$
You should be able to see that when we differentiate that we multiply by the power and then subtract one from the power.
Examples
$y$ $\frac{dy}{dx}$ $x^5$ $5x^4$ $3x^7$ $21x^6$ $5x^{-3}$ $-15x^{-4}$
Questions<|endoftext|>
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Lead: An Introduction
Lead is a natural elemental metal that has many uses and has been used through antiquity. Though it has many uses it is extremely toxic to humans and as such, has been regulated by government agencies throughout the world. Awareness is growing, but people are still exposed to lead. Lead: an introduction to help you protect yourself and your family.
While lead is not easily absorbed by the skin, lead absorption by inhalation of fumes or by ingestion of lead contaminated materials is dangerous to everyone. Generally, lead poisoning occurs slowly, as slow accumulation of lead builds up in bone and tissue after repeated exposure. It is, however, particularly toxic to children and pregnant women. Adults absorb only 10% to 15% of the lead ingested while children and pregnant women absorb roughly 50%. Lead has a slow half-life in the human body, as lead that has accumulated in the bones and teeth only diminishes by half over a period of 30 years.
Lead Poisoning Symptoms
Lead affects practically all systems within the human body. People afflicted with lead poisoning show symptoms of: Headaches, Irritability, Abdominal Pain, Vomiting, Anemia, Weight Loss, Poor Attention Span, Noticeable Learning Difficulty, Slowed Speech Development, Hyperactivity and Impotence.
The effects of lead poisoning include: Reading/Learning Disabilities, Speech/Language Handicaps, Lowered IQ, Neurological Deficits, Behavior Problems, Mental Retardation, Kidney Disease, Heart Disease, Stroke, Weakness in the extremities, Peripheral Nerve Damage, High Blood Pressure, Convulsions, Coma and Death.
Children and Lead Exposure
Children may risk higher exposures since they are more likely to get lead dust on their hands while crawling and then put their fingers or other lead-contaminated objects into their mouths. Blood lead levels as low as 10 micrograms per deciliter can impair mental and physical development in children. As such, the Centers for Disease Control and Prevention (CDC) have established 5 micrograms per deciliter of blood as the reference level to identity children with blood lead levels much higher than most children’s levels.
As such, childhood blood tests are, by law, reported to the California Department of Public Health (CDPH). Should a child be found to have levels of lead in the blood that are considered unhealthy, the CDPH can legally force a building owner to perform a lead risk assessment and abatement on a building to determine that has Lead Materials, Lead Based Paint (LBP), and/or Lead-contaminated Dust or Soil in the vicinity.
Lead Assessment & Abatement
It behooves the concerned building owner to take proper renovation or proper abatement steps before it is forced upon them by the CDPH. Not only for the health of the occupants but for legal and financial reasons as well.
Who is Healthy Building Science?
Environmental Testing Services at HBS
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Your email is never shared or sold.<|endoftext|>
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# Question 8099a
Nov 11, 2015
$\text{1.3 g/mL}$
#### Explanation:
The first thing to do here is pick a sample of this solution. To make calculations easier, let's say that we pick a $\text{1.00-L}$ sample of this concentrated phosphoric acid solution.
Now, a solution's molarity is defined as the number of moles of solute, which in your case is phosphoric acid, divided by the volume of the solution - expressed in liters.
$\textcolor{b l u e}{\text{molarity" = "moles of solute"/"liters of solution}}$
This means that you can use this volume sample and the solution's molarity to determine how many moles of phosphoric acid you have
$c = \frac{n}{V} \implies n = C \cdot V$
$n = {\text{12.2 M" * "1.00 L" = "12.2 moles H"_3"PO}}_{4}$
Use phosphoric acid's molar mass, which tells you what the exact mass of one mole of a compound is, in order to determine how many grams of phosphoric acid you have in this sample
12.2color(red)(cancel(color(black)("moles"))) * "97.995 g"/(1color(red)(cancel(color(black)("mole")))) = "1195.5 g H"_3"PO"_4
You know that this solution is 90% phosphoric acid by mass, which is equivalent to saying that you get $\text{90 g}$ of phosphoric acid for every $\text{100 g}$ of solution.
The mass of the solution that contains $\text{1195.5 g}$ of phosphoric acid will thus be
1195.5color(red)(cancel(color(black)("g H"_3"PO"_4))) * "100 g solution"/(90color(red)(cancel(color(black)("g H"_3"PO"_4)))) = "1328.3 g solution"#
Now you know the volume of the solution, which we've chosen to be equal to $\text{1.00 L}$, and the mass of the solution, which is $\text{1328.3 g}$, so the density of the solution will be - expressed in grams per mililiter
$\textcolor{b l u e}{\text{density" = "mass"/"volume}}$
$\rho = \text{1328.3 g"/(1.00color(red)(cancel(color(black)("L")))) * (1color(red)(cancel(color(black)("L"))))/"1000 mL" = "1.3283 g/mL}$
I'll leave the answer rounded to two sig figs, despite the fact that you only have one sig figs for the concentration of the solution
$\rho = \textcolor{g r e e n}{\text{1.3 g/mL}}$
SIDE NOTE I recommend doing the calculations with different samples of the solution, the result must come out the same regardless of what volume you pick.<|endoftext|>
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#### Recommended Lessons and Courses for You
Lesson Transcript
Instructor: Damien Howard
Damien has a master's degree in physics and has taught physics lab to college students.
Explore how we tell when two variables are in quadratic or inverse relationships in this lesson. Once you understand the basics, we'll go over a couple of examples where these relationships show up in a physics class.
## Relationships Between Variables
When you're working a physics lab for a class, you'll often find yourself making graphs of a couple variables. You change one of the variables yourself, and track the corresponding change in the other. For instance, you might be tasked with placing a sealed container filled with gas in a pot of water, and measuring the change in pressure of the gas as the water is heated. Here, the temperature of the water is the variable you are changing, and the gas pressure is the second variable you are tracking. When you make a graph of pressure vs. temperature you are exploring the relationship between these two variables.
In an introductory physics course, there are four different common relationships between variables you are bound to run into: they are linear, direct, quadratic, and inverse relationships. Here, we'll go over both quadratic and inverse relationships, and a couple examples of places they pop up in a physics course.
A quadratic relationship is a mathematical relation between two variables that follows the form of a quadratic equation. To put it simply, the equation that holds our two variables looks like the following:
Here, y and x are our variables, and a, b, and c are constants. If you didn't have this equation, and only had some data points for a graph, you'd be able to tell it's a quadratic relation if the graph's curve forms a parabola, which on a graph looks like a dip or a valley. Even if we have an equation like this where b and c both equal zero, it's still considered quadratic. If that happens, we get the simplest form of a quadratic relationship:
This works because it turns out that it's the x squared component that's absolutely necessary for a relationship to be quadratic. So, unlike b and c, a must never equal zero because that will remove the x squared from the formula since zero multiplied by anything is zero.
One of the first places you'll encounter a quadratic relation in physics is with projectile motion. This makes sense if you think about how a projectile travels through the air over time. Imagine you're tossing a baseball straight up in the air. Let's try visualizing this with a height vs. time graph. Over time the ball goes up to a maximum height, and then back down to the starting height again when you catch it.
We can see our graph creates an upside-down parabola, which is the sort of thing you might expect from a quadratic relation. To make absolutely sure the relation between height and time is quadratic, we'll also look at the vertical equation for projectile motion that deals with position and time:
Does it look familiar? Let's try rearranging the equation a bit:
You might not know this yet, but in this equation the only two variables are height (y) and time (t). Everything else is constant over the course of a single throw. So, the equation we're seeing here is really in the exact form of a quadratic equation:
## Inverse Relationships
You might remember that in a direct relationship, as one variable increases, the other increases, or as one decreases, the other decreases. In an inverse relationship, instead of the two variables moving in the same direction they move in opposite directions, meaning as one variable increases, the other decreases.
Often in a physics course, the type of inverse relationship you'll run across is an inversely proportional relationship. For inversely proportional relationships, we specify that as one variable increases the other decreases at the same rate.
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# Question ffde9
Jun 20, 2017
See the explanation
#### Explanation:
It all depends on the correct interpretation of the question. I have given two interpretations to demonstrate some mathematical processes. Even if they are not the correct solutions $\textcolor{m a \ge n t a}{\text{the methods are important.}}$
You state $m + \frac{2}{3} + \frac{1}{4} m - 1$ and you use the word 'solve'. This implies that you wish to determine the value of $m$.
As there is no equals sign it is not possible to 'solve' for $m$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Taking the work on from your part 'solution'}}$
$\textcolor{g r e e n}{12 m + 8 = 3 m - 1}$
subtract $\textcolor{red}{3 m}$ from both sides
$\textcolor{g r e e n}{12 m \textcolor{red}{- 3 m} + 8 \text{ "=" } 3 m \textcolor{red}{- 3 m} - 1}$
$\textcolor{g r e e n}{\text{ "9m" "color(white)(.)+8" "=" "0" } - 1}$
Subtract $\textcolor{red}{8}$ from both sides
color(green)(9m+8color(red)(-8)" "=" "-1color(red)(-8)#
$\textcolor{g r e e n}{9 m \text{ "+0" "=" } - 9}$
Divide both sides by $\textcolor{red}{9}$
$\textcolor{g r e e n}{\frac{9}{\textcolor{red}{9}} m \text{ "=" } \frac{- 9}{\textcolor{red}{9}}}$
but $\frac{9}{9} = 1 \mathmr{and} \frac{- 9}{9} = - 1$
$\textcolor{g r e e n}{1 m = - 1}$
but writing $1 m$ is bad practice so write this as just $m$
$m = - 1$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Assuming that we have an expression and not an equation}}$
Simplifying the expression$\to m + \frac{2}{3} + \frac{1}{4} m - 1$
Note that $m$ is the same as $1 m$ which is also the same as $\frac{4}{4} m$
$m + \frac{2}{3} + \frac{1}{4} m - 1 \text{ "->" } \frac{4}{4} m + \frac{2}{3} + \frac{1}{4} m - 1$
$\text{ "->" } \frac{4}{4} m + \frac{1}{4} m + \frac{2}{3} - 1$
$\text{ "->" } \frac{5}{4} m + \frac{2}{3} - 1$
Note that $- 1$ is the same as $- \frac{3}{3}$
$\frac{5}{4} m + \frac{2}{3} - 1 \text{ "->" } \frac{5}{4} m + \frac{2}{3} - \frac{3}{3}$
$\text{ "->" "5/4m-1/3 larr" Simplified}$<|endoftext|>
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What is it?
Dyslipidaemia involves unhealthy levels of fat in your blood and can cause atherosclerosis.
This can include an elevation of cholesterol or triglyceride levels, or a decrease of high-density lipoprotein levels in the blood. High cholesterol levels, together with other lipid and metabolic disorders, may lead to increased cardiovascular risks.
Dyslipidaemias can be classified as follows:
- Hypercholesterolemia: an increase in cholesterol levels
- Hypertriglyceridemia: an increase in triglyceride levels
- Hyperlipidaemia: a combined increase in both cholesterol and triglyceride levels
What are the symptoms?
Dyslipidaemias are generally asymptomatic and are diagnosed due to the conditions they contribute to, such as coronary heart disease, ischemia, peripheral arterial disease, pancreatitis and tendinous xanthomas.
Dyslipidaemias can also cause altered sensation in the limbs, shortness of breath, or a feeling of disorientation.
How is it diagnosed?
Dyslipidaemia is diagnosed by taking a blood test and checking triglyceride and cholesterol levels in the blood. It may also be necessary to measure glucose levels, TSH (Thyroid-Stimulating Hormone) levels and the amount of protein in the urine.
What causes dyslipidaemia?
There are two main causes for dyslipidaemia:
- Primary (or genetic) causes: genetic disorders may lead to an excessive or insufficient production of cholesterol and triglycerides;
- Secondary causes: these may be linked to an unhealthy lifestyle, such as smoking and overconsumption of foods rich in cholesterol, trans-fats and saturated fats. Dyslipidaemias can also be caused by diabetes, kidney and liver conditions, hypothyroidism, HIV or certain kinds of medication.
How can it be prevented?
You can reduce your risk of dyslipidaemia by leading a healthy lifestyle, including:
- eating more fibre and complex carbohydrates
- decreasing your cholesterol and saturated fat intake
- exercising regularly
- maintaining a healthy weight
What is the treatment?
People affected by dyslipidaemia often need to be treated for hypertension and diabetes as well. In order to reduce cholesterol levels you may be given medication or food supplements.
Which doctor should I talk to?
If you think you may have dyslipidaemia, you can talk to a dietician. If you have diabetes you can also talk to your endocrinologist or diabetes specialist.<|endoftext|>
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Dynamics: Force and Newton’s Laws of Motion
- Understand the definition of force.
Dynamics is the study of the forces that cause objects and systems to move. To understand this, we need a working definition of force. Our intuitive definition of force—that is, a push or a pull—is a good place to start. We know that a push or pull has both magnitude and direction (therefore, it is a vector quantity) and can vary considerably in each regard. For example, a cannon exerts a strong force on a cannonball that is launched into the air. In contrast, Earth exerts only a tiny downward pull on a flea. Our everyday experiences also give us a good idea of how multiple forces add. If two people push in different directions on a third person, as illustrated in [link], we might expect the total force to be in the direction shown. Since force is a vector, it adds just like other vectors, as illustrated in [link](a) for two ice skaters. Forces, like other vectors, are represented by arrows and can be added using the familiar head-to-tail method or by trigonometric methods. These ideas were developed in Two-Dimensional Kinematics.
[link](b) is our first example of a free-body diagram, which is a technique used to illustrate all the external forces acting on a body. The body is represented by a single isolated point (or free body), and only those forces acting on the body from the outside (external forces) are shown. (These forces are the only ones shown, because only external forces acting on the body affect its motion. We can ignore any internal forces within the body.) Free-body diagrams are very useful in analyzing forces acting on a system and are employed extensively in the study and application of Newton’s laws of motion.
A more quantitative definition of force can be based on some standard force, just as distance is measured in units relative to a standard distance. One possibility is to stretch a spring a certain fixed distance, as illustrated in [link], and use the force it exerts to pull itself back to its relaxed shape—called a restoring force—as a standard. The magnitude of all other forces can be stated as multiples of this standard unit of force. Many other possibilities exist for standard forces. (One that we will encounter in Magnetism is the magnetic force between two wires carrying electric current.) Some alternative definitions of force will be given later in this chapter.
To investigate force standards and cause and effect, get two identical rubber bands. Hang one rubber band vertically on a hook. Find a small household item that could be attached to the rubber band using a paper clip, and use this item as a weight to investigate the stretch of the rubber band. Measure the amount of stretch produced in the rubber band with one, two, and four of these (identical) items suspended from the rubber band. What is the relationship between the number of items and the amount of stretch? How large a stretch would you expect for the same number of items suspended from two rubber bands? What happens to the amount of stretch of the rubber band (with the weights attached) if the weights are also pushed to the side with a pencil?
- Dynamics is the study of how forces affect the motion of objects.
- Force is a push or pull that can be defined in terms of various standards, and it is a vector having both magnitude and direction.
- External forces are any outside forces that act on a body. A free-body diagram is a drawing of all external forces acting on a body.
Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
What properties do forces have that allow us to classify them as vectors?
- the study of how forces affect the motion of objects and systems
- external force
- a force acting on an object or system that originates outside of the object or system
- free-body diagram
- a sketch showing all of the external forces acting on an object or system; the system is represented by a dot, and the forces are represented by vectors extending outward from the dot
- a push or pull on an object with a specific magnitude and direction; can be represented by vectors; can be expressed as a multiple of a standard force<|endoftext|>
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Finding distinct equivalence classes.
I am going through some practice questions and am having trouble to finding distinct equivalence classes and this is my understanding so far.
Let S be a nonempty subset of Z, and let R be a relation defined on $S$ by $xRy$ if $3 | (x + 2y)$.
Find the distinct equivalence classes for $S=\{−7, −6, −2, 0, 1, 4, 5, 7\}$
Now, I understand that for $3 | (x + 2y) \equiv$ $$x\equiv -2y(mod 3)$$
So that means, I would have to find combinations of elements from the set S such that $-2y$ must have a remainder $x$ when divided by 3? However for one of the solutions given, $$[5] = \{5,-7\}$$
If I am not mistaken, for $-2(5) = - 10; -2(-7) = 14$ and $$-10(mod3)=2$$ $$14(mod3)=2$$.
So, then $[2] = \{5,-7\}$ right? Or am I missing something that I don't understand. Also, from further observation, anything divided by 3 can only have remainder of 0,1 and 2. Then [5] = ... doesn't make sense. If anyone could clarify, that would be helpful! Thanks.
So that means, I would have to find combinations of elements from the set S such that $−2y$ must have a remainder $x$ when divided by $3$?
No; this means that $x$ and $-2y$ must have the same remainder when divided by $3$. For example, both $5$ and $-2(-7)$ have the remainder $2$ when divided by $3$, so $5$ and $-7$ are in the same class.
So, then $[2] = \{5,-7\}$ right?
$2$ doesn't belong to $S$, so technically the "class of $2$", which you denoted $[2]$, is meaningless in this context.
Then $[5]$ doesn't make sense
By definition, $[5]$ means "the equivalence class that contains $5$." Hopefully the above remarks make it clearer that this does makes sense.
• $[5]={x∈S: xR5}={x∈S: x≡-10(mod3)} = {x ∈ S : 3 | (x +10)} = \{5,-7\}$ Is that correct? – misheekoh Dec 15 '15 at 20:03
• @misheekoh You want to replace all your instances of $Z$ with $S$ since here $S$ is the set on which the equivalence relation is defined. So by definition $[5] = \{ x \in S : xR5 \} = \{ x \in S : 3 |(x+10) \} = \{5, -7 \}$. But you have the right idea. – Alex Provost Dec 15 '15 at 20:05
• So the idea is that for [5], the remainder of the elements in that class will have a remainder of 2 then right? – misheekoh Dec 15 '15 at 20:12
• @misheekoh Yes, because $x \equiv -10 \mod 3 \iff x \equiv 2 \mod 3$. – Alex Provost Dec 15 '15 at 21:10
It's easier if you convert $3 \mid x+2y$ to $x + 2y \equiv 0 \pmod 3$. We then "solve" for $y$ in terms of $x$. $y \equiv x \pmod 3$. In other words, $$xRy \iff x \equiv y \pmod 3$$
It helps to make a table of $s \pmod 3$ for all $s \in S$.
\begin{array}{|c|rrrrrrrr|} \hline s & −7 & −6 & −2 & 0 & 1 & 4 & 5 & 7 \\ s \pmod 3 & 2 & 0 & 1 & 0 & 1 & 1 & 2 & 1 \\ \hline \end{array}
The partition of $S$ is therefore $\big\{\{-6,0\}, \{-2,1,4,7 \}, \{-7,5\} \big\}$<|endoftext|>
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Deep learning recently returned to the headlines when
Although the terms are sometimes used synonymously, deep learning and machine learning are not the same thing. Deep learning is a kind of machine learning just as cycling is a kind of exercise. Understanding deep learning is easier if you have a basic idea of what machine learning is all about.
Machine learning is often described as a type of AI where computers learn to do something without being programmed to do it. To take a simple example, you could program a computer to identify an animal as a cat by writing code that tells the program to say "cat" when it sees a picture of a particular cat. This works well if the only cat the program will ever see is the cat in the picture. It doesn't work very well if the program is going to see a lot of different animals, including a lot of different cats, and it has to pick out the cats from all of the other animals.
Machine learning programs can be trained in a number of different ways. In one type of training, the program is shown a lot of pictures of different animals and each picture is labeled with the name of the animal; the cats are all labeled "cat". The program will eventually learn that the animals that look like cats are called "cats" without ever being programmed to call a picture of a cat a "cat".
The program does this by learning combinations of features that tend to appear together. Cats have visual features - such as their body shape, long whiskers, and the way their faces look - that make them visually different from other animals. The program learns to associate this distinctive combination of features with the word "cat". This learning process is usually called constructing a model of a cat.
Once it has constructed the cat model, a machine learning program tests the model by trying to identify the cats in a set of pictures it hasn't seen before. The program measures how well it did at identifying the new cats and uses this information to adjust the model so it will do a better job of picking out cats the next time it tries. The new model is then tested, its performance is evaluated, and it receives another adjustment. This iterative process continues until the program has built a model that can identify cats with a high level of accuracy.
Deep learning carries out the machine learning process using an artificial neural net that is composed of a number of levels arranged in a hierarchy. The network learns something simple at the initial level in the hierarchy and then sends this information to the next level. The next level takes this simple information, combines it into something that is a bit more complex, and passes it on the the third level. This process continues as each level in the hierarchy builds something more complex from the input it received from the previous level.
Continuing the cat example, the initial level of a deep learning network might use differences in the light and dark areas of an image to learn where edges or lines are in a picture of a cat. The initial level passes this information about edges to the second level which combines the edges into simple shapes like a diagonal line or a right angle. The third level combines the simple shapes into more complex objects likes ovals or rectangles. The next level might combine the ovals and rectangles into rudimentary whiskers, paws and tails. The process continues until it reaches the top level in the hierarchy where the network has learned to identify cats. While it was learning about cats, the network also learned to identify all of the other animals it saw along with the cats.
Why deep learning is useful
Deep learning has attracted a lot of attention because it is particularly good at a type of learning that has the potential to be very useful for real-world applications. The earlier introduction to machine learning described a training method in which all the pictures that are used to train the program are labeled with the name of the thing in the picture. In the cat example, the pictures of cats are all labeled "cat". Each iterative step in testing and refining the model involves comparing the label on a picture with the label the program assigned to the picture to determine whether the program labeled the picture correctly. This method of training is called supervised learning.
Supervised learning is relatively fast and demands relatively less computational power than some other training techniques that are used in machine learning. It has an important drawback for real-world applications, however. An immense amount of information about people is gathered everyday from social media, hardware and software service agreements, app permissions and website cookies. This information has the potential to be very valuable to businesses at all levels. The problem is that all of this data is unlabeled and can't be used to train machine learning programs that depend on supervised learning. A person is needed to label the data and the labeling process is time-consuming and expensive.
Deep learning networks can avoid this drawback because they excel at unsupervised learning. The key difference between supervised and unsupervised learning is that the data are not labeled in unsupervised learning. Even though the pictures of cats don't come with the label "cat", deep learning networks will still learn to identify the cats. Other forms of machine learning are not nearly as successful with unsupervised learning.
The ability to learn from unlabeled or unstructured data is an enormous benefit for those interested in real-world applications. Deep learning unlocks the treasure trove of unstructured big data for those with the imagination to use it .
How deep learning is useful
Deep learning networks can be successfully applied to big data for knowledge discovery, knowledge application, and knowledge-based prediction. In other words, deep learning can be a powerful engine for producing actionable results.
A good way to get a handle on how deep learning can be useful is by taking a look at some of the companies that are doing interesting things with deep learning systems.
ViSENZE develops commercial applications that use deep learning networks to power image recognition and tagging. Customers can use pictures rather than keywords to search a company's products for matching or similar items.
Skymind has built an open-source deep learning platform with applications in fraud detection, customer recommendations, customer relations management and more. They provide set-up, support and training services.
Atomwise applies deep learning networks to the problem of drug discovery. They use deep learning networks to explore the possibility of repurposing known and tested drugs for use against new diseases.
Descartes Labs is a spin-off from the Los Alamos National Laboratory. They analyze satellite imagery with deep learning networks to provide real-time insights into food production, energy infrastructure and more.
These examples are just a small sample of the many companies that are using deep learning to do innovative and exciting things. For more, see "Thirteen Companies That Use Deep Learning to Produce Actionable Results".
The discovery and recognition of patterns and regularities in the world around us lies at the heart of scientific and technological progress. It's how we advance and how we innovate. It's also an area where deep learning excels. The question isn't whether or not deep learning is useful, it's how can you use deep learning to improve what you're already doing, or to gain new insights from the data you already have.
Update: This post was updated on April 5 to remove the reference to Ersatz, a deep-learning company that is now out of business.
Kevin Murnane covers science, technology and video games for Forbes. His blogs are The Info Monkey & Tuned In To Cycling and he's The Info Monkey on Facebook & @TheInfoMonkey on Twitter.<|endoftext|>
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# How do you write 10^-2*10^-8 using only positive exponents?
##### 1 Answer
Jan 15, 2017
See entire process below.
#### Explanation:
First, we will use this rule of exponents to combine the terms:
${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$
${10}^{\textcolor{red}{- 2}} \cdot {10}^{\textcolor{b l u e}{- 8}} \to {10}^{\textcolor{red}{- 2} + \textcolor{b l u e}{- 8}} \to {10}^{-} 10$
Now, we can use this rule of exponents to put this into a term with a positive exponent:
${x}^{\textcolor{red}{a}} = \frac{1}{x} ^ \textcolor{red}{- a}$
${10}^{\textcolor{red}{- 10}} \to \frac{1}{10} ^ \textcolor{red}{- - 10} \to \frac{1}{10} ^ 10$<|endoftext|>
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Diseases of the heart valves are grouped according to which valve or valves are involved plus the amount of blood flow that is disrupted by the problem. The most common and serious valve problems happen in the mitral and aortic valves. Diseases of the tricuspid and pulmonary valves are fairly rare.
The mitral valve regulates the flow of blood from the upper-left chamber (the left atrium) to the lower-left chamber (the left ventricle).
Mitral Valve Prolapse
The mitral valve regulates the flow of blood from the upper-left chamber (the left atrium) to the lower-left chamber (the left ventricle). Mitral valve prolapse (MVP) means that one or both of the valve flaps (called cusps or leaflets) are enlarged, and the flaps’ supporting muscles are too long. Instead of closing evenly, one or both of the flaps collapse or bulge into the left atrium. MVP is often called click-murmur syndrome because when the valve does not close properly, it makes a clicking sound and then a murmur.
What causes MVP? MVP is one of the most common forms of valve disease. It is also genetic running in families.
What are the symptoms? Most people with MVP do not have symptoms. When symptoms do happen, they may include
Shortness of breath, especially when lying down.
Trouble breathing after exercise.
Most of the time, MVP is not a serious condition. Some patients say they feel palpitations or sharp chest pain.
Mitral Regurgitation is also called mitral insufficiency or mitral incompetence. This happens when the mitral valve allows a backflow of blood into the heart’s upper-left chamber (the left atrium). Mitral regurgitation can take years to reveal itself.
If it goes on long enough, it may cause a buildup of pressure in the lungs or cause the heart to enlarge. In time, this will lead to symptoms.<|endoftext|>
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At Meet A Creature, we know that our educational animal workshops for children in primary schools are a great way for getting children involved and an excellent tool for learning. But what else comes with an animal workshop? What can children learn from these educational visits?
Teaches children how to care for animals
One of the main things your pupils can learn from animal workshops is about caring for animals. The key is that we introduce each of our animals so that children know how the animal behaves and other key information. After this, the children will handle them. This is an excellent way for children to know how to care for and respect animals, as well as understanding the importance of caring for other children or adults. It can also help children mature and become the respectful and compassionate pet owners of the future.
Great way of building confidence
Some children will be excited of the idea of holding animals and learning about them. However, there will always be some pupils that will not be as confident. With the support of our experienced Ranger, pupils will be able to gain confidence and join in with the animal handling at their own pace. We aim to include everyone and take the time to help children build confidence with the animals, which can help their overall confidence and child development.
As well as this, it gives children the confidence of experiencing something that is out of their comfort zone and something they might have never experienced before. This can help children develop other skills in the future, for instance by approaching new tasks with a positive attitude because they believe they can succeed.
Learning the concept of teamwork
Animal workshops are a great way for getting all of your pupils working together, enjoying themselves and understanding one another. For example, there might be a few pupils that are not as comfortable with animals as some other pupils. Because our animal handling workshops are a group activity, everyone works together, making the whole experience more beneficial for child development.
At Meet A Creature, we understand the importance of children having contact with as many safe animals as possible during their developing years. If you are interested in our educational animal workshops, then please do not hesitate to contact us on 07432 560371 and we’ll be more than happy to help. If you haven’t already, check out our Facebook and Twitter and watch some of our videos on YouTube!<|endoftext|>
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Chapter 2: Climate and Culture: Creating a Common Thread
Summary: Climate and culture are the foundation for everything we do. It determines how our relationships function, how we make decisions and solve problems as a staff, and how people feel when they enter our buildings. This chapter focuses on little changes to create a commonality between people that can positively impact a culture and climate like creating a common language, expectations, and belief system.
School Culture, School Climate: They Are Not the Same Thing
Read, Write, Lead: breakthrough strategies for schoolwide literacy success by Regie Routman
The First Days of School by Wong and Wong (now updated)
- List and define one to two terms regularly used in your district that may not be defined for stakeholders. These could be being used in initiatives, your mission/vision, etc.
- How will you explicitly and effectively communicate these definitions to stakeholders?
- What do you see as values you desire your district to have? School? Classroom?
- What do you see as what your district/school/classroom actually valuing (how do those compare)?
- What specifically do you do (or not do) that proves those are the values?
- What values do your initiatives support?<|endoftext|>
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The What, Why, and How of Down Syndrome
In the United States, approximately 400,000 people have Down syndrome.
For just a second, I'm going to take you back to your ninth-grade biology class and explain just a little bit about how the cell works.
In the nucleus of the cell, there are things that are called genes. Genes carry information that eventually decide what our hair color is going to be, what color our skin is going to be, and, really, everything about “us.”
The genes are clumped together into a structure that either resembles an X or a Y, hence the XY and XX business when it comes to having children.
In every nucleus of every cell in the entire body, there are 23 pairs of XY chromosomes if you're a male and XX chromosomes if you're a female—46 chromosomes in total.
Sometimes, however, chromosome 21 may have a third copy made. This disorder is what causes some issues in the genetic makeup of the cell and causes Down syndrome.
According to the National Down Syndrome Society, Down syndrome is the most common genetic condition, being that 1 in 691 children are born with it. In the United States alone, approximately 400,000 people have Down syndrome.
How to spot Down syndrome
In order to identify Down syndrome, diagnoses are given at birth as well as prenatally. When testing for any genetic alterations in the fetus, two options are given: a diagnostic test or a screening test.
Diagnostic tests may cause some apprehension in some mothers. The test is invasive and has a 1% chance of causing a miscarriage. Despite the likelihood of causing miscarriage, the tests are just about as close to 100% accurate as can be.
Screening tests include the typical ultrasound with the accompaniment of a blood test. When looking for indicators of Down syndrome during an ultrasound, doctors will look for what they call “markers,” or common characteristics that the doctor could associate with Down syndrome. Even though the diagnostic test is most often seen as the more accurate of the two tests, the accuracy rate is still extremely high.<|endoftext|>
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# What is the value ofx ifx^(4/5)=(2^8)/(3^8)?
Mar 18, 2017
$x = \frac{1024}{59049}$
#### Explanation:
If $x > 0$ and $a , b$ are any real numbers, then:
${\left({x}^{a}\right)}^{b} = {x}^{a b}$
So we find:
$x = {x}^{1} = {x}^{\frac{4}{5} \cdot \frac{5}{4}} = {\left({x}^{\frac{4}{5}}\right)}^{\frac{5}{4}} = {\left({2}^{8} / {3}^{8}\right)}^{\frac{5}{4}} = {\left({\left(\frac{2}{3}\right)}^{8}\right)}^{\frac{5}{4}} = {\left(\frac{2}{3}\right)}^{8 \cdot \frac{5}{4}} = {\left(\frac{2}{3}\right)}^{10} = {2}^{10} / {3}^{10} = \frac{1024}{59049}$
Mar 18, 2017
$x = {\left(\frac{2}{3}\right)}^{10}$
#### Explanation:
In general if ${x}^{a} = {p}^{b}$ then
$\textcolor{w h i t e}{\text{XXX}} x = {\left({x}^{a}\right)}^{\frac{1}{a}} = {\left({p}^{b}\right)}^{\frac{1}{a}}$
In this case
$\textcolor{w h i t e}{\text{XXX")a = 4/5color(white)("XX")rarrcolor(white)("XX}} \frac{1}{a} = \frac{5}{4}$
$\textcolor{w h i t e}{\text{XXX}} p = \frac{2}{3}$
and
$\textcolor{w h i t e}{\text{XXXXXX}}$after noting that $\frac{{2}^{8}}{{3}^{8}} = {\left(\frac{2}{3}\right)}^{8}$
$\textcolor{w h i t e}{\text{XXX}} b = 8$
So
$\textcolor{w h i t e}{\text{XXX}} x = {\left({\left(\frac{2}{3}\right)}^{8}\right)}^{\frac{5}{4}} = {\left(\frac{2}{3}\right)}^{\frac{8 \cdot 5}{4}} = {\left(\frac{2}{3}\right)}^{10}$
Mar 18, 2017
$x = 0.01734152$
#### Explanation:
${x}^{\frac{4}{5}} = \frac{{2}^{8}}{3} ^ 8$
Make $x$ radical,
$\sqrt[5]{{x}^{4}} = {2}^{8} / {3}^{8}$
Multiply both sides by the index of 5,
${\left(\sqrt[5]{{x}^{4}}\right)}^{5} = {\left({2}^{8} / {3}^{8}\right)}^{5}$
${x}^{4} = {2}^{40} / {3}^{40}$
Root both sides by index of 4,
$\sqrt[4]{{x}^{4}} = \sqrt[4]{{2}^{40} / {3}^{40}}$
${\left({x}^{4}\right)}^{\frac{1}{4}} = {\left({2}^{40} / {3}^{40}\right)}^{\frac{1}{4}}$
$x = {2}^{10} / {3}^{10}$
$x = \frac{1024}{59049}$
Hence $x = 0.01734152$.<|endoftext|>
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It is known since one century that nuclear energy is of the order of one million times more concentrated, in the same mass, than chemical energy.The radius of the atomic nucleus is, indeed, of the order of one million times smaller than that of a molecule. This is because the chemical and nuclear binding energies are of the same electric nature : they obey both to the Coulomb's laws, in 1/r where r is the radius of either a nucleus or a molecule. No need of a mysterious "strong force". According to Einstein, the energy of an object of mass m is E = mc². According to Bohr, the chemical energy is α²mec² . According to Schaeffer, the nuclear energy is αmc², intermediate. m is the considered mass, me the mass of the electron, α = 1/137 the fine structure constant and c the velocity of light. The so-called "strong force" (also referred to as the strong nuclear interaction or force) is the hypothetical force binding together the protons and the neutrons in an atomic nucleus. The word strong comes from the fact that the nuclear energy is around one million times greater than the chemical energy, for the same volume, dramatically demonstrated at Hiroshima. It is also known that the binding energy is around 1% of the mass energy mc². There are two well known fundamental forces in nature, the universal gravitation (formulated by Newton) and the electromagnetic force (formulated by Maxwell). It is fashionable to add two more hypothetical forces : the strong and weak nuclear forces. Very little is known about these forces.
According to the Bohr scheme, the electrons gravitate around the nucleus. The shell model assumes that the nucleons also gravitate (they don't on the graph) around a hypothetical force center. Indeed the nucleus has no nucleus around which the nucleons may orbite.
The origin of the strong force
The Rutherford scattering experiment, one century old now, consisted to collide alpha particles from a radioactive element on an atomic nucleus. The alpha particles from the radioactive element striking gold foils are scattered in different orientations. In a constant orientation, the number of particles per solid angle, called cross-section dσ/dΩ, are mesured as a function of the kinetic energy obtained by varying the number of gold foils:
The problem was to find n. Bieler, a Rutherford student, imagined in 1924 a magnetic attraction equilibrating an electrostatic repulsion between the protons (this is valid for the nuclear energy as I have shown elsewhere). Here the minus sign is wrong. Indeed, the electric interaction discovered by Rutherford explains the diffusion of the α particle by an atomic nucleus. For high energies, it doesn't work, thus "explained by a mysterious "strong force". In fact, the magnetic force replaces the electric force, thus the electric 1/r Coulomb's law is replaced by the magnetic 1/r³ Poisson's law. The sign should be the same, thus positive instead of negative as hypothesized. The graph shows the electric part (1/r law) discovered one century ago by Rutherford and the not so anomalous magnetic part (1/r³ law), discovered by me:
The binding energy per nucleon is given by the formula
BE/A = (Z mp + N mn - M)/A
where Z is the atomic number, N the neutron number and A = Z + N is the atomic mass number. M, mp, and mn are the masses of one nucleus (approximately of an atom), one proton and one neutron.
At a time when the neutron was not discovered, Aston used the packing fraction given by the formula:
f = (M - A)/A
where M is the mass of the atom measured experimentally and A the atomic number.
The Shell Model or Independent-particle Model or Hartree Model
The official mainstream physics model of the atomic nucleus is the shell model which the atomic model of Bohr and continuators adapted to the atomic nucleus.
The electrons are replaced by the nucleons. There are two kinds of nucleons, protons, electrically charged, and neutrons, uncharged. Contrarily to the atom, the nucleus has no nucleus. This deficiency is circumvented by the bold assumption that each nucleon experiences a central attractive force which can be ascribed to the average effect of all the other nucleons in the nucleus. On this assumption, each nucleon behaves as though it were movins independently in a central field, which is described as short-range potential well. Secondly, this potential is assumed to be the same for all values of l, the angular momentum quantum number of the nucleons. In the assumed central potential, each nucleon is imagined to be capable of describing an orbit of well-defined energy and angular momentum, in a manner analogous to the behavior of atomic electrons.
This assumption seems to be in conflict with the strong interaction between nucleons, as seen experimentally, and in nuclear reactions generally. This weak interaction paradox was solved by using the Pauli exclusion principle. The expected strong interaction may be present but unable to manifest itself because all the quantum states into which the nucleon might be scattered are already occupied…
The main characteristic of the shell model is the so-called "magic numbers" (Z and/or N = 2, 8, 20, 28, 50, 82, 126) corresponding to the atomic levels.
The Liquid drop Model or Semi-empirical Mass Formula
The liquid drop model is the antithesis of the independent-particle model. The interactions are assumed to be strong instead of weak. The initial assumptions are (Evans p. 365) :
1. The nucleus is like a droplet of incompressible matter, and all nuclei have the same density.
2. Forces between nucleons are considered to be spin-independent as well as charge-independent if the coulomb force is turned off.
3. These nuclear forces have a short-range character and are effective only between nearest neighbors. Each nucleon interacts with all its nearest neighbors.
4. The volume or exchange energy is proportional to to the number of nucleons A for A ≥ 16, giving a radius R0 A1/3, where R0 is a constant.
5. The surface energy is like the surface tension of a liquid due to the fact that nucleons at the surface have fewer near neighbors than nucleons that are deep within the nuclear volume. We can expect a deficit of binding energy for these surface nucleons. A simple calculation shows that the surface energy is proportional to A2/3.
6. The Coulomb repulsion energy between protons is the only known long-range force in nuclei. The total nuclear charge Ze is assumed to be spread approximately uniformly throughout the nuclear volume. Again assuming a constant-density nuclear radius R0 A 1/3 and applying the Coulomb law for the potential, the Coulomb energy is proportional to Z2/A1/3.
7. The asymmetry energy, the deficit in energy dependent on the neutron excess or deficit, is (N - Z)2/A = (A - 2Z)2/A.
8. The pairing energy δ is a correction to take into account the pairing of N and Z.
The complete Bethe-Weizsäcker formula is
The following graph shows the Bethe-Weizsäcker curve (in blue), compared with the experimental data (in red) for the N = Z nuclei. The BW formula is unable to represent the binding energies of even Z and even N. There is also no distinction for same A nuclei with even N - odd Z and even Z - odd N (mirror nuclei).
Atomic number in abscissas and binding energy per nucleon in ordinates
Four-shell structure or α particle model
This type of models have been initiated by Gamow who observed that the nuclei with atomic masses multiple of 4 have larger binding energies. Indeed, it can be observed peaks of the binding energies for even A, Z and N. It is maximum for both N and Z even. The addition of an α-particle adds a tetrahedron to the structure already existing and so three bonds are alloted per addition.
For example, the maximum of ³Li is smaller than for both ²He and ⁴Be. For given Z, peaks appear for even N. They are greater if Z is also even. The amplitude diminishes when Z increases. The peaks are small but detectable even for the heavy nuclides:
Binding energies of 2,000 nuclides (Excel, to be downloaded)
The following curve shows the helium isotopes total binding energies, experimental and "ab initio", calculated by a supercomputer . It shows that the total binding energy is practically constant, except for the pairing effect, for the N>2 isotopes of helium. The discrepancy with the experimental values is attributed to the "3 body force". In fact, a better approximation is obtained by assuming that the excess neutrons are unbounded (halo nuclei). The red horizontal line is nearer to the experimental values (in red) than the super computer calculated orange line. Moreover, it seems that 4H (alpha particle) has not been calculated at all : the -28 MeV is the experimental value of its total binding energy. Even the simplest bounded nucleus (2H, the deuteron or heavy hydrogen) has never been calculated "ab initio". The fundamental laws and constants of the so-called "strong force" are still inexisting after one century of nuclear physics.
Figure 5. ORNL's Jaguar, the world's second fastest computer, enables certain nuclear calculations only dreamt of a few years ago. As an example, Jaguar was used for the first ever ab initio computation of neutron-rich helium nuclei using coupled cluster theory (shown in the figure on the side of the computer). The figure shows the binding energy of these nuclei, while the inset indicates the width, related to lifetime. Experimental data are marked in red. The calculated masses show a systematic deviation from experiment; this can be attributed to a three-body force, missing in the calculation. (original legend)
Energy per proton
Dividing the total binding energy by the the proton number, one obtains the following curves. From selected elements of the atomic mass table, one sees that the binding energy per proton tends towards a limit for each chemical element :
This limit is maximum for Fe and neighbors. For heavier nuclides, the binding energy decreases, due to the Coulomb repulsion between the protons.
Electromagnetic theory of the nuclear force
It is believed since almost a century that the strong force cannot be electromagnetic. This is incorrect because the nucleus has no nucleus or, in other words, the angular momentum has no fixed point like the atom. Therfore such a nucleus cannot be stable. The proof is given by my calculations ("Electromagnetic Theory of the Binding Energy of the Hydrogen Isotopes", J Fusion Energy (2011) 30 :377-381 here).
Deuteron binding energy
For the simple case of the deuteron, I have obtained the following formula:
If you have studied electromagnetism you will recognize the Coulomb attractive force and the magnetic repulsive force. Graphically, it gives the electromagnetic nuclear potential similar to other phenomenological potentials but truly ab initio because it contains only universal constants:
The formula in the graph is the same as the previous one but with different universal constants. One may recognize the famous mc² formula for the mass energy. It is multiplied by the fine structure constant α = 1/137. RP is the proton Compton radius. gp and gn are the Landé factors of the proton and of the neutron.
A better precision is obtained with a graphical resolution taking into account both electric charges in the neutron. Comparison is given below :
The curve at the right is the same as above. The scales are to be divided by 100. This is the first ab initio calculation of the binding energy of a nucleon (ab initio means a calculation using only fundamental constants, without adjusting parameters, and a well established theory e.g. Maxwell theory of electromagnetism).
Binding energy of the hydrogen isotopes
The binding energies of all the hydrogen isotopes have been calculated assuming the following structures:
Simplified calculation for H and He nuclei with N > 2
This method has been simplified for the hydrogen (giving the same results) and helium isotopes with N > 2. It is assumed that the total binding energy is the same when the number of nucleons is larger than 2, due to the small binding energy of the excess neutrons. These nuclei are called halo nuclei. The 4He binding energy is too low, probably due to the neglect of the positive charge of the neutron. More precise calculations are necessary for 4He. One can see that the binding energies of the N>2 isotopes are parallel to the experimental curves, justifying the approximation of almost zero binding energy of the last neutron (halo nuclei).
Nuclear to chemical energy ratio
The electromagnetic theory of the nuclear energy shows that it is αmc² or 1/137 of the mass energy, known to be of the order of 1%.
It is well known that the chemical energy is given by the Rydberg constant from the Bohr theory of the atom, ½α²mec² where me is the masses of the electron and α the fine structure constant.
The electromagnetic theory gives also the nuclear to chemical energy ratio, for the same weight, as
α-1mp/me = 137 x 1836 = 250,000
where mp and me is the mass of the proton. This formula, a consequence of the electromagnetic theory, explains for the first time why the nuclear energy is up to one million times more concentrated than the chemical energy, for the same volume.
Simple derivation of the nuclear to chemical energy ratio
It is known since one century that radium releases a huge energy, one million times larger than any combustion energy, according to Pierre Curie and others. This is the reason why nuclear energy is expressed in MeV and chemical energy in eV. "The energy stored by the binding energy of the outer electrons to the nucleus is, inversely, very much smaller (a hundred thousand or even a million times smaller), than that stored in the binding of nucleons in the nucleus. I have often been asked why it is that the smallest particles carry the largest energy. The precise analysis of this relation would lead us too far." Using this suggestion, we shall compare theoretically the radius and energy ratios of the hydrogen atom and the deuteron nucleus. The Bohr radius of the hydrogen atom is :
is the fine structure constant, representing the strength of coupling between radiation and matter, appearing also in the nuclear cross section. h is Planck's constant, me, the electron mass and c, the light velocity.
H atom and 2H nucleus. - Comparison between electron and neutron distances. a0 is the Bohr radius and RP is the proton Compton radius identified with the proton radius.
There exists no formula, using fundamental constants only, for the radius of a nucleon. Therefore we shall use the proton Compton radius RP, five times smaller than the experimental value of the proton radius but of the same order of magnitude, knowing that the binding energy per nucleon varies from one to ten times from deuteron to iron :
where mp is the proton mass. In the deuteron, the distance between the centers of the neutron and the proton being 2RP (figure), one obtains an expression of the ratio of the separation distances of the electron and of the neutron from the proton :
The separation energy of a neutron from a proton is 2.2 MeV and 13.6 eV for an electron from a proton, giving a 163,000 ratio. The Bohr formula of the binding energy of the fundamental state of the hydrogen atom is :
Multiplying by a0 / (2RP) from equation above, one obtains the binding energy of the deuteron :
This value was already obtained with an electromagnetic method. It is 30 % smaller than the experimental value, 2.2 MeV. A more precise result with a three body formulation (unpublished) gives a precision of 5 %.
Dropping the 1/4 coefficient, one obtains a mean value of the order of magnitude of the nuclear binding energy per nucleon :
which is coincidentally almost that of the alpha particle, 7.02.
This calculation explains why the binding energy of the nuclei is between 0.1 % and 1 % of the mass energy, or, per nucleon, between 1 MeV for the deuteron and 10 MeV for iron.This simple calculation confirms that the order of magnitude of the binding energy of a nucleus can be found theoretically from first principles and fundamental constants without ad hoc constants.
Poster: (click to enlarge or, better, unload it)<|endoftext|>
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The Sun orbits the center of the Milky way at a speed of about 230 km/s, taking about 250 million years to go around the galaxy once. It is a period of times sometimes referred to as a galactic year. But the Sun does not move in a simple circle or ellipse as the planets move around the Sun. This is due to the fact that the mass of the galaxy is not concentrated at a single point, but is instead spread across a plane with spiral arms and such. As a result, while the Sun orbits the galaxy it also moves up and down across the galactic plane. While the Sun is above the plane, the mass of the galaxy works to move it downward, and when below the plane the mass pulls it upward. As a result the Sun oscillates through the galaxy, crossing the galactic plane once every 30 million years or so.
There has been a great deal of speculation that this oscillatory motion could have implications for life on Earth, such as triggering cometary bombardments and causing mass extinctions. There is little evidence to support this idea, since mass extinctions don’t strongly follow a 30 million year cycle, and studies of impacts on the Moon show no correlation either. But now a new study in Scientific Reports shows what seems to be a relation between galactic motion and Earth’s temperature.
The paper looked at temperature measurements of the Phanerozoic, which is the geologic period covering the last 540 million years. It covers everything from the Cambrian up to the present, which is most of the period in which complex life has been on Earth. Specifically, they looked at what is known as delta-O-18 measurements, which is a measurement of the oxygen 18 isotope relative to oxygen 16 within calcium carbonate deposits. These deposits were made by shelled organisms. Since the evaporation of water prefers O16 over O18 due to its smaller mass, delta-O-18 provides an indicator for geologic temperatures.
The team looked at 24,000 delta-O-18 measurements covering the Phanerozoic, and looked for a correlation between O18 levels and the position of the Sun relative to the galactic plane. What they found was a correlation with a confidence of 99.9%. So it seems fairly clear that our galactic position has had an effect on geologic temperatures. What isn’t clear is what could cause such a variation. The authors suggest that the motion may result in a variation of gamma rays striking the upper atmosphere, which could lead to changes in atmospheric temperature. At this point that it still pretty speculative.
Just to be clear, this paper looked at variations over long geologic scales. The motion of the Sun through the galaxy and any resulting temperature variation has no effect on the current warming trend we observe due to rising CO2 levels. Global warming, as it is often called, is not a galactic effect.
Paper: Nir J. Shaviv, et al. Is the Solar System’s Galactic Motion Imprinted in the Phanerozoic Climate? Scientific Reports 4, Article number: 6150 (2014)<|endoftext|>
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Are you feeling difficulty in calculating the elapsed time? Don’t worry we are here to help you to overcome the difficulties in calculating the time. Before that, you have to know what the elapsed time is. Elapsed time is the calculation of time that passes from starting off a program till its end.
It is very important to know how to calculate the elapsed time to know time. There are different techniques to calculate the elapsed time. One method is the time interval between the start of an event to the end of the event. Another method is the number line on which we break up the time intervals. We suggest the students of 5th grade go through the article and solve the given problems. Learn the concept with suitable examples and score good marks in the exams.
See More: Adding and Subtracting Time
## What is Elapsed Time in Math?
Elapsed time is the amount of time duration from the start of the event to the finish of the event. In simple words, the elapsed time is the time that goes from one time to another time.
### How to Calculate Elapsed time?
1. For solving the elapsed time first we would find the starting time and ending time.
2. Second, counts the hours and minutes between the starting point to noontime and from noontime to ending time.
3. The third is finding out the elapsed time by adding durations.
### Elapsed Time Examples
Example 1.
Find the elapsed time from 7.00 am to 8.00 pm?
Solution:
Given that,
Starting time = 7.00 am
Finishing time = 8.00 pm
The difference between 7.00 am to 12 noon = 5 hours
The difference between 12 noon to 8.00 pm = 8 hours
Duration time = 5 hours + 8 hours = 13 hours.
Duration time is 8 hours
Example 2.
Find the elapsed time from 7 hours 30 minutes to 3 hours 20 minutes?
Solution:
7 hours 30 minutes – 3 hours 20 minutes
7 hours – 3 hours = 4 hours
30 minutes – 20 minutes = 10 minutes
The elapsed time = 4 hours + 10 minutes = 4 hours 10 minutes
Example 3.
What time would it be 3 hours 30 minutes after 8 am?
Solution:
Given that,
Present time = 3 hours 30 minutes
After time = 8 am = 8 hours
Therefore 3 hours 30 minutes + 8 hours
Time would be 11 hours 30 minutes.
Example 4.
The starting time is 11.25 am and the finishing time is 3.40 pm. What is the duration of the time?
Solution:
Given that,
Starting time = 11.25 am
Finishing time = 3.40 pm
The difference between 11.25 am to 12 noon = 35 minutes
The difference between 12 noon to 3.40 pm = 3 hours 40 minutes.
Duration time = 35 minutes + 3 hours 40 minutes
35 minutes + 40 minutes = 1 hour 15 minutes
3 hours + 1 hour 15 minutes = 4 hours 15 minutes
Therefore, the Duration time is 4 hours 15 minutes.
Example 5.
If the movie starts at 4 pm and ends at 8 pm. How long is the movie?
Solution:
Given that,
Starting time of the movie = 4 pm
Finishing time of the movie = 8 pm
Duration of the movie time = 4 pm – 8 pm = 4
Thus, the duration of the movie is 4 hours.
### FAQs on Elapsed Time
1. What is elapsed time?
An elapsed time is the amount of time taken to travel or to start from one place to another place.
2. How do you calculate elapsed time?
1. First count in minutes from earlier time to the nearest hour.
2. Count on hours to the hour nearest to the later time.
3. Then count in minutes to reach the later time.
3. How do you subtract elapsed time?
In order to subtract the time, subtract the minutes and then subtract the hours.<|endoftext|>
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In the short story, “The Fall of the House of Usher,” by Edgar Allen Poe, setting is used extensively to do many things. The author uses it to convey ideas, effects, and images. It establishes a mood and foreshadows future events. Poe communicates truths about the character through setting. Symbols are also used throughout to help understand the theme through the setting. Poe uses the setting to create an atmosphere in the reader’s mind. He chose every word in every sentence carefully to create a gloomy mood. For example, Usher’s house, its windows, bricks, and dungeon are all used to make a dismal atmosphere.
The “white trunks of decayed trees,” the “black and lurid tarn,” and the “vacant, eyelike windows” contribute to the collective atmosphere of dispair and anguish. This is done with the words black, lurid, decayed, and vacant. The narrator says that the Usher mansion had “an atmosphere which had no affinity with the air of heaven. ” It was no where near being beautiful, holy, or clean. He uses descriptive words such as decayed, strange, peculiar, gray, mystic, Gothic, pestilent, dull and sluggish to create the atmosphere. Poe’s meticulous choice of words creates a very effective atmosphere in the story.
Another important way Poe uses the setting is to foreshadow events in the story. Roderick Usher’s mansion is on example of this. There is a “barely perceptible fissure” in the masonry. It is a small crack in “The House of Usher” which the narrator defines as “both the family and the family mansion. ” This foreshadows an event that will ruin the house and the family. The fissure divides the house. Roderick and Madeline die, destroying the family. The narrator says there is a “wild inconsistency between [the masonry’s] still perfect adaptation.. and the crumbling condition of the individual stones.
This is also symbolic. The stones represent the individual people of the Usher family, and the entire mansion stands for the whole family. The “wild inconsistency” makes the reader aware that something later in the story will make the inconsistency” clear or consistent. From far away, no one knows that the House of Usher is in despair. The “fabric gave little token of instability”– or the mansion itself did not tell of the turmoil it concealed. The story takes place in autumn, a season associated with death.
When the story’s tension is about to reach its crescendo, a storm comes up, a “rising tempest. This is a symbol for the “tempest” brewing in Roderick Usher’s mind. Poe’s use of foreshadowing is just enough to clue the reader into what will happen, but not enough to give it away. Character traits are displayed through how the setting affects, influences, and reveals the characters.
The narrator is affected by the gloomy atmosphere of the Usher mansion. He is “sucked in” to Usher’s “dream world,” the world he created after living alone in his dismal house for years. Usher’s house itself is a symbol for Usher. It is isolated like Usher. There are many “intricate passages,” like the many facets of his mind.
One of the rooms had windows which “feeble gleams of encrimsoned light… served to render sufficiently distinct the more prominent objects around. ” The windows stand for Usher’s eyes, the light is reality. He lives in his own world he created. Reality enters his brain only in “feeble gleams of light. ” “The eye… struggles in vain to reach the remoter angles of the chamber.. ” The reality does not reach all of his brain. These quotes show that Usher is only half in the real world, half in his own world. The books Usher read, his art, and music all reveal his personality. He played “long improvised dirges” on the guitar.
The narrator describes his painting as “phantasmagoric. ” The books he reads are about death, magic, mysticism, the occult, and torture. His favorite is a book of vigils for the dead. All these things show that Usher is unstable and obsessed with death. Through the setting, Edgar Allan Poe is able to foreshadow events, establish an atmosphere, and reveal character traits. Although the reader may not notice all the numerous devices used, they contribute to giving the story depth. Noticed or not, Poe utilizes the setting to its’ full capacity to create the mood, characters and foreshadowing.<|endoftext|>
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# Simple Pendulum
Goal: To determine the value of the gravitational acceleration by using a simple pendulum.
• Theory: A simple pendulum consists of an object with negligible size hanging on a string. When the object is deflected from its equilibrium, it oscillates back and forth. The time for one complete oscillation is called the period of oscillations, T. For small angles of deflection, the period of the simple pendulum is given by the following formula:
By re-arranging the above equation,
,
we see that the graph T2 vs. L should be a straight line with a slope:
Thus, the gravitational acceleration can be determined by the slope of the T2 vs. L graph. In order to increase the accuracy of the final result, for any given length, L, the period T must be measured several times. The average of these value will be T for that particular L. We repeat the procedure for several different lengths and plot the T2 vs. L graph.
• Preliminary Setup:
• You need:
• A bead and a long string
• A measuring tape (or ruler) and a watch. If you don't have a watch, you can use this Java Applet Stopwatch.
• Scissors, a pencil and a scotch tape
• Experimental setting:
• Cut approximately 1 meter long string.
• Tie one end of the string to the bead and the other to the pencil.
• Tape the pencil onto a table or any other surface, so that the bead is hanging freely. This is your simple pendulum.
• Activity 1: Determine the value of the gravitational acceleration.
• Determine the period of oscillations for a given length.
• Measure the length of the string from the pencil end to the bead. Write down the length in your lab notebook.
• Pull the bead approximately 5 cm to the side from its equilibrium position and let it go.
• Measure the time for 20 oscillations, t, and write it down
• Repeat the procedure seven times and record your data in your lab notebook
• Average the seven values and determine the period of oscillations by dividing the time t by 20.
• Measure the period for different lengths.
• Decrease the length of the string by approximately 10 cm. Measure the new length, L, and write it down.
• Measure the period for the new length following the outlined procedure above.
• Repeat the steps until you have determined the period of oscillations for 5 different lengths.
• Record the data in the following format:
L (m) t1 (s) t2 (s) t3 (s) t4 (s) t5 (s) t6 (s) t7 (s) tav (s) Tav (s) Tav2 (s)2
• Results:
• Plot T2 vs. L. You can directly plot the graph on a Graphing Paper or you can use a spreadsheet or any other program.
• Find the slope of the graph. You can use your calculator, a spreadsheet, or you can go to this website for a quick calculation of the slop using the linear regression methods. If you choose the latter, clear the data and type in your own data. The slope of the line is given by "m" in the box below the graph.
• Determine the gravitational acceleration g from the slope. Calculate the percent error, 100% . |gmeasured - 9.8|/(9.8)
Acknowledgements.
Back to: Physics Lab Home T. Stantcheva Home NVCC Last modified: Thu Jun 2 16:54:19 EDT 2011<|endoftext|>
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In humans and other animals, glands are tissues or organs that produce substances that are necessary for the functioning of other tissues or organs. They remove specific substances from the blood, change or concentrate them, and then either release them for further use or eliminate them. This article discusses the glands of the human body.
Glands are classified by the manner in which they secrete their substances. There are two main types: exocrine and endocrine. Exocrine glands secrete substances onto an external or internal body surface, typically through a small duct, or tube. For instance, sweat glands release sweat through ducts onto the surface of the skin. Salivary, lachrymal (tear), and intestinal glands are other examples of exocrine glands.
Endocrine glands are ductless. They secrete substances called hormones into the fluid that surrounds the body’s cells. The hormone may act on nearby tissue, or the bloodstream or lymph system may pick the hormone up and transport it to the target organ or tissue. The adrenals, thyroid, parathyroid, pituitary, hypothalamus, pineal, and ovaries are endocrine glands. Mixed exocrine and endocrine glands, which secrete in both ways, include the liver, testes, and pancreas.
The volume of the substances produced varies. Endocrine glands release extremely small amounts because hormones are powerful substances. Digestive secretions such as saliva are produced in larger volumes.
Glands become active in response to specific stimulation that may occur near the gland or some distance away. The activity of sweat glands, for example, depends largely on the temperature of adjacent skin, but the growth of egg cells in the ovaries is regulated by a hormone from the pituitary gland on the underside of the brain.
Several types of exocrine gland are found in the skin. The basic type of sweat gland secretes sweat, or perspiration, onto the surface of the skin when the body temperature rises. Sweat consists mainly of water. As it evaporates, the skin is cooled. Another type of sweat gland, concentrated in the underarms and genital area, secretes small amounts of a thick, fatty substance into hair follicles. As bacteria break down this secretion, a pungent odor may be produced. Sebaceous glands secrete an oily substance called sebum, usually into hair follicles. The sebum helps keep the skin flexible and prevents too much water from being lost or absorbed. The sebaceous glands are found in the skin all over the body except on the palms of the hands and the soles of the feet. They are most numerous on the face and scalp. In the skin of the ear canal, special glands produce a waxy substance called cerumen, or earwax. Earwax is thought to protect the ears by trapping foreign particles and slowing the growth of microbes.
The lachrymal glands produce tears, a watery substance that moistens and protects the eyes. The main glands are located above the outer corner of each eye, behind the upper eyelid.
The mammary glands are located in the chest and are normally fully developed and functional only in females. After childbirth, hormones cause these glands to produce milk, with which the mother can feed the child.
The digestive system has numerous exocrine glands. Humans have three major pairs of salivary glands, which secrete a fluid called saliva into the mouth. Saliva has many functions, including moistening and softening food. Glands in the stomach and intestines produce digestive juices, which include enzymes that break down food. They also secrete mucus, which protects the organs themselves from being digested by the digestive juices. Goblet cells in the intestines make mucin, the main component of mucus. The exocrine glands of the pancreas also produce digestive enzymes, and the liver produces bile, which aids in the digestion of fats.
The goblet cells are unique in that each one consists of only a single cell. Most exocrine glands are multicellular. They may have a simple tubelike duct or one that is coiled or has many branches. Some glands have many ducts.
Exocrine glands may secrete their product continuously, periodically, or only once. The substance secreted may be oily, watery, gelatinous, or granular. The very act of secreting can cause changes in the gland’s cells. Merocrine glands, such as basic sweat glands, lose no cytoplasm, so the gland’s cells remain intact. However, the cells of holocrine glands, such as sebaceous glands, completely disintegrate and are extruded along with the secretion. The dead, secreted cells are then replaced with new ones. In apocrine glands, such as mammary glands and underarm sweat glands, the secretions gather at the tips of the cells and are released by being pinched off.
The activities of the endocrine glands form one of the most complex systems in the body. Although each gland has its own unique function, the glands of the endocrine system are interdependent, and the function of one depends on the activity of another. The hypothalamus produces several hormones, including those that regulate pituitary activity. The pituitary gland produces its own hormones that regulate growth and stimulate other endocrine glands. The adrenal glands, thyroid gland, testes, and ovaries are dependent upon pituitary stimulation. The hormones these glands produce govern metabolism, blood pressure, water and mineral balance, and reproductive functions, and they help defend against injury.
The term hormone is derived from a Greek word meaning “to stir up.” Hormones act as chemical messengers in creating a communication chain that links the body systems together, thus controlling and integrating the functions of the body.
Some endocrine glands are coordinated on the principle of negative feedback, in which the rise or fall of one hormone can trigger an increase or decrease of another. For example, increased adrenocorticotropic hormone (ACTH) causes the adrenal glands to release more cortisol, which raises blood sugar levels. This in turn causes ACTH levels to decrease. Other substances in the bloodstream also affect hormone secretion. The amount of calcium in the blood, for example, regulates the release of parathormone (PTH) from the parathyroid glands, and the amount of sugar in the blood regulates insulin release. (For a fuller discussion of the endocrine system, see hormones.)
Disorders of the endocrine system can result from a gland producing too much or too little of a hormone or from abnormal reactions to a hormone. Given the complex interaction of the endocrine glands, a disorder of one gland often affects other glands and the entire functioning of the body. Diabetes mellitus, for example, is the result of insulin deficiency, which causes high levels of blood sugar. Water is not properly processed by the kidneys when insulin is deficient, so frequent urination causes dehydration and excessive thirst. In addition, weight loss from the breakdown of fat stores causes excessive hunger. In contrast, a disorder of an exocrine gland is usually limited and seldom affects other glands or distant organs. An exception is cystic fibrosis, a genetic disorder in which the exocrine glands lining the pancreas and lungs produce thick mucous secretions that interfere with normal functioning of these organs.<|endoftext|>
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Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :
# Isosceles Triangle Theorems
You may have already learnt about the properties and types of triangles. One of the special types of a triangle is the isosceles triangle. An isosceles triangle is a triangle which has two equal sides, no matter in what direction the apex (or peak) of the triangle points. Some pointers about isosceles triangles are:
• It has two equal sides.
• It has two equal angles, that is, the base angles.
• When the third angle is 90 degree, it is called a right isosceles triangle.
In this article, we have given two theorems regarding the properties of isosceles triangles along with their proofs.
## Isosceles Triangle Theorems and Proofs
Theorem 1: Angles opposite to the equal sides of an isosceles triangle are also equal.
Proof: Consider an isosceles triangle ABC where AC = BC.
We need to prove that the angles opposite to the sides AC and BC are equal, that is, ∠CAB = ∠CBA.
We first draw a bisector of ∠ACB and name it as CD.
Now in ∆ACD and ∆BCD we have,
AC = BCÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (Given)
∠ACD = ∠BCD                                                   (By construction)
CD = CDÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (Common to both)
Thus, ∆ACD ≅∆BCD                     (By SAS congruence criterion)
So, ∠CAB = ∠CBA                       (By CPCT)
Hence proved.
Theorem 2: Sides opposite to the equal angles of a triangle are equal.
Proof: In a triangle ABC, base angles are equal and we need to prove that AC = BC or ∆ABC is an isosceles triangle.
Construct a bisector CD which meets the side AB at right angles.
Now in ∆ACD and ∆BCD we have,
∠ACD = ∠BCD                                                   (By construction)
CD = CD Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (Common side)
∠ADC = ∠BDC = 90°                      (By construction)
Thus, ∆ACD ≅ ∆BCD                     (By ASA congruence criterion)
So, AC = BC Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (By CPCT)
Or ∆ABC is isosceles.<|endoftext|>
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# Determinant $\mathbb{I}+\vec{b}\vec{b}^T$.
I have to solve the determinant $$\det(\mathbb{I}+\vec{b}\vec{b}^T)$$. (The result shloud be 1.)
We can use the sum rule for rows to get.
$$\det(\mathbb{I}+\vec{b}\vec{b}^T)= \begin{vmatrix} 1 & 0 & \ldots& 0\\ b_1 b_2 & & & \\ \vdots & & (\mathbb{I}+\vec{b}\vec{b}^T)_{(2\ldots n)\times (2\ldots n)} &\\ b_1 b_n & & & \end{vmatrix} + \begin{vmatrix} b_1^2 & b_2 b_1 & \ldots& b_n b_1\\ b_1 b_2 & & & \\ \vdots & & (\mathbb{I}+\vec{b}\vec{b}^T)_{(2\ldots n)\times (2\ldots n)} &\\ b_1 b_n & & & \end{vmatrix}=:\det(A)+\det(B),$$ where $$(\mathbb{I}+\vec{b}\vec{b}^T)_{(2\ldots n)\times (2\ldots n)}$$ is the submatrix with the first row and column deleted. Using the Laplace decomposition formula on $$\det (A)$$, we have the same problem with reduced dimension, so $$\det(A)=1$$ by induction, if $$\det(B)=0.$$
So we have to show that $$\det(B)=0.$$ Using the sum rule again, we have $$\det(B)=\begin{vmatrix} b_1^2 & b_2 b_1 & b_3 b_1& \ldots& b_n b_1\\ b_1 b_2 & 1 &0 & \ldots& 0\\ b_1 b_3&b_2 b_3&&\\ \vdots & \vdots & & (\mathbb{I}+\vec{b}\vec{b}^T)_{(3\ldots n)\times (3\ldots n)} &\\ b_1 b_n & b_2 b_n & & \end{vmatrix}+\begin{vmatrix} b_1^2 & b_2 b_1 & b_3 b_1& \ldots& b_n b_1\\ b_1 b_2 & b_2^2 &0 & \ldots& 0\\ b_1 b_3&b_2 b_3&&\\ \vdots & \vdots & & (\mathbb{I}+\vec{b}\vec{b}^T)_{(3\ldots n)\times (3\ldots n)} &\\ b_1 b_n & b_2 b_n & & \end{vmatrix}$$ It is clear that the second term is zero, as the first two columns are linearly dependent (multiples of each other). But I do not see why the first term should vanish.
Did I make a mistake or is the first term in the second equation vanishing?
• Have you tried a basis in which $b_i=|b|\delta_{i1}$?
– J.G.
Commented Oct 17, 2021 at 8:56
• Commented Oct 17, 2021 at 9:22
• by matrix determinant lemma, $\det(I+bb^T) = 1 + b^Tb$. Commented Oct 17, 2021 at 20:35
I will assume that $$b \in \mathbb{R}^n$$.
The result is not necessarily $$1$$. For example, take $$b = e_1$$. Then $$I + e_1e_1^T = \text{diag}(2, 1, 1, \dots, 1)$$ so $$\det(I + e_1e_1^T) = 2$$.
I'll give a hint: To find $$\det(I + bb^T)$$, it is convenient to diagonalize $$I + bb^T$$ first. $$I + bb^T$$ is easily diagonalized: Pick a basis $$\{v_2, v_3, \dots, v_n\}$$ of $$\text{span}(b)^\perp$$. Now consider the basis $$\{b, v_2, \dots, v_n\}$$ of $$\mathbb{R}^n$$. What is the matrix representation of $$I + bb^T$$ with respect to this basis?
• $bb^t$ is not necessarily diagonalizable if $b\in\mathbb C^n$, e.g. $b = (1, i)$ (or $(1,1)\in\mathbb F_2^2$ if finite fields are allowed). But the the last eigenvalue can always be computed using the trace. Commented Oct 17, 2021 at 20:20
• @Justauser Yes I am assuming $b \in \mathbb{R}^n$. I will add that to the post. Commented Oct 17, 2021 at 20:26
• This is in fact OK for me, I am working with a matrix in O(n). The result seems to be $1+|b|^2$, as this is the eigenvalue when applied to $b$, the remaining entries are $1$. Commented Oct 17, 2021 at 20:33<|endoftext|>
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Notes on Algebraic Expressions | Grade 7 > Compulsory Maths > Algebraic Expressions | KULLABS.COM
Notes, Exercises, Videos, Tests and Things to Remember on Algebraic Expressions
Please scroll down to get to the study materials.
• Note
• Things to remember
• Videos
• Exercise
• Quiz
#### Algebraic Expression
Generally, Algebraic expressions are the symbol or a combination of symbols used in algebra containing one or more numbers, variables, and arithmetic operations. 2xyz, 2x +yz, 2x + y, etc. are the examples of Algebraic Expressions.
Evaluation of Algebraic Expressions
When we replace the variable of a term or expression with numbers, the value of the terms or expression is obtained. It is called the evaluation of a term or expression. For examples,
If x = 3, y = 4 and z = 2, then
2xyz = 2 × 3 × 4 × 2 = 48
Law of Indices
While performing the operations of multiplication and division of algebraic expression it needs to work out indices of the same bases under the certain rules. These are called the law of indices.
1. Product law of indices
In product law of indices, when the same base is multiplied we should add their indices. When am and an are two terms then aa + n. For examples,
2 × 2 = 21 + 1 = 22
2×2 = 24+2 = 26
2. Quotient law of indices
In quotient law of indices, when the base is divided by another same base, power should be subtracted. Like, am ÷ an = am-n. For examples,
32÷3 = $$\frac{3^2}{3}$$ =$$\frac{3×3}{3}$$ = 3 = 32-1
3. Power law of indices
In power law of indices, when a base with some power has another power, the powers are multiplied. For examples,
(22)2 = 22 × 22= 22+2= 24
4. Law of Zero indices
In this law of indices, the value of a base with power 0 is always1. If a0 is any term with a base a and power 0, then a0 = 1. For example,
20 = 1, 50 = 1
Addition and Subtraction of algebraic expression
To add and subtract the algebraic expression, there should be the like and unlike terms. Unlike terms are those which do not have the same base and like terms are those which have the same base. While adding and subtracting algebraic term, we should add or subtract the coefficients of
like terms. For example,
1. 2x + 3x = 5x
2. 7x - 5x = 2x
Multiplication of algebraic expressions
When the coefficients of the terms of the terms are multiplied and the power of the same bases are added then it is called Multiplication of algebraic expressions. For example,
1. 2x2×3x = 6x3
2. 3a2×3a2 = 9a4
Multiplication of polynomials by monomials
In this case, in each term of polynomials is separately multiplied by the monomial. For example
Multiply, (b+c) by x.
Here, x× (b+c) = bx +cx
Multiplication of polynomials
In this case, each term of polynomials is separately multiplied by each term of another polynomial. Then, the product is simplified. For examples,
(a+b) by (x+y)
Here, (x+y)(a+b)
or, x(a+b) = y(a+b)
or, ax+bx+ay+by
Some special products formulae
1. The product of (a+b) and (a+b) (square of binomials).
Let's multiply (a+b) by (a+b)
(a+b)×(a+b) = a(a+b) + b(a+b)
(a+b)2 = a2 + 2ab + b2
= a2 + 2ab + b2
Thus, (a+b)2 = a2 +2ab +b2, then
a2 + b2 = (a+b)2 - 2ab
2. The product of (a-b) and (a-b)
Let's multiply (a-b) by (a-b)
(a-b)×(a-b) = a(a-b) - b(a-b)
(a-b)2 = a2 - ab - ab + b2
= a2 - 2ab + b2
THus, (a-b)2 = a2 - 2ab + b2
Here, if (a-b)2 = a2 - 2ab + b2, then,
a2 + b2 = (a-b)2 + 2ab
3. The product of (a+b), (a+b) and (a+b) (cube of binomials)
Lets, find the products of (a+b)3
(a+b)3 = (a+b) (a+b) (a+b)
(a+b) (a+b)2
(a+b)(a2+2ab+b2)
a(a2+2ab+b2) + b(a2+2ab+b2)
a3 + 2a2b +ab2 + a2b + 2ab2 +b3
a3 +3a2b + 3ab2 + b3
∴(a+b)3 = (a3 +3a2b + 3ab2 + b3)
4. The product of (a-b), (a-b) and (a-b)
Lets, find the products of (a-b)3
(a-b)3 = (a-b) (a-b) (a-b)
(a-b) (a-b)2
(a-b)(a2-2ab+b2)
a(a2-2ab+b2) - b(a2-2ab+b2)
a3- 3a2b +ab2 + a2b + 2ab2- b3
a3- 3a2b + 3ab2- b3
∴(a-b)3 = (a3-3a2b + 3ab2- b3)
We can express these formulaes in other following forms
we have (a3 +3a2b + 3ab2 + b3) =(a+b)3
a3 + b3 + 3ab (a+b) =(a+b)3
a3 + b3 =(a+b)3 - 3ab (a+b)
∴ a3 + b3 =(a+b)3- 3ab (a+b)
Also,(a3-3a2b + 3ab2- b3) = (a-b)3
a3 - b3 - 3ab(a-b) = (a-b)3
a3 - b3 - 3ab(a-b) = (a-b)3 + 3ab(a-b)
∴ a3- b3 =(a-b)3+ 3ab (a-b)
5. The product of (a+b) and (a+b)
Let's multiply (a-b) by (a-b)
(a+b)(a-b) = a(a-b) + b(a-b) = a2 - ab + ab - b2 = a2 - b2
Thus, (a+b) (a-b) = a2 -b2
Division of algebraic expressions
While dividing a monomial by another monomial, divide the coefficient of dividend by the coefficient of of divisor. Then substract the power of the base of divisor from the power of the same base of dividend. For examples,
18x4y3 by 6x2y2
or, 18x4y3÷ 6x2y2 = $$\frac{18x^4y^3}{6x^2y^2}$$
or, 3x4-2y3-2 = 3x2y
Division of polynomials by monomials
In this case each term of a polynomial is separately dividend by the monomial. For example,
(12x4 - 15x3)÷ 3x2
= $$\frac{24x^4}{3x^2}$$ - $$\frac{15x^3}{3x^2}$$
= 4x4-2 - 5x3-2
= 4x2 - 5x
Division of polynomials by polynomials
In this case, at first we should arrange the terms of divisor and dividend in descending or ascending order of power of common bases. Then we should atart the division dividing the term of dividend with the highest power.
• (a+b)2 = a2 + 2ab + b2
• a2 + b2 = (a+b)2 - 2ab
• (a-b)2 = a2 - 2ab + b2
• a2 + b2 = (a-b)2 + 2ab
• (a+b)3 = (a3 +3a2b + 3ab2 + b3)
• a3 - b3 = (a-b)3 + 3ab (a-b)
• (a+b) (a-b) = a2 -b2
.
### Very Short Questions
Divisor Dividend 2 1296 2 648 2 324 2 162 3 81 3 27 3 9 3
$$\therefore$$ 1296 = 2×2×2×2×3×3×3×3
= 24×34
=(2×3)4
= 6 ans.
Solution:
($$\frac{16}{81}$$)$$\frac{1}{4}$$
= ($$\frac{2^4}{3^4}$$)$$\frac{1}{4}$$
= ($$\frac{2}{3}$$)4×$$\frac{1}{4}$$
= ($$\frac{2}{3}$$)1
= $$\frac{2}{3}$$
Solution:
= (4x - 7) (2x2 + 3x - 5)
= 4x(2x2 + 3x - 5) - 7 (2x2 + 3x - 5)
= 8x3 + 12x2 - 20x - 14x2 - 21x + 35
= 8x3 - 2x2 - 41x + 35
Solution:
Here, (a+b) = 2
$$\therefore$$ (a+b)3 = 23
or, a3 + 3a2b + 3ab2 + b3 = 8
or, a3 + b3 + 3ab2 = 8
or, a3 + b3 + 6ab = 8
So, the required value of a3 + b3 + 6ab is 8.
0%
-140
-180
-160
-120
12
8
6
6x
5x
4x
3x
-5pq
15pq
-15pq
5pq
• ### What will be the final products of (a - b)3.
a3 + 3a2b + 3ab2 + b3
a3 - 3a2b + 3ab2 - b3
a3 - 3a2b - 3ab2 - b3
a3 - 3a2b + 3ab2 + b3
8
16
12
20
4x
2x
6x
5x
67
69
68
66
• ### Find the squares of the (y - 3).
y2 + 9
(y - 3)2
y- 6y + 9
(y + 3) (y - 3)
18.32cm2
23.12 cm2
16.36 cm2
21.16 cm2
## ASK ANY QUESTION ON Algebraic Expressions
No discussion on this note yet. Be first to comment on this note<|endoftext|>
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# Probability Fraction Calculator
Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD
Last updated: Jan 18, 2024
Following the definition of probability, we can easily calculate probability as a fraction: with our tool, it will be super easy, barely an inconvenience. If you need to calculate the probability as a fraction for multiple events, you are in the right place! Keep reading for a quick explanation of the math behind the calculations, examples, and applications of the fractional representation of probability.
## What is the probability of an event?
The probability of an event is the measure of the frequency with which said event happens out of a total possible amount of outcomes. If you are dealing with coin tosses, for example, you may find out that head is the result in $495$ out of $1000$ tosses.
There are many ways to express probability, but in general, they all stem from its representation as the ratio between the occurrences of a given outcome and the total number of events happening:
$n_{\mathrm{outcome}}:n_{\mathrm{total}}$
We are used to seeing this ratio expressed as a decimal number, the result of the division of the two members, or as a percentage (the same result, multiplied by $100$. There's, however, an additional way to express probability, and it may come in handy in specific situations: in the next section, we will learn how to calculate probability in fraction form.
## Probability as a fraction
To express probability as a fraction, simply write the number of events that resulted in the desired outcome as the numerator of the fraction and the total number of realizations as the denominator.
You can easily calculate the fraction form of probability with the following formula:
$P(\mathrm{A}) = \frac{n_{\mathrm{A}}}{n_{\mathrm{total}}}$
Where:
• $P(\mathrm{A})$ — The probability of the outcome $\mathrm{A}$;
• $n_{\mathrm{A}}$ — The number of times the event had outcome $\mathrm{A}$; and
• $n_{\mathrm{total}}$ — The total number of events from which we consider the selected outcome.
To calculate the probability as a fraction, follow these steps:
1. Find the number of outcomes and the total number of repetitions.
2. Write the number of outcomes as the numerator and the total number of repetitions as the denominator.
3. Calculate the greatest common divisor of these two numbers. Visit Omni's GCF calculator if you need a refresh on the topic!
4. If the GCF is larger than $1$, divide both the numerator and the denominator of the probability fraction by its value: you will obtain the reduced form of the probability.
## How do I calculate probability as a fraction: the case of multiple events.
In case of multiple outcomes of an event (think of a die and its six faces), we can still calculate the probability as a fraction; however, we need to introduce some small modifications to the process!
Say that you are dealing with an event with possible outcomes $\mathrm{A}$, $\mathrm{B}$, and $\mathrm{C}$. Each of the outcomes happened with the following results:
$\begin{split} \mathrm{A}& \rightarrow n_{\mathrm{A}}\\ \mathrm{B}& \rightarrow n_{\mathrm{B}}\\ \mathrm{C}& \rightarrow n_{\mathrm{C}}\\ \end{split}$
If we sum the occurrences, we find the total number of "realizations":
$n_{\mathrm{total}} = n_{\mathrm{A}}+ n_{\mathrm{B}}+ n_{\mathrm{C}}$
Then, knowing that probability can be a fraction, for the first outcome we write the following expressions:
$P(\mathrm{A}) = \frac{n_{\mathrm{A}}}{n_{\mathrm{total}}}$
For the second and third outcomes, we have, respectively:
$P(\mathrm{B}) = \frac{n_{\mathrm{B}}}{n_{\mathrm{total}}}$
And:
$P(\mathrm{C}) = \frac{n_{\mathrm{C}}}{n_{\mathrm{total}}}$
As a fundamental rule, if the considered outcomes span all the realizations of the event, the following rule holds:
$P(\mathrm{A})+P(\mathrm{B})+P(\mathrm{C})=1$
Calculating the probability of multiple events as a fraction is closely related to another way to represent the same quantities! Think about it: if your outcomes span all the possible events, then their fraction will sum to unity. If we compare unity to a full angle, we can represent the partition of the outcomes as sectors in a pie chart. To understand this comparison even better, visit our three related tools:
With a graphical representation, the analogy will be clear!
## FAQ
### Can probability be a fraction?
Yes: since we define probability as the ratio between the number of events that resulted in a given outcome and the total number of events, we can write these two numbers as the numerator and denominator of a fraction. The fractional representation of probability gives us a quick indication of the magnitude of the probability since its value easily compares to the unit fraction 1.
### How do I find probability in fractions?
To calculate probability in fraction form, follow these easy steps:
1. Find the number of events that resulted in the desired outcome. We call this number nA.
2. Find the total number of realizations, nTOT.
3. Define the fractional form of the probability of the event A as:
P(A) = nA/nTOT
4. Calculate the greater common factor of nA and nTOT. If it's different from 1, divide both numbers by the factor, and find the reduced form of the fraction.
### What is the fraction form of the probability of the results of a coin toss?
1/2 for heads and 1/2 for tails. What do these numbers mean? Take the first fraction:
• The 2 at the denominator is the total number of tosses;
• The 1 at the numerator is the number of tosses resulting in heads.
However, getting two tails or two heads is not so unlikely. Try tossing 1000 times. Heads may be the result of, say, 504 tosses. The fraction 504/1000 is similar in value to 1/2. Repeating the event gives a more accurate fractional representation of the probability.
### How do I calculate the fractional form of the probability of multiple events?
To calculate the fractional form of the probability of multiple events:
1. Define the number of events with defined outcomes: nA, nB, nC, and so on.
2. Calculate the number of total events: nTOT = nA + nB + nC + ...
3. Write the desired number of fractions in the form nA/nTOT, nB/nTOT, nC/nTOT, etc.
4. Simplify the fractions, if possible, using the greater common factor between the respective numerator and the denominator.
Davide Borchia
Enter values in each group
Outcome A
Outcome B
Outcome C
Outcome D
Outcome E
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# Lecture 5 Principal Minors and the Hessian
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1 Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Principal minors Principal minors Let A be a symmetric n n matrix. We know that we can determine the definiteness of A by computing its eigenvalues. Another method is to use the principal minors. Definition A minor of A of order k is principal if it is obtained by deleting n k rows and the n k columns with the same numbers. The leading principal minor of A of order k is the minor of order k obtained by deleting the last n k rows and columns. For instance, in a principal minor where you have deleted row 1 and 3, you should also delete column 1 and 3. Notation We write D k for the leading principal minor of order k. There are ( ) n k principal minors of order k, and we write k for any of the principal minors of order k. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
2 Principal minors Two by two symmetric matrices Let A = ( ) a b b c be a symmetric 2 2 matrix. Then the leading principal minors are D 1 = a and D 2 = ac b 2. If we want to find all the principal minors, these are given by 1 = a and 1 = c (of order one) and 2 = ac b 2 (of order two). Let us compute what it means that the leading principal minors are positive for 2 2 matrices: Let A = ( ) a b b c be a symmetric 2 2 matrix. Show that if D1 = a > 0 and D 2 = ac b 2 > 0, then A is positive definite. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Principal minors Leading principal minors: An example If D 1 = a > 0 and D 2 = ac b 2 > 0, then c > 0 also, since ac > b 2 0. The characteristic equation of A is λ 2 (a + c)λ + (ac b 2 ) = 0 and it has two solutions (since A is symmetric) given by λ = a + c 2 ± (a + c) 2 4(ac b 2 ) 2 Both solutions are positive, since (a + c) > (a + c) 2 4(ac b 2 ). This means that A is positive definite. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
3 Principal minors Definiteness and principal minors Theorem Let A be a symmetric n n matrix. Then we have: A is positive definite D k > 0 for all leading principal minors A is negative definite ( 1) k D k > 0 for all leading principal minors A is positive semidefinite k 0 for all principal minors A is negative semidefinite ( 1) k k 0 for all principal minors In the first two cases, it is enough to check the inequality for all the leading principal minors (i.e. for 1 k n). In the last two cases, we must check for all principal minors (i.e for each k with 1 k n and for each of the ( n k) principal minors of order k). Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Principal minors Definiteness: An example Determine the definiteness of the symmetric 3 3 matrix A = One may try to compute the eigenvalues of A. However, the characteristic equation is det(a λi ) = (1 λ)(λ 2 8λ + 11) 4(18 4λ) + 6(6λ 16) = 0 This equations (of order three with no obvious factorization) seems difficult to solve! Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
4 Principal minors Definiteness: An example (Continued) Let us instead try to use the leading principal minors. They are: D 1 = 1, D 2 = = 14, D = = 109 Let us compare with the criteria in the theorem: Positive definite: D 1 > 0, D 2 > 0, D 3 > 0 Negative definite: D 1 < 0, D 2 > 0, D 3 < 0 Positive semidefinite: 1 0, 2 0, 3 0 for all principal minors Negative semidefinite: 1 0, 2 0, 3 0 for all principal minors The principal leading minors we have computed do not fit with any of these criteria. We can therefore conclude that A is indefinite. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Principal minors Definiteness: Another example Determine the definiteness of the quadratic form Q(x 1, x 2, x 3 ) = 3x x 1 x 3 + x 2 2 4x 2 x 3 + 8x 2 3 The symmetric matrix associated with the quadratic form Q is A = Since the leading principal minors are positive, Q is positive definite: D 1 = 3, D 2 = = 3, D = = 3 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
5 Optimization of functions of several variables The first part of this course was centered around matrices and linear algebra. The next part will be centered around optimization problems for functions f (x 1, x 2,..., x n ) in several variables. C 2 functions Let f (x 1, x 2,..., x n ) be a function in n variables. We say that f is C 2 if all second order partial derivatives of f exist and are continuous. All functions that we meet in this course are C 2 functions, so shall not check this in each case. Notation We use the notation x = (x 1, x 2,..., x n ). Hence we write f (x) in place of f (x 1, x 2,..., x n ). Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Stationary points of functions in several variables The partial derivatives of f (x) are written f 1, f 2,..., f n. The stationary points of f are the solutions of the first order conditions: Definition Let f (x) be a function in n variables. We say that x is a stationary point of f if f 1(x) = f 2(x) = = f n(x) = 0 Let us look at an example: Let f (x 1, x 2 ) = x 2 1 x 2 2 x 1x 2. Find the stationary points of f. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
6 Stationary points: An example We compute the partial derivatives of f (x 1, x 2 ) = x 2 1 x 2 2 x 1x 2 to be f 1(x 1, x 2 ) = 2x 1 x 2, f 2(x 1, x 2 ) = 2x 2 x 1 Hence the stationary points are the solutions of the following linear system 2x 1 x 2 = 0 x 1 2x 2 = 0 Since det ( ) = 5 0, the only stationary point is x = (0, 0). Given a stationary point of f (x), how do we determine its type? Is it a local minimum, a local maximum or perhaps a saddle point? Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 The Hessian matrix Let f (x) be a function in n variables. The Hessian matrix of f is the matrix consisting of all the second order partial derivatives of f : Definition The Hessian matrix of f at the point x is the n n matrix f 11 (x) f 12 (x)... f 1n (x) f f 21 (x) f 22 (x)... f 2n (x) = (x) f n1 (x) f n2 (x)... f nn(x) Notice that each entry in the Hessian matrix is a second order partial derivative, and therefore a function in x. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
7 The Hessian matrix: An example Compute the Hessian matrix of the quadratic form f (x 1, x 2 ) = x 2 1 x 2 2 x 1 x 2 We computed the first order partial derivatives above, and found that Hence we get f 1(x 1, x 2 ) = 2x 1 x 2, f 2(x 1, x 2 ) = 2x 2 x 1 f 11(x 1, x 2 ) = 2 f 12(x 1, x 2 ) = 1 f 21(x 1, x 2 ) = 1 f 22(x 1, x 2 ) = 2 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 The Hessian matrix: An example (Continued) The Hessian matrix is therefore given by ( ) f 2 1 (x) = 1 2 The following fact is useful to notice, as it will simplify our computations in the future: Proposition If f (x) is a C 2 function, then the Hessian matrix is symmetric. The proof of this fact is quite technical, and we will skip it in the lecture. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
8 Convexity and the Hessian Definition A subset S R n is open if it is without boundary. Subsets defined by inequalities with < or > are usually open. Also, R n is open by definition. The set S = {(x 1, x 2 ) : x 1 > 0, x 2 > 0} R 2 is an open set. Its boundary consists of the positive x 1 - and x 2 -axis, and these are not part of S. Theorem Let f (x) be a C 2 function in n variables defined on an open convex set S. Then we have: 1 f (x) is positive semidefinite for all x S f is convex in S 2 f (x) is negative semidefinite for all x S f is concave in S 3 f (x) is positive definite for all x S f is strictly convex in S 4 f (x) is negative definite for all x S f is strictly concave in S Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Convexity: An example Is f (x 1, x 2 ) = x 2 1 x 2 2 x 1x 2 convex or concave? We computed the Hessian of this function earlier. It is given by ( ) f 2 1 (x) = 1 2 Since the leading principal minors are D 1 = 2 and D 2 = 5, the Hessian is neither positive semidefinite or negative semidefinite. This means that f is neither convex nor concave. Notice that since f is a quadratic form, we could also have used the symmetric matrix of the quadratic form to conclude this. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
9 Convexity: Another example Is f (x 1, x 2 ) = 2x 1 x 2 x x 1x 2 x 2 2 convex or concave? We compute the Hessian of f. We have that f 1 = 2 2x 1 + x 2 and f 2 = 1 + x 1 2x 2. This gives Hessian ( ) f 2 1 (x) = 1 2 Since the leading principal minors are D 1 = 2 and D 2 = 3, the Hessian is negative definite. This means that f is strictly concave. Notice that since f is a sum of a quadratic and a linear form, we could have used earlier results about quadratic forms and properties of convex functions to conclude this. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Convexity: An example of degree three Is f (x 1, x 2, x 3 ) = x 1 + x x 3 3 x 1x 2 3x 3 convex or concave? We compute the Hessian of f. We have that f 1 2, f 2 2 x 1 and f This gives Hessian matrix f (x) = x 3 Since the leading principal minors are D 1 = 0, D 2 = 1 and D 3 = 6x 3, the Hessian is indefinite for all x. This means that f is neither convex nor concave. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
10 Extremal points Definition Let f (x) be a function in n variables defined on a set S R n. A point x S is called a global maximum point for f if and a global minimum point for f if f (x ) f (x) for all x S f (x ) f (x) for all x S If x is a maximum or minimum point for f, then we call f (x ) the maximum or minimum value. How do we find global maxima and global minima points, or global extremal points, for a given function f (x)? Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 First order conditions Proposition If x is a global maximum or minimum point of f that is not on the boundary of S, then x is a stationary point. This means that the stationary points are candidates for being global extremal points. On the other hand, we have the following important result: Theorem Suppose that f (x) is a function of n variables defined on a convex set S R n. Then we have: 1 If f is convex, then all stationary points are global minimum points. 2 If f is concave, then all stationary points are global maximum points. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
11 Global extremal points: An example Show that the function has a global maximum. f (x 1, x 2 ) = 2x 1 x 2 x x 1 x 2 x 2 2 Earlier, we found out that this function is (strictly) concave, so all stationary points are global maxima. We must find the stationary points. We computed the first order partial derivatives of f earlier, and use them to write down the first order conditions: f 1 = 2 2x 1 + x 2 = 0, f 2 = 1 + x 1 2x 2 = 0 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Global extremal points: An example (Continued) This leads to the linear system 2x 1 x 2 = 2 x 1 2x 2 = 1 We solve this linear system, and find that (x 1, x 2 ) = (1, 0) is the unique stationary point of f. This means that (x 1, x 2 ) = (1, 0) is a global maximum point for f. Find all extremal points of the function f given by f (x, y, z) = x 2 + 2y 2 + 3z 2 + 2xy + 2xz + 3 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
12 Global extremal points: Another example We first find the partial derivatives of f : f x = 2x + 2y + 2z, f y = 4y + 2x, f z = 6z + 2x This gives Hessian matrix f (x) = We see that the leading principal minors are D 1 = 2, D 2 = 4 and D 3 = 8, so f is a (strictly) convex function and all stationary points are global minima. We therefore compute the stationary points by solving the first order conditions f x = f y = f z = 0. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25 Global extremal points: More examples (Continued) The first order conditions give a linear system 2x + 2y + 2z = 0 2x + 4y = 0 2x + 6z = 0 We must solve this linear system. Since the 3 3 determinant of the coefficient matrix is 8 0, the system has only one solution x = (0, 0, 0). This means that (x 1, x 2, x 3 ) = (0, 0, 0) is a global minimum point for f. Find all extremal points of the function f (x 1, x 2, x 3 ) = x x x x x x 2 3 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
13 Global extremal points: More examples The function f has one global minimum (0, 0, 0). See the lecture notes for details. So far, all examples where functions defined on all of R n. Let us look at one example where the function is defined on a smaller subset: Show that S = {(x 1, x 2 ) : x 1 > 0, x 2 > 0} is a convex set and that the function f (x 1, x 2 ) = x1 3 x 2 2 defined on S is a concave function. See the lecture notes for details. Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and the Hessian October 01, / 25
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