RealLife Math Phần 5 pot

Geometry 244 REAL-LIFE MATH edges cubed; and there are 27 smaller cubes; so the volume of the main cube is equal to the volume of one small cube multiplied by 27. The multitude of mathematical facts that can be illustrated (and even discovered) while playing with a Rubik’s Cube is amazing. Initially and when in solved form, each of the six faces of the cube is its own color: green, blue, red, orange, yellow, or white. As the layers are rotated, the colored faces are shuffled. The goal of the puzzle is to restore each face to a single color after thorough shuffling. Numerous strategies have been developed for solving a Rubik’s Cube, all of which involve some degree of geometric reasoning. Some strategies can be simulated by computer programs, and many contests take place to compare strategies based on the average number of moves required to solve ran- domized configurations. The top strategies can require less than 20 moves. Possibly the most daunting fact about the 3 ϫ 3 ϫ 3 Rubik’s Cube is that 43,252,003,274,489,856,000 different combinations of colors can be created on the faces of the cube. That’s more than 43 quintillion combinations, or 43 million multiplied by a million, and then multiplied by a million again. Keep in mind that the original 3 ϫ 3 ϫ 3 cube is among the smallest and least complicated of Rubik’s puzzles! SHOOTING AN ARROW The aim of archery is to shoot an arrow and hit a target. The three main components involved in shooting an arrow—the bow, the arrow, and the target—are thor- oughly analyzed in order to optimize accuracy. The act of shooting an arrow provides an excellent exploration of vectors (as may be deduced by the fact that vectors are usually represented by arrows in mathematical figures). The intended path of the arrow, the forces that alter this path, and the true path taken by the arrow when released can all be represented as vectors. In fact, the vector that represents the true path taken by the arrow is the sum of the vectors produced by the forward motion of the arrow and the vectors that represent the forces that disrupt the motion of the arrow. Gravity, wind, and rain essentially add vectors to the vector of the intended path, so that the original speed and direction of the arrow is not maintained. When an arrow is aimed directly at a target and then released, it begins to travel in the direction of the target with a specific speed. However, the point at which an arrow is directly aimed is never the exact point hit by the arrow. Gravity immediately adds a downward force to the forward force created by the bow, pulling the arrow down and reducing its speed. Gravity is constant, so the vector used to represent this force always points straight toward the ground with the same magnitude (length). If gravity is the only force acting on an arrow flying toward its target, then the point hit will be directly below the pointed at which the arrow is aimed; how far below depends on the distance the arrow flies. Any amount of wind or rain moving in any direction has a similar affect on the flight of the arrow, further altering the speed and direction of the arrow. To determine the point that the arrow will actually hit involves moving from the intended target in the direction and length of the vectors that represent the additional forces, similar to the way that addition of vectors is represented on a piece of graph paper. Though the addition of vectors in three-dimensional space is the most prominent application of geometry found in archery, geometric concepts can be unearthed in all aspects of the sport. The bow consists of a flexible strip of material (e.g., wood or light, pliable metal) held at a precise curvature by a taught cord. The intended target and the actual final location of the arrowhead—whether on a piece of wood, a bail of hay, or the ground—can be thought of as theoretical points in space. The most popular target is made of circles with different radial dis- tances from the same center, called concentric circles. If feathers are not attached at precise angles and positions near the rear of the arrow, they will not properly stabilize the arrow and it will wobble unpredictably in flight. In these ways and more, geometric reasoning is essential to every release of an arrow. STEALTH TECHNOLOGY Radar involves sending out radio waves and waiting a brief moment to detect the angles from which waves are reflected back. An omnidirectional radar station on the ground detects anything within a certain distance above the surface of Earth, essentially creating a hemisphere of detection range. A radar station in the air (e.g., attached to a spy plane), can send out signals in all directions, detecting any object within the spherical boundary of the radar’s range. The direction and speed of an object in motion can be determined by changes in the reflected radio waves. Among other things, radar is used to detect the speed of cars and baseballs, track weather patterns, and detect passing aircraft. Most airplanes consist almost entirely of round surfaces that help to make them aerodynamic. For example, a cross-section of the main cabin of a passenger plane (parallel to the wingspan or a row of seats) is somewhat circular; so when the plane flies relatively near a radar station on the ground, it provides a perfect reflecting surface for radio waves at all times. To illustrate this, consider Geometry REAL-LIFE MATH 245 someone holding a clean aluminum can parallel to the ground on a sunny day. If he looks at the can, he will be able to see the reflection of the Sun no matter how the can is turned or moved, as long as it remains parallel to the ground. However, if the can were traded for a flat mirror, he would have to turn the mirror to the proper angle or move it to the correct position relative to his eyes in order to reflect the Sun into his face. The difficulty of accurately reflecting the sun using the flat mirror provides the basis for stealth technology. To avoid being detected by radar while sneaking around enemy territories, the United States military has developed aircraft—including the B-2 Bomber and the F-117 Nighthawk—that are specially designed to reflect radio waves at angles other than directly back to the source. The underside of an aircraft designed for stealth is essentially a large flat surface; and sharp transitions between the various parts of the aircraft create well- defined angles. The danger of being detected by radar comes into play only if the aircraft is directly above a radar station; a mistake easily avoided with the aid of devices that warn pilots and navigators of oncoming radio waves. Potential Applications ROBOTIC SURGERY While the idea of a robot operating on a human body with metallic arms wielding powerful clamps, prodding rods, probing cameras, razor-sharp scalpels, and spinning saws could make even the bravest of patients squeamish, the day that thinking machines perform vital operations on people may not be that far away. Multiple robotic surgical aids are already in develop- ment. One model is already in use in the United States and another, currently in use in Europe, is waiting to be approved by the U.S. Food and Drug Administration (FDA). All existing models require human input and control. Initial instructions are input via a computer work- station using the usual computer equipment, including a screen and keyboard. A control center is also attached to the computer and includes a special three-dimensional viewing device and two elaborate joysticks. Cameras on the ends of some of the robotic arms near or inside the patient’s body send information back to the computer system, which maps the visual information into mathematical data. This data is used to recreate the three- dimensional environment being invaded by the robotic arms by converting the information into highly accurate geometric representations. The viewing device has two goggle-like eyeholes so that the surgeon’s eyes and brain perceive the images in three dimensions as well. The images can be precisely magnified, shifting the perception of the surgeon to the ideal viewpoint. Once engrossed in this three-dimensional representation, the surgeon uses the joysticks to control the various robotic appendages. Pressing a button or causing any slight movement in the joysticks sends signals to the computer, which translates this information into data that causes the precise movement of the surgical instruments. These types of robotic systems have already been used to position cameras inside of patients, as well as perform gallbladder and gastrointestinal surgeries. Immediate goals include operating on a beating heart without creating large openings in the chest. By programming robotic units with geometric knowledge, humans can accurately navigate just about any environment, from the inside of a beating human heart to the darkest depths of the sea. By combining spacecraft, telescopes, and robotics, scientists can send out robot aids that explore the reaches of the Universe while receiving instructions from Earth. When artificial intelligence becomes a practical reality, scientists in all fields will be able to send out unmonitored helpers to explore any environment, perform tasks, and report back with pertinent information. With the rise of artificial intelligence, robots might soon be programmed to detect any issues inside of a living body, and perform the appro- priate operations to restore the body to a healthy state without any human guidance. From the first incision to the final suture, critical decisions will be made by a thinking robotic surgeon. THE FOURTH DIMENSION Basic studies in geometry usually examine only three dimensions in order to facilitate the investigation of the properties of physical objects. To say that anything in the Universe exists only in three dimensions, however, is a great oversimplification. As humans perceive things, the Universe has a fourth dimension that can be studied in the same way as the length, width, and height of an object. This fourth dimension is time, and has just as much influence on the state of an object as its physical dimensions. Similar to the way that a cylinder can be seen as a two-dimensional circle extended into a third dimension, a can of soda thrown from one person to another can be seen as a three-dimensional object extending through time, having a different distinct position relative to the things around it at every instant. This is the fundamental concept behind the movement of objects. If there were truly only three dimensions, things could not move Geometry 246 REAL-LIFE MATH or change. But just as a circular cross-section of a cylinder helps to shed light on its three-dimensional properties, studying snapshots of objects in time makes it possible to understand their structure. As perceived by the people of Earth, time moves at a constant rate in one direction. The opposite direction in time, involving the moments of the past, only exists in the forms of memory, photography, and scientific theory. Altering the perceived rate of time—in the opposite direction or in the same direction at an accelerated speed—has been a popular fantasy in science fiction for hundreds of years. Until the twentieth century, the potential of time travel was considered by even the most brilliant scientists to lie much more in the realm of fiction. In the last hun- dred years, however, a string of scientists have delved into this fascinating topic to explore methods for manipulating time. The idea of time as a malleable (changeable) dimension was initiated by the theory of special relativity proposed by Albert Einstein (1879–1955) in the early twentieth century. An important result of the theory of special relativity is that when things move relative to each other, one will perceive the other as shrinking in the direction of relative motion. For example, if a car were to drive past the woman in the chair, its length would appear to shrink, but not its height or width. Only the dimension measured in the direction of motion is affected. Of course, humans never actually see this happen because we do not see things that move quickly enough to cause a visible shrinking in appearance. Something would have to fly past the woman at about 80% the speed of light for her to notice the shrinking, in which case she would probably miss the car altogether, and would surely have no perception of its dimensions. Similar to the manner in which the length of an object moving near the speed of light would seem to shrink as perceived by a relatively still human, time would theoretically seem to slow down as well. However, time would not be affected in any way from the point of view of the moving object, just as physical measurements only seem to shrink from the point of view of someone not moving at the same speed along the same path. If two people are flying by each other in space, to both of these people it will seem that the other is the one moving. So while one could theoretically see physical shrinking and a slowing of the watch on the other’s arm, the other sees the same affects in the other person. Without a large nearby reference point, it is easy to feel like the center of the universe, with the movement, mass, and rate of time all- dependent upon the local perception. All of these ideas about skewed perception due to speed of relative motion are rather difficult to grasp because none of it can be witnessed with human eyes, but recall that the notion of Earth as a sphere moving in space was once commonly tossed aside as mystical nonsense. Einstein’s theory of relativity explains events in the Uni- verse much more accurately than previous theories. For example, relativity corrects the inaccuracies of English mathematician Isaac Newton’s (1642–1727) proposed laws of gravity and motion, which had been the most acceptable method for explaining the forces of Earth’s gravity for hundreds of years. Just as humans can now film the Earth from space to visually verify its spherical nature, its path around the sun, and so forth, the future may very well bring technology that can vividly verify the theories that have been evolving over the last century. For now, these theories are supported by a number of experiments. In 1972, for example, two precise atomic clocks were synchronized, one placed on a high-speed airplane, and the other left on the ground. After the airplane flew around and landed, the time indicated by the clock on the airplane was behind that of the clock on the ground. The amount of time was accurately explained and predicted by the theory of relativity. Inconsistencies in experiments involving the speed of light dating back to the early eigh- teenth century can be accurately accounted for by the theory of relativity as well. To travel into the past would require moving faster than the speed of light. Imagine sitting on a spacecraft in outer space and looking through a telescope at someone walking on the surface of Earth. New light is continually reflecting off of Earth and the walker, entering the telescope. However, if the spacecraft were to begin moving away from Earth at the speed of light, the walker would appear to freeze because the spacecraft and the light would be moving at the same speed. The same vision would be following the telescope and no new information from Earth would reach it. The light waves that had passed the spacecraft just before it started moving would be traveling at the same speed directly in front of the spacecraft. If the spacecraft could speed up just a little, it would move in front of the light of the past, and the viewer would again see events from the past. The walker would appear to be moving backward as the spacecraft continued to move past the light from further in the past. The faster the spacecraft moved away from Earth, the faster everything would rewind in front of the viewer’s eyes. Moving much faster than the speed of light in a large looping path that returned to Earth could land the viewer on a planet full of dinosaurs. Unfortunately, moving faster than the speed of light is considered to be impossible, so traveling backward in time is out of the Geometry REAL-LIFE MATH 247 question. The idea of traveling into the future at and accelerated rate, on the other hand, is believed to be theoretically possible; but the best ideas so far involve flying into theoretical objects in space, such as black holes, which would most likely crush anything that entered and might not even exist at all. The interwoven relationship of space and time is often referred to as the space-time continuum. To those who possess a firm understanding of the sophisticated ideas of special relativity, the four dimensions of the universe begin to reveal themselves more plainly; and to some, the fabric of time is begging to be ripped in order to allow travel to other times. While time travel is not likely to be realized in the near future, every experiment and theory helps the human race explain the events of the past, and predict the events of the future. Where to Learn More Books Hawking, Stephen. A Brief History of Time: From the Big Bang to Black Holes. New York: Bantam, 1998. Pritchard, Chris. The Changing Shape of Geometry. Cambridge, UK: Cambridge University Press, 2003. Stewart, Ian. Concepts of Modern Mathematics. Dover Publica- tions, 1995. Web sites Utah State University. “National Library of Virtual Manipula- tives for Interactive Mathematics.” National Science Foun- dation. April 26, 2005. Ͻhttp://matti.usu.edu/nlvm/ nav/topic_t_3.htmlϾ (May 3, 2005). Key Terms Angle: A geometric figure formed by two lines diverging from a common point or two planes diverging from a common line often measured in degrees. Area: The measurement of a surface bounded by a set of curves as measured in square units. Cross-section: The two-dimensional figure outlined by slicing a three-dimensional object. Curve: A curved or straight geometric element generated by a moving point that has extension only along the one-dimensional path of the point. Geometry: A fundamental branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, surfaces, and solids. Line: A straight geometric element generated by a moving point that has extension only along the one- dimensional path of the point. Point: A geometric element defined only by an ordered set of coordinates. Segment: A portion truncated from a geometric figure by one or more points, lines, or planes; the finite part of a line bounded by two points in the line. Vector: A quantity consisting of magnitude and direction, usually represented by an arrow whose length represents the magnitude and whose orientation in space represents the direction. Volume: The amount of space occupied by a three- dimensional object as measured in cubic units. 248 REAL-LIFE MATH Overview In its most straightforward definition, graphing is the act of representing mathematical relationships or quantities in a visual form. Real-life applications can range from records of stock prices to calculations used in the design of spacecraft to evaluations of global climate change. Fundamental Mathematical Concepts and Terms In basic mathematics, graphs depict how one variable changes with respect to another and are often referred to as charts or plots. The graphs can be either empirical, meaning that they show measured or observed quantities, or they can be functional. Examples of empirical measurements are the speed shown on the speedome- ter of a car, the weight of a person shown on a bathroom scale, or any other value obtained by measurement. Func- tion plots, in contrast, show pure mathematical relationships known as functions, such as y ϭ b ϩ m, x, or y ϭ x 2 . In these examples, each value of x corresponds to a specific value of y and y is said to be a function of x. Mathematicians and computer scientists sometimes refer to graphs in a different sense when they are analyz- ing possible ways to connect points (also known as ver- tices or nodes) in space using networks of lines (also known as edges or arcs). The body of knowledge related to this kind of analysis is known as graph theory. Graph theory has applications to the design of many kinds of networks. Examples include the structure of the elec- tronic links that comprise the Internet, determining the most economical route between two points connected by a complicated network of roads (or railroads, air routes, or shipping routes), electrical circuit design, and job scheduling. In order to accurately represent empirical or functional relationships between variables, graphs must use some method to scale, or size, the information being plotted. The most common way to do this relies upon an idea developed by the French mathematician René Descartes (1596–1650) in the seventeenth century. Descartes created graphs by measuring the value of one variable along an imaginary line and the value of the second variable along another imaginary line perpendicular to the first. Each of the lines is known as an axis, and it has become standard practice to draw and label the axes rather than using only imaginary lines. Other kinds of coordinate systems exist and are useful for special applications in science and engineering, but the Graphing Graphing REAL-LIFE MATH 249 majority of graphs encountered on a daily basis use a set of two perpendicular axes. In most graphs, the dependent variable is plotted using the vertical axis and the independent variable is plotted using the horizontal axis. For example, a graph showing measured rainfall on each day of the year would commonly show the rainfall on the vertical axis because it is dependent upon the day of the year and is, therefore, the dependent variable. Time, represented by the day of the year, is the independent variable because its value is not controlled by the amount of rainfall. Likewise, a graph showing the number of cars sold in the United States for each of the past ten years will usually have the years shown along the horizontal axis and the number of cars sold along the vertical axis. There are some excep- tions to this general rule. Atmospheric scientists measuring the amount of air pollution at different altitudes or geologists measuring the chemical composition of rocks at different depths beneath Earth’s surface often choose to create graphs in which the independent variable (in these cases, altitude or depth) is shown on the vertical axis. In both cases the dependent variable is being measured ver- tically, so it makes sense to make graphs having the same orientation. BAR GRAPHS Bar graphs are used to show values associated with clearly defined categories. For example, the number of cars sold by a dealer each month, the numbers of homes sold in different cities during a certain year, or the amount of rainfall measured each day during a one-year period can all be shown on bar graphs. The categories are shown along one axis and the values are represented by bars drawn perpendicular to the category axis. In some cases bar graphs will contain a value axis, but in other cases the value axis may be omitted and the values indicated by a number just above or next to each bar. The term “bar graph” is sometimes restricted to graphs in which the bars are horizontal. In that case, graphs with vertical bars are called column graphs. One bar is drawn for each category on a bar graph, and the height or length of the bar is proportional to the value being shown. For example, the following set of numbers could reflect the average price of homes sold in different parts of Santa Barbara County, California, in February 2005: Area 1, $334,000; Area 2, $381,000; Area 3, $308,000; Area 4, $234,000; Area 5, $259,950. If these figures were plotted on a bar graph, the tallest bar would cor- respond to the price for Area 2. The absolute height of this A computer chip (which contains billions of pure light converting proteins) is shown in the foreground. The chip may one day be a power source in electronics such as mobile phones or laptops. In the background is a graph which displays gravity forces that can separate light-electricity converting protein from spinach. Researchers at MIT say they have used spinach to harness a plant’s ability to convert sunlight into energy for the first time, creating a device that may one day power laptops, mobile phones and more. AP/WIDE WORLD PHOTOS. REPRODUCED BY PERMISSION. Graphing 250 REAL-LIFE MATH bar does not matter, because the largest value will control the values of all the other bars. The height of the bar for Area 1, which has the second most expensive homes, would be 334,000 / 381,000 ϭ 88% as tall as the bar representing Area 2. Similarly, the bar representing Area 3 would be 308,000 / 381,000 ϭ 81% as tall as the Area 2 bar. See Figure 1, which depicts the bar graph reflecting the average price of homes sold in different parts of Santa Barbara County, California, in February 2005. Bar graph categories can represent virtually anything for or about which data can be collected. In Figure 1, the categories represent different parts of a county for which real estate sales records are kept. In other cases bar graph categories represent a quantity such as time, such as the rainfall measured in New York City on each day of Feb- ruary 2005, with each bar representing one day. Scientists and engineers often use specialized forms of bar graphs known as stem graphs, in which the bars are replaced by lines. Using lines instead of bars can help to make the graph more readable when there are many categories; for example, the sizes of the largest floods along the Rio Grande during the past 100 years would require 100 bars or stems. More often than not, the kinds of data collected by scientists and engineers dictate that the categories involve some measure of distance or time (for example, the year in which each flood occurred). As such, they are usually ordered from smallest to largest. Stem graphs can also have small open or filled circles at the end of each stem. Unless the legend for the graph specifies otherwise, the circles are used simply to make the graph more readable and do not have any significance of their own. Histograms are specialized bar graphs in which each category represents a range of possible values, and the values plotted perpendicular to the category axis represent the number of occurrences of each category. An important characteristic of a histogram is that each category does not represent just one value or attribute, but rather a range of values that are grouped together into a single category or bin. For example, suppose that in a group of 100 people there are 20 who earn annual salaries between $20,000 and $30,000, 40 who earn annual salaries between $30,001 and $40,000, 30 who earn annual salaries between $40,001 and $50,000, and 10 who earn annual $200000 $400000 0 $100000 Santa Barbara Goleta Montecito Median Home Prices Summerland Carpinteria $300000 Figure 1. Graphing REAL-LIFE MATH 251 salaries between $50,001 and $60,000. The bins in a histogram showing this salary distribution would be $20,000 to $30,000, $30,001 to $40,000, $40,001 to $50,000, and $50,001 to $60,000. The height of each bin would be proportional to the number of people whose salaries fall into that bin. The tallest bar would represent the bin with the most occurrences, in this case the $30,001 to $40,000. The second tallest bar would represent the $40,001 to $50,000 category, and it would be 30/40 ϭ 75% as tall as the tallest bin. The width of each bin is proportional to the range of values that it represents. Therefore, if each class interval is the same size then all of the bars on a histogram will be the same width. A histogram containing bars with different widths will have unequal class intervals. Some bar graphs use more than one set of bars in order to convey several sets of information. Continuing with the home price example from Figure 1, the bars showing the 2005 prices could be supplemented with bars showing the average home sales prices for the same areas in February 2004. Figure 2 allows readers to quickly compare prices and see how they changed between 2004 and 2005. Each category has two bars, one for 2004 and one for 2005, filled with different colors, patterns, or shades of gray to distinguish them from each other. A third kind of bar graph is the stacked bar graph, in which different types of data for each category are represented using bars stacked on top of each other. The bottom bar in each of the stacks will generally have a different height, which makes it difficult to compare values among categories for all but the bottom bars. For this rea- son, stacked bar graphs can be difficult to read and should generally be avoided. LINE GRAPHS Line graphs share some similarities with bar graphs, but use points connected by straight lines rather than bars to represent the values being graphed. As with bar graphs, the categories on a line graph can represent either some kind of measurable quantity or more abstract qual- ities such as geographic regions. Line graphs are constructed much like bar graphs. In line graphs, values for each category are known or measured, and the categories are placed along one axis. The values are then scaled along the value axis, and a point, sometimes represented by a symbol such as a circle or a square, is drawn to represent the value for each category. The points are then connected with straight line seg- ments to create the line graph. One of the weaknesses of line graphs is that they can imply some kind of connection between categories, which may or may not be the intention of the person creating the graph. In a bar chart, each category is represented by a bar that is completely separate from its $200000 2004 $400000 0 $100000 Average Home Sales Price $300000 2005 Santa Barbara Goleta Montecito Summerland Carpinteria Figure 2. Graphing 252 REAL-LIFE MATH neighbors. Therefore, no connection or relationship between adjacent categories is implied by the graph. A line graph implies that the value varies continuously between adjacent categories because the points are connected by lines. If there is no real connection between the values for adjacent categories, for example the home sales prices used in the Figure 1 bar graph example, then it may be better to use a bar graph or stem graph than a line graph. Like bar graphs, line graphs can be combined to create multiple line graphs. Each line represents a different value associated with each category. For example, a multiple line graph might show different household expenses for each month of the year (rent, heat, water, groceries, etc.) or the income and expenses of a business for each quarter of a particular year. Rather than being placed side-by-side as in a multiple bar graph, however, multiple line graphs are placed on top of each other and the lines are distinguished by different colors or patterns. If only two sets of values are being graphed and their values are significantly different, two value axes may be used. As shown in Figure 3, each value axis corresponds to one of the sets of values and is labeled accordingly. AREA GRAPHS Area graphs are line graphs in which the area between the line and the category axis is filled with a color or pattern, and are used when there is a need to show both the values associated with each category and the total of all the values. As Figure 4 shows, the values are represented by the height of the colored area, whereas the total is represented by the amount of area that is colored. If the total area beneath the lines is not important, then a bar graph or line graph may be a better choice. Area graphs can also be stacked if the objective is to show information about more than one set of values. The result is much like a stacked bar graph. PIE GRAPHS Pie graphs are circular graphs that represent the relative magnitudes of different categories of data using angular wedges resembling slices of pie. The size of each wedge, which is measured as an angle, is proportional to the relative size of the value it represents. If the data are given as percentages that add up to 100%, then the angular increment of each wedge is its Graphs are often used as visuals representing finances. AP/WIDE WORLD PHOTOS. REPRODUCED BY PERMISSION. [...]... this example: 0, 0 .5, 1, 1 .5, 2, 2 .5, and 3 These values will be the abscissae Substitute each abscissa into the function (in this case y ϭ x2) and calculate the value of the function for that value, which will produce the ordinates 0, 0. 25, 1, 2. 25, 4, 6. 25, and 9 Finally, plot a point for each corresponding abscissa and ordinate, or (0,0), (0 .5, 0. 25) , (1,1), (1 .5, 2. 25) , (2,4), (2 .5, 6. 25) , and (3,9) Because... be used during that time 35 30 25 20 15 10 5 Figure 8 The most noticeable property of an x-y graph is that it consists of points rather than bars or lines Lines can be added to x-y plots but they are in addition to the points and not a replacement for them Line graphs can also have points added as an embellishment and can therefore 254 R 0 0 1 2 3 4 Rainfall (millimeters) 5 6 E A L - L I F E M A T... be 0.18 ϫ 360Њ ϭ 65 , and the wedge representing all other stores would (1.00 Ϫ 0.30 Ϫ 0.18) ϫ 360Њ ϭ 0 .52 ϫ 360Њ ϭ 187Њ Figure 5 depicts a representative pie graph 1960 1980 2000 2020 Year 2040 New sources 2060 Figure 4: Stacked area graph showing different sources of water (values) by year (categories) Percentages of Computers Sold 52 % 30% Store B Store A All Others 18% Figure 5 253 Graphing GANTT... 9, 20 05) National Oceanic and Atmospheric Administration “Figures.” Climate Modeling and Diagnostics Laboratory Ͻhttp://www.cmdl.noaa.gov/gallery/cmdl_figuresϾ (March 9, 20 05) Weisstein, E.W “Function Graph.” Mathworld Ͻhttp:// mathworld.wolfram.com/FunctionGraph.htmlϾ (March 9, 20 05) Friendly, Michael “The Best and Worst of Statistical Graphics.” Gallery of Data Visualization 2000 Ͻhttp://www .math. .. stores to calculate the percentage for that store If Store A sold 1 ,50 0 computers, Store B sold 900 computers, and all other stores combined sold 2,600 computers, then the total number of computers sold would be 5, 000 The percentage sold by Store A would be 1 ,50 0 /5, 000 ϭ 0.30, or 30% Similar calculations produce results of 18% for Store B and 52 % for all other stores combined (just as in the previous example)... computer is programmed to decide which blobs in the image are potatoes, how big each potato is, and whether the potatoes that are big enough for baking are also the right shape All these steps involve imaging mathematics DANCE S T E G A N O G R A P H Y A N D D I G I TA L WAT E R M A R K S Dance and other motions of the human body can be described mathematically This knowledge can then be used For thousands... simplest of all R E A L - L I F E M A T H Message 1 Message 2 Eight possible message pairs 0 0 0 0 1 1 1 1 00 10 01 11 00 10 01 11 0 00 0 10 0 01 0 11 1 00 1 10 1 01 1 11 Figure 1 4 3 .5 3 2 .5 H(N ) 2 1 .5 1 0 .5 0 1 2 3 4 6 5 7 8 9 10 N Figure 2 The information content of a single message selected from N equally likely messages: H(N) ϭ log2N Units of H(N) are bits functions, as shown in Figure 3 The equation... that blob a face, a potato, or a bomb in the luggage? If it’s a face, whose face is it? Is that dark patch in the satellite photograph a city, a lake, or a plowed field? Such questions are answered using a wide array of mathematical techniques that reduce images to representation of pixels by numbers that are then subject to mathematical analysis and operations 263 Imaging OPTICS Mathematics and imaging... health of a human fetus; in agriculture, mathematical techniques like grayscale statistical analysis, gray-scale spatial texture analysis, and frequency spectrum texture analysis can be applied to them in order to decide the degree of marbling Different mathematics are applied to the sorting of another food item that often appears at mealtime with meat: potatoes Potatoes that are the right size and shape... without graphs illustrating the degrees of interconnection between different nodes Applied mathematicians also use graph theory to help design the most efficient networks possible under a given set of constraints Oil Saturation and Cumulative Production 18 Oil Saturation 177 0 2 15 177 182 1694 1441 0 169 1 0.7 0.6 0 .5 0.4 0.3 0.2 0.1 0.0 Oil Production (Barrels) Scientific visualization is a form of graphing . the ordinates 0, 0. 25, 1, 2. 25, 4, 6. 25, and 9. Finally, plot a point for each corresponding abscissa and ordinate, or (0,0), (0 .5, 0. 25) , (1,1), (1 .5, 2. 25) , (2,4), (2 .5, 6. 25) , and (3,9). Because. 6. Analysis Flowcharting Coding Testing Debugging Planned 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Actual Application Design Steps Figure 7. 10 20 30 40 0 0123 Rainfall (millimeters) Increase in Water Level (centimeters) 456 5 15 25 35 Figure 8. Graphing REAL-LIFE. of Santa Barbara County, California, in February 20 05: Area 1, $334,000; Area 2, $381,000; Area 3, $308,000; Area 4, $234,000; Area 5, $ 259 , 950 . If these figures were plotted on a bar graph,

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