K. Lee Lerner và Brenda Wilmoth Lerner, Biên tập Dự án biên tập Kimberley A. McGrath biên tập Luann Brennan, Condino Meggin M., Madeline Harris, Paul Lewon, Elizabeth Manar biên tập Dịch Vụ Hỗ Trợ Andrea Lopeman © 2006 Thomson Gale.
Ve c t o r s Crowe, Michael J A History of Vector Analysis Notre Dame, IN: Web sites University of Notre Dame Press, 1967 Olive, Jenny “Working With Vectors.” September 2003 Tallack, J.C Introduction to Vector Analysis Cambridge, UK: Ͻhttp://www.netcomuk.co.uk/~jenolive/homevec.htmlϾ Cambridge University Press, 1970 (March 1, 2005) Periodicals Roal, Jim “Automobile physics.” AllFordMustangs.com July 2003 Slauterbeck, James “Gender differences among sagittal Ͻhttp://www.allfordmustangs.com/Detailed/29.shtmlϾ (March 7, 2005) plane knee kinematic and ground reaction force character- istics during a rapid sprint and cut maneuver.” Research “Vector Math for 3D Computer Graphics.” Central Connecticut Quarterly for Exercise and Sport Vol 75, No 1 (2004): State University, Computer Science Department July 2003 31–38 Ͻ h t t p : / / c h o r t l e c c s u c t s t a t e u e d u / Ve c t o r L e s s o n s / vectorIndex.htmlϾ (March 1, 2005) 574 REAL-LIFE MATH Overview An object’s volume describes the amount of space it contains Calculations and measurements of volume are used in medicine, architecture, science, construction, and business Gasses and liquids such as propane, gasoline, and water are sold by volume, as are many groceries and construction materials Fundamental Mathematical Concepts Volume and Terms UNITS OF VOLUME Volume is measured in units based on length: cubic feet, cubic meters, cubic miles, and so on A cubic meter, for instance, is the amount of volume inside a box 1 meter (m) tall, 1 m wide, and 1 m deep Such a box is a 1-meter cube, so this much volume is said to be one “cubic” meter An object doesn’t have to be a cube to con- tain a cubic meter: one cubic meter is also the space inside a sphere 1.24 meters across Cubic units are written by using exponent notation: that is, 1 cubic meter is written “1 m3.” This is why raising any number to the third power—that is, multiplying it by itself three times, as in 23 ϭ 2 ϫ 2 ϫ 2—is called “cubing” the number VOLUME OF A BOX There are standard formulas for calculating the vol- umes of simple shapes The simplest and most commonly used of these is the formula for the volume of a box (By “box,” we mean a solid with rectangular sides whose edges meet at right angles—what the language of geometry also calls a “cuboid,” “right prism,” or “rectangular paral- lelepiped.”) To find the volume of a box, first measure the lengths of its edges If the box is L centimeters (cm) long, W cm wide, and H cm high, then its volume, V, is given by the formula V ϭ L cm ϫ W cm ϫ H cm This can be written more shortly as V ϭ LWH cm3 The units of length used do not make any difference to the formula for volume: inches or feet will do just as well as centimeters For example, a room that is 20 feet (ft) long, 10 ft wide, and 12 feet high has volume V ϭ 20 ft ϫ 10 ft ϫ 12 ft ϭ 2,400 ft3 (cubic feet) VOLUMES OF COMMON SOLIDS There are standard formulas for finding the volumes of other simple solids, too Figure 1 shows some of these formulas REAL-LIFE MATH 575 Volume Solid Shape Dimensions Formula for Volume box Length L, width W, height H cube Length = width= height = L V = LWH V = L3 sphere radius R cylinder radius R, height H V = 43 π R3 Base radius R, height H V = π R2H cone base area A, height H V = 13 π R2H pyramid distance from center of torus to V = 13 AH torus (doughnut) center of tube D, radius of tube R V = 2DR2 Figure 1: Standard formula to calculate volume In all these formulas, three measures of length are V= 1R multiplied—not added This means that whenever an A3 object is made larger without changing its shape, its vol- ume increases faster than its size as measured using a which, if we multiply both sides by A, becomes ruler or tape measure For example, a sphere 4 m across (a sphere with a radius of 2 m) has a volume of V ϭ 4/3 V = AR 23 ϭ 33.5 m3, whereas a sphere that is twice as wide 3 (radius of 4 m) has a volume of V ϭ 4/3 43 ϭ 268.1 m3 Doubling the radius does not double the volume, but This means that when we increase the radius R of a makes it 8 times larger In general, since the radius is sphere, area and volume both increase, but volume cubed in calculating the volume, we say that a sphere’s increases by the increased area times R/3 Volume volume “increases in proportion to” or “goes as” the cube increases faster than area This fact has important conse- of its radius This is true for objects of all shapes, not just quences for real-world objects For example, how easily an spheres: Increasing the size of an object without changing animal can cool itself depends on its surface area, because its shape makes its volume grow in proportion to the its surface is the only place it can give heat away to the air; cube (third power) of the size increase but how much heat an animal produces depends on its volume, because all the cubic inches of flesh it contains The formula for an object’s volume can be compared must burn calories to stay alive Therefore, the larger an to the formula for its area The area of a sphere of radius animal gets (while keeping the same shape), the fewer R, for example, is A ϭ 4 R2 The radius appears only as square inches of heat-radiating skin it has per pound: its a squared term (R2) in this formula, whereas in the vol- volume increases faster than its area A large animal in a ume formula it appears as a cubed term (R3) Dividing cold climate should, therefore, have an easier time staying the volume formula by the area formula yields an inter- warm And in fact, animals in the far North tend to be big- esting and useful result: ger than their close relatives farther south Polar bears, for example, are the world’s largest bears They have evolved 4 V 3 π R3 to large size because it is easier for them to stay warm On = the other hand, a large animal in a hot climate has a harder time staying cool This is why elephants have big ears: the A 4πR2 ears have tremendous surface area, and help the elephant stay cool Crossing out terms that are the same on the top and bottom of the fraction, we have REAL-LIFE MATH 576 Volume Volume can be described in terms of an amount of the flat surfaces and areas It could be generalized to three space an object assumes, such as water in a bucket dimensions—that is, to ordinary space It had now become possible to calculate exactly the volumes of complexly- ROYALTY-FREE/CORBIS shaped objects, as long as their surfaces could be described by mathematical equations The next great revolution in volume calculation came with computers Since computers can add many numbers very quickly, they have made it possible to cal- culate areas and volumes for complex shapes even when the shapes cannot be described by nice, neat mathemati- cal equations Today, the calculation of volumes of simple shapes is still routine in many fields, but the use of calcu- lus and computers for complex shapes such as airplane wings and the human brain is increasingly common A Brief History of Discovery Real-life Applications and Development PRICING Weights, lengths, areas, and volumes were the earliest measurements made by humankind Not only are they eas- Volume is closely related to density, which is how ier to measure than other physical quantities, like velocity much a given volume of a substance weighs For instance, and temperature, but they have an immediate money the density of gold is 19.3 grams per cubic centimeter, value Measuring lengths, builders can build more complex that is, one cubic centimeter of gold weighs 19.3 grams, structures, such as temples; measuring area, landowners which is 19.3 times as much as one cubic centimeter of can know how much land, exactly, they are buying and sell- water Silver, platinum, and other metals all have different ing; measuring volume, traders can tell how much grain a densities This fact is used by some jewelry makers to basket holds, or how much water a cistern (holding tank) decide how much to charge for their jewelry holds Therefore it is no surprise to find that the Egyptians, Sumerians, Greeks, and ancient Chinese all knew the con- Different metals not only have different densities, cept of volume and knew many of the standard equations they have different costs: at a 2005 price of about $850 per for calculating it In 250 B.C (over 2,200 years ago), the ounce, for example, platinum cost about twice as much as Greek mathematician Archimedes wrote down formulas gold So when a jewelry maker uses a blend of gold and for the volume of a sphere and cylinder In approximately platinum in a piece of jewelry, they need to know exactly 100 B.C., the Chinese had formulas for the volumes of how much of each they have used in order to know how cubes, cuboids, prisms, spheres, cylinders, and other shapes much to charge for the piece Now, a blend of two metals (using, like the Greeks, approximate values for ranging (called an alloy) has a density that is somewhere between from rough to excellent) the densities of the two original metals Therefore, deter- mining the average density of a piece (say, a ring) will tell Such formulas are useful but do not give any way of a manufacturer how much gold and platinum it contains, exactly calculating the volume of a shape whose surface is regardless of how complicated the piece is Volume and not described by flat planes or by circles (as are the curved weight together are used to determine density The fin- sides of a cylinder, or the surface of a sphere) New progress ished piece is suspended in water by a thread Any object in the calculation of volumes had to wait almost 2,000 submerged in water experiences an upward force that years, until the invention of the branch of mathematics depends only on the volume of water the object displaces known as calculus in the 1600s One of the two basic math- Therefore, by weighing the piece of jewelry as it hangs in ematical operations of calculus is called “integration.” Inte- water, and comparing that weight to its weight out gration, as it was first invented, allowed mathematicians to of water, the jeweler can measure exactly what weight of exactly calculate the area under any mathematically water it displaces Since the density of water is known defined curve or any part of such a curve; it was soon (1 gram per cubic centimeter), this water weight tells discovered, however, that integration was not restricted to the jeweler the exact volume of the piece Finally, know- ing both the volume of the piece and its weight, the REAL-LIFE MATH jeweler can calculate its density by the equation density 577 Volume ϭ weight/volume The jewelry maker’s wholesale price why forgetting is one of Alzheimer’s first symptoms But will be determined partly by this calculation, and so will instead of waiting for memory to fail badly, doctors can the retail price in the store measure the volume of the hippocampus using MRI A shrinking hippocampus can be observed at least 4 years MEDICAL APPLICATIONS before Alzheimer’s disease is bad enough to diagnose from memory loss alone In medicine, volume measurements are used to char- acterize brain damage, lung function, sexual maturity, Pollution’s Effects on Teenagers Polychlorinated aro- anemia, body fat percentage, and many other aspects of matic hydrocarbons (PCAHs) are a type of toxic chemi- health A few of these uses of volume are described below cal that is produced by bleaching paper to make it white, improper garbage incineration, and the manufacture of Brain Damage from Alcohol Using modern medical pesticides (bug-killing chemicals) These chemicals, imaging technologies such as magnetic resonance imaging which are present almost everywhere today, get into the (MRI), doctors can take three-dimensional digital pic- human body when we eat and drink In 2002 scientists in tures of organs inside the body, including the brain Com- Belgium studied the effects of PCAHs on the sexual mat- puters can then measure the volumes of different parts of uration of boys and girls living in a polluted suburb They the brain from these digital pictures, using geometry and compared how early boys and girls in the polluted suburb calculus to calculate volumes from raw image data went through puberty (grew to sexual maturity) com- pared to children in cleaner areas They found that high MRI volume studies show that many parts of the levels of PCAH-related chemicals in the blood signifi- brain shrink over time in people who are addicted to alco- cantly increased the chances of both boys and girls of hol The frontal lobes—the wrinkled part of the brain sur- having delayed sexual maturity Once again, volume face that is just behind the forehead—are strongly affected measurements proved useful in assessing health The It is this part of the brain that we use for reasoning, mak- researchers estimated the volume of the testicles as a way ing judgments, and problem solving But other parts of the of measuring sexual maturity in boys, while they assessed brain shrink, too, including structures involved in memory sexual maturity in girls by noting breast development and muscular coordination Alcoholics who stop drinking This study, and others, show that some pollutants can may regain some of the lost brain volume, but not all MRI injure human health and development even in very low studies also show that male and female alcoholics lose the concentrations Testicular volume measurements are also same amount of brain volume, even though women alco- used in diagnosing infertility in men holics tend to drink much less Doctors conclude from this that women are probably even more vulnerable to brain Body Fat Doctors speak of “body composition” to refer damage from alcohol than are men to how much of a person’s body consists of fat, muscle, and bone, and where the fat and muscle are located on the Diagnosing Disease Almost half of Americans alive today body Measuring body composition is important to mon- who live to be more than 85 years old will suffer eventu- itoring the effects of diet and exercise programs and ally from Alzheimer’s disease Alzheimer’s disease is a loss tracking the progress of some diseases Volume measure- of brain function In its early stages, its victims sometimes ment is used to measure some aspects of body composi- have trouble remembering the names for common tion For example, the overall density of the body can be objects, or how they got somewhere, or where they parked used estimate what percentage of the body consists of fat their car; in its late stages, they may become incurably Measuring body density requires the measurement of the angry or distressed, forget their own names, and forget body’s weight—which can be done easily, using a scale— who other people are Doctors are trying understand the and two volumes causes of Alzheimer’s disease and develop treatments for it All agree that preventing the brain damage of The first volume needed is the volume of the body as Alzheimer’s—starting treatment in the early stages—is a whole Since the body is not made of simple shapes like likely to be much more effective than trying to treat the cubes and cylinders, its volume cannot be found by tak- late stages But how can Alzheimer’s be detected before it ing a few measurements and using standard geometric is already damaging the mental powers of the victim? formulas Instead, its volume must be measured by sub- merging it in water The body’s overall volume can then Recent research has shown that the part of the brain be found by measuring how much the water level rises or, called the hippocampus, which is a small area of the brain alternatively, by weighing the body while it is underwater located in the temporal lobe (just below the ear), is the to see how much water it has displaced (Underwater first part of the brain to be damaged by Alzheimer’s The weighing is the same method used to measure the density hippocampus helps the brain store memories, which is REAL-LIFE MATH 578 Volume of jewelry containing mixed metals, as described earlier in COMPRESSION RATIOS IN ENGINES this article.) The body’s overall volume is equal to the water displaced Internal combustion engines are engines that burn mixtures of fuel and air inside cylinders Almost all However, doctors want to know the weight of the engines that drive cars and trucks are of this type In an solid part of the body; the air in the lungs does not count internal combustion engine, the source of power is the And even when a person has pushed all the air they can cylinder: a round, hollow shaft sealed at one end and with out of their lungs, there is still some left, the “residual a plug of metal (the piston) that can slide back and forth lung volume.” Residual lung volume must therefore also inside the shaft When the piston is withdrawn as far as it be measured, as well as overall body volume This is done will go, the cylinder contains the maximum volume of air using special machines that measure how much gas that it can hold: when the piston is pushed in as far as it remains in the lungs when the person exhales The body’s will go, the cylinder contains the minimum volume of air true, solid volume is approximately calculated by sub- To generate power, the cylinder is filled with air at its tracting the residual lung volume from the body’s water maximum volume Then the piston is pushed along the displacement volume cylinder to compress the air This makes the air hotter, according to the well-known Ideal Gas Law of basic Dividing the body’s weight by its true, non-air vol- physics—just how hot depends on how small the mini- ume gives its density This is used to estimate body fat mum volume is Fuel is squirted into the small, hot vol- percentage by a standard mathematical formula ume of air inside the cylinder The mixture of fuel and air is then ignited (either by sheer heat of compression, as in BUILDING AND ARCHITECTURE a diesel engine, or by a spark plug, as in a regular engine) and the expanding gas from the miniature explosion Many building materials are purchased by area or pushes the piston back out of the cylinder The ratio of volume Area-purchased materials include flooring, sid- the cylinder’s largest volume to its smallest is the “com- ing, roofing, wallpaper, and paint Volume-purchased pression ratio” of the engine: a typical compression ratio materials include concrete for pouring foundations and would be about 10 to 1 Engines with high compression other structures, sand or crushed rock, and grout (a kind ratios tend to burn hotter, and therefore more efficiently of thin cement used to fill up masonry joints) All these They are also more powerful Unfortunately, there is a materials are ordered by units of the cubic yard (One dilemma: burning very hot (high compression ratio) cubic yard equals about 765 cubic meters.) In practice, allows the nitrogen in air to combine with the oxygen, simple volume formulas for boxes and cylinders are used forming the pollutant nitrogen oxide; burning relatively to calculate how many cubic yards of cement must be cool (low compression ratio) allows the carbon in the fuel ordered to build simple structures like housing founda- to combine only partly with the oxygen in the air, form- tions A simple foundation, shaped like a box without a ing the pollutant carbon monoxide (rather than the non- top, can be broken into three slab-shaped boxes, namely poisonous greenhouse gas carbon dioxide) the four walls and the floor Multiplying the length by the width by the thickness of each of these slabs gives a vol- GLOWING BUBBLES: ume: the sum of these volumes is the cubic yardage that SONOLUMINESCENCE the cement truck must deliver For concrete columns, the formula for the volume of a cylinder is used For complex When small atoms come together to make a single structures with curving shapes, a computer uses calculus- heavier atom, energy is released This process is called based methods to calculate volumes based on digital “fusion” because in it, two atoms fuse into one All stars, blueprints for the structure including the Sun, get their energy from fusion Some nuclear weapons are also based on fusion But fusion is The same principle is used in designing machine difficult to control on Earth, because atoms only fuse parts It is necessary to know the volume of a machine under extreme heat If fusion could be controlled, rather part while it is still just a drawing in order to know what than exploding as a bomb, it could be used to generate its weight will be: its weight must be known to calculate electricity Many billions of dollars have been spent on how much it will weigh, and (if it is a moving part) how trying to figure out how to make atoms trapped inside much force it will exert on other parts when it moves For magnetic fields fuse—so far without success parts that are not too complicated in shape, the volume of the piece is calculated as a sum of volumes of simple ele- Yet there is a new possibility Some reputable scientists ments: box, cylinder, cone, and the like Computers take claim that they can produce fusion using nothing more over when it is necessary to calculate the volumes of expensive or exotic than a jar full of room-temperature pieces with strange or curvy shapes 579 REAL-LIFE MATH Volume liquid bombarded by sound waves This claim—which has guess of 1.6 ft (about 50 centimeters) with the ocean con- not yet been tested by other researchers—is related to the tinuing to rise Hundreds of millions of people live near effect called “sonoluminescence,” which means “sound- sea level worldwide, and their homes might be flooded or light.” Sonoluminescence depends on changes in volume at greater risk from flooding during storms Also, many of bubbles in liquid Under certain conditions, tiny bub- small island nations might be completely flooded bles form and disappear in any liquid that is squeezed and stretched by strong sound waves; when the bubbles col- Sea level rises when the volume of water in the ocean lapse, they can emit flashes of light This happens as fol- increases There are two ways in which a warmer Earth lows: Pummeled by high-frequency sound waves, a bubble causes the volume of water in the ocean to increase First, forms and expands When the bubble collapses, its radius there is the melting of ice Ice exists on Earth mostly in decreases very rapidly as its surface moves inward at sev- the form of glaciers perched on mountain ranges and the eral times the speed of sound Because the volume of a ice caps at the north and south poles Second, there is the sphere is proportional to the cube (third power) of its volume increase of water as it gets warmer Like most radius, when a bubble’s radius decreases to 1/10 of its substances, water expands as it gets warmer: a cubic cen- starting value, its volume decreases to (1/10)3 ϭ 1/1,000 timeter of seawater gains about 00021 cubic centimeters of its starting value (These are typical figures for the col- of volume if it is made 1 degree Centigrade warmer lapse of a sonoluminescence bubble.) This decrease in Therefore, the oceans get bigger just by getting warmer In volume squeezes the gas inside the bubble, and, according fact, the International Panel on Climate Change predicts to laws of physics, when a gas is squeezed its temperature that most of the sea-level rise that will occur in this cen- goes up Also, the compression happens very quickly— tury will be caused by water expansion, rather than by ice too quickly for much heat to escape from the bubble melting and increasing the mass of the sea Calculations Therefore, the bubble’s rapid shrinkage causes a fast rise of the volume of water that will be added to the ocean by in temperature inside the bubble The temperature has melting glaciers and icecaps and by thermal expansion been shown to rise to tens of thousands of degrees, and are at the heart of predicting the effects of global warm- may reach over two hundred thousand degrees Such heat ing on sea levels rivals that at the heart of the Sun and makes the gas in the bubble glow It may also do something else: in 2002 sci- WHY THERMOMETERS WORK entists at Oak Ridge National Laboratory claimed to have detected neutrons flying out of a beaker of fluid in which The fact that liquids expand as they get warmer sonoluminescence was occurring Neutrons would be a (until they start to boil) is used to measure temperature sign that fusion was occurring If it is, then there is a close in old-fashioned mercury or colored-alcohol thermome- resemblance between bubble fusion and the diesel ters Geometry is used to amplify or multiply the expan- engines found in trucks: both devices work by rapidly sion effect: a thin cylinder connected attached to a sphere decreasing the volume of a gas in order to heat it to the (the “bulb”) The bulb is full of liquid If the radius of the point where energy is released In a diesel engine, the thermometer bulb is rB, then its volume (VB, for “volume, energy is released by a chemical reaction In a fusion bub- bulb”) is given by the standard volume formula for a ble, it would be released by a nuclear reaction sphere as As of 2005, the reality of bubble fusion had been nei- V = 4πr3 ther proved nor disproved If it is proved, it might even- tually mean that producing electricity from fusion could B 3B be done more cheaply than scientists had ever before dreamed Describing changes in bubble volume mathe- If the cylinder’s radius is rC, then the volume of liq- matically is basic to all attempts to understand and con- uid in the cylinder (VC, for “volume, cylinder”) is given by trol sonoluminescence and bubble fusion the standard volume formula for a cylinder as VC ϭ rC2H, where H is the height of the fluid in the cylinder SEA LEVEL CHANGES We read the temperature from a thermometer of this type by reading H from marks on the cylinder One of the potential threats to human well-being from possible global climate change is the rising of sea There is room in the cylinder for more liquid, but levels The International Panel on Climate Change pre- there is no room in the sphere, which is full If the ther- dicts that ocean levels will rise by 3.5 inches to 34.5 inches mometer contains a liquid that has a “volume thermal (about 9 to 88 centimeters) by the year 2100, with a best coefficient” of ␣ ϭ 0001, a cubic centimeter of the liquid will gain 0001 cubic centimeters of volume if it is warmed 580 by 1 degree Centigrade Say that the thermometer starts REAL-LIFE MATH Volume out with no fluid in the cylinder and the bulb perfectly the cube of the change in size, the larger the size change, full Then the temperature of the thermometer goes up by the more misleading the picture 1ЊC This causes the volume of the fluid in the bulb, VB before it is warmed, to increase by 0001VB But this extra Look carefully at any illustration that shows growing volume has nowhere to go in the bulb, which is full, so it or shrinking two-dimensional or three-dimensional goes up the cylinder The amount of fluid in the cylinder objects to illustrate one-dimensional data (plain old is then VC ϭ rC2H ϭ 0001VB If we divide both sides of numbers that are getting larger or smaller) Does the art- this equation by rC2, we find that work exaggerate? H = 001VB SWIMMING POOL MAINTENANCE π rc 2 Everyone who owns a swimming pool knows that Because VB is on top of the fraction, making it bigger they have to add chemicals to keep the water healthy for makes H bigger That is, the bigger the bulb, the bigger swimming It’s not enough to just dump in a bucket or the change in the height of the fluid in the cylinder when two of aluminum sulfate or calcium hypochlorite, the temperature goes up Since rC is on the bottom of the though—the dose has to be proportioned to the volume fraction, making it smaller also makes H bigger That is, of water in the pool the narrower the cylinder, the bigger the change in the height of the fluid in the cylinder when the temperature Some pools have simple, box-like shapes: their vol- goes up This is why thermometers have very narrow ume can be calculated using the standard formula for the cylinders attached to fat bulbs—so it is easy to see how far volume of a box, volume equals length times width times the fluid goes up or down the cylinder when the temper- height A standard formula can also be used for a circular ature changes pool with a flat volume, which is simply a cylinder of water Many pools have more complex shapes, though, MISLEADING GRAPHICS and even a rectangular pool often has a deep end and a shallow end The deep and shallow ends may be flat, with Many newspapers and magazines think that statistics a step between, or the bottom of the pool may slope are dull, and so they have the people who work in their Some pools are elliptical (shaped like a stretched circle), graphics departments make them more visually appeal- and an elliptical pool may also have a sloping bottom ing For example, to illustrate money inflation (how a Euro or a dollar buys less every year), they will show you To calculate the correct chemical dose for a swimming a picture of shrinking bill—a big bill, then a smaller bill pool, it is necessary, then, to take some measurements A below it, and a smaller below that, and so forth Or, to pool with a complex shape has to be divided into sections illustrate the increasing price of oil, they will show you a with simpler shapes, and the volumes of the separate pieces picture of a row of oil barrels, each bigger than the last calculated and added up More complex formulas are needed for, say, the volume of an elliptical pool with a slop- Such pictures can create a very false impression, ing bottom; calculus is needed to find these formulas For- because it is usually the lengths of the dollar bills or the tunately for the owners of complexly shaped pools, oil barrels (or whatever the object is), not their areas or volume-calculation computer software exists that will cal- volumes, that matches the statistic the art is trying to culate a pool’s volume given the basic measurements of its communicate So, to show the price of oil going up by shape For an elliptical pool with a sloping bottom, you 10%, a publication will often show two barrels, one 10% would need to measure the length of the pool, the width of taller and wider than the other But the equation for the the pool, the maximum depth, and the minimum depth volume of a barrel, which is a cylinder, is V ϭ r 2H, where r is the radius of the barrel and H is its height BIOMETRIC MEASUREMENTS Increasing r or H by 10% is the same as multiplying it by 1.1, so increasing the dimensions of the barrel by 10% On average, men’s brains tend to be larger than shows us a barrel whose volume is Vbigger ϭ (1.1 r)2 women’s, occupying more volume and weighing more (1.1)H If we multiply out the factors of 1.1, we find that Before the invention of modern medical imaging Vbigger ϭ 1.331V—that is, the volume of the larger barrel machines like CAT (computerized axial tomography) in the picture, the amount of oil it would contain, is not scanners, brain volumes were measured by measuring the 10% larger but 33.1% larger Because volume increases by volumes of men’s and women’s skulls after they were dead Beads, seeds, or ball bearings were poured into the empty REAL-LIFE MATH skull to see how much the skull would hold, then they were weighed More beads, seeds, or bearings meant more 581 Volume brain volume Today, brain volume can be measured in volume of water that will be added by snowmelt and rain- living people using computer software that uses three- fall during a given period of time This indicates how dimensional medical scans of the brain to count how much water will arrive, and when and how fast, in various many cubic centimeters of volume the brain occupies rivers or lakes But the fact that men, on average, have slightly larger Hydrogeologists and weather scientists use complex brains (about 10% larger) does not mean that men are mathematical equations, satellite data, soil-test data, and smarter than women To begin with, a bigger brain does not computer programs to predict runoff volumes Some of mean a more intelligent mind, and there is great individual the factors that they must take into account include rain variation among people of both sexes Some famous schol- amount, intensity, duration, and location; soil type and ars have been found, after death, to have brains only half the wetness; snowpack depth and location; temperature size of other scholars People of famous intelligence, like and sunshine; time of year; ground slope; and the type Einstein, usually do not have larger-than-average brain vol- and health of the vegetation covering the ground All this ume Second, about half of the average size difference is information goes into a mathematical model of the accounted for by the fact that men tend to be larger than stream, lake, or reservoir basin into which the water is women Brain size goes, on average, with body size: taller, draining Given the exact shape of the basin receiving the more muscular men tend to have larger brains than smaller, water, water volume can be translated into water depth less muscular men Elephants and whales have larger brains In some places, water can be drained from reservoirs to several times larger than those of human beings, but are not make room for the volume of water that has been forecast more intelligent To some extent, therefore, men have larger to flow from higher ground, thus preventing floods brains only because their bodies are larger, too Where to Learn More In the nineteenth and early twentieth centuries, brain-volume measurements were used to justify laws Books that allowed only men to vote and hold some other legal Tufte, Edward R The Visual Display of Quantitative Information rights This is a classic case of accurate measurements being interpreted in a completely misleading way Cheshire, CT: Graphics Press, 2001 RUNOFF Web sites “Causes of Sea Level Rise.” Columbia University, 2005 Runoff is water from rain or melting snow that runs off the ground into streams and rivers instead of soaking Ͻhttp://www.columbia.edu/~epg40/elissa/webpages/ into the ground Scientists and engineers who study flood Causes_of_Sea_Level_Rise.htmlϾ (April 4, 2005) control, sewage management, generating electricity from “Making a River Forecast.” US National Weather Service, rivers, shipping goods on rivers, or recreation on rivers Sep 21, 2004 Ͻhttp://www.srh.noaa.gov/wgrfc/resources/ make determinations of water volume to estimate supply making_forecast.htmlϾ (April 6, 2005) To make an educated guess, they initially estimate the “Volume.” Mathworld 2005 Ͻhttp://mathworld.wolfram.com/ Volume.htmlϾ (April 4, 2005) 582 REAL-LIFE MATH ... example of inverse proportion It Sunny Mathville Metlock can be worked out using the unitary method: 10 people: Oneville 10 hours; person: 100 hours; people: 50 hours Figure Even... tion happens to involve mathematics Mathematical modeling is considered to be the process of turning real- life problems into the more abstract and rigorous lan- guage of mathematics It generally... have ideas about how mathematical ability to solve the problem It is for this rea- to make the interface look and how to program the son that mathematics, whether through mathematicians, operating