18.1 Geometric Similarity

1. Your digital camera probably takes pictures with an aspect ratio of 4 to 3, meaning that the longer side is 4/3 times as long (in pixels) as the shorter side. For example, you probably can take a “small” picture with 640 pixels by 480 pixels, or perhaps a “large” picture with 2592 pixels by 1944 pixels (for a total of pixels, or just a little more than 5 megapixels). Photographic prints from your digital camera are available in various sizes of paper, quoted in inches: , and .

2. A natural application of this idea is to pizza. The prices at Vince’s pizza restaurant in Beloit, Wisconsin, are $7.57, $8.71, $9.49, $11.13, and $12.63, respectively, for small (10-in.), medium (12-in.), large (14-in.), extra-large (16-in.), and XX-large (18-in.) cheese pizzas.

3. The NBC Nightly News on June 21, 2010, featured a story about food portions, stating that dinner plates were 9 in. in diameter in the 1960s but now they are 12 in. in diameter, “making room for one-third more food.”

4. Dollhouses and their furnishings are usually built to a scale of exactly 1 in. to 1 ft, meaning that an item 1 ft long in a real house is 1 in. long in a dollhouse.

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5. Shellac O’Lean, a freshman on your school’s football team, is 6 ft tall and weighs 150 lb. Since he would rather play basketball, he plans to grow to 7 ft 1 in., the same height as Shaquille O’Neal, his idol. If Shellac scales geometrically as he grows, how much can he expect to weigh? (O’Neal weighs 325 lb.)

6. At our house, we have a 10-in. frying pan and a 12-in. frying pan; the 12-in. one weighs a lot more, cooks food more slowly, and never gets as hot. Suppose that a 10-in. frying pan weighs 1 lb, apart from its handle. How much would a geometrically similar 12-in. frying pan weigh? How much would it weigh if it had the same thickness of metal as the 10-in. pan?

7. The human figures in Lego sets are 4 cm tall (without hats or helmets).

8. Recent dollar coins (Presidential, Susan B. Anthony, Sacagawea) have been largely rejected by the public, which finds them too small and too light. Suppose that you are appointed to the U.S. Citizens Coinage Advisory Committee and commissioned to design a new $5 coin. (Whom should it depict?) The sole requirement is that it be made of the same material as the quarter but weigh 4 times as much. A quarter can be described geometrically as a circular cylinder approximately 24.26 mm in diameter and 1.75 mm thick. Because your new dollar should weigh 4 times as much, it needs to have 4 times the volume of a quarter. [The formula for the volume of a cylinder is , where is the height and is the diameter.]

9. Criticize the following statement and write a correct version.

Murders by firearms in Britain are 30 times fewer per capita than in the US.

—ragingbull.com/forum/topic/449338

10. Criticize the following statement and write a correct version.

Energizer Ultimate Lithium … designed to last up to 9x longer, which means up to 9x less waste.

—Formerly stated on the Energizer website

11. Criticize the following statement and write a correct version.

Cadmium chloride is a nasty chemical… used as a coating for thin-film solar cells because it increases the efficiency of converting sunlight to energy…. [In tests] magnesium chloride … yielded comparable efficiency…. Magnesium chloride is also nontoxic, abundant and costs about 300 times less than cadmium chloride.

—Scientific American 311(3) (September 2014): 20

12. Criticize the following statement and write a correct version.

Antibiotics for children that are covered by private insurance cost five times more in the United States than in the United Kingdom.

—consumer.healthday.com/public-health-information-30/health-cost-news-348/briefs-emb-12-19-antibiotic-costs-pharmacotherapy-bumc-release-batch-1071-683273.html

13. Abuses of the language of comparison aren’t hard to find. For example, the phrase “times less than” occurs more than 1 million times in documents on the Internet. Search on the Internet to find an abuse of “times” and “less than” together, figure out what the author meant to say, and write it in correct language.

14. The phrase “times more than” occurs suprisingly more frequently on the Internet than “times less than"— more than 29 million times. Search on the Internet to find an abuse of “times” and “more than” together, figure out what the author meant to say, and write it in correct language.

18.2 How Much Is That in …?

15. In 1991, Edward N. Lorenz, a meteorologist who was an early researcher into chaos and dynamical systems (discussed in Chapter 23), received the Kyoto Prize in Basic Sciences, consisting of a gold medal and 45 million Japanese yen (¥). If at the time, what was the value of the cash award in 1991 U.S. dollars? (In Chapter 21, we show how to convert such an amount to its value in today’s dollars.)

16. An average adult in the U.S. consumes about 2500 food calories per day; one food calorie has the same energy as . A gallon of gasoline contains 33.7 kWh of energy. Compare the energy consumption of a car that gets 30 mpg and is driven about 30 mi/day (say, 12,000 mi/yr) with the food energy consumption of its driver. What do you conclude?

17. The Toyota Prius recovers energy to its battery when the car brakes. In the “classic” Prius model (2002-2003), for each 50 watt-hours (Wh) of energy recovered, an icon appears on the dashboard screen. A gallon of gasoline contains 33.7 kWh of energy. Suppose gasoline costs $3.00/gal. How many icons would have to appear before the car saves $1 in gasoline cost? (We neglect consideration of the relative efficiency of the car in converting gasoline or recovered electrical energy into mechanical motion.)

18. About of U.S. land could be devoted to wind energy (“without raising too many hackles,” according to Englishman David J. C. Mackay, Sustainable Energy—Without the Hot Air, UIT Cambridge, 2009, p. 234). The windmills would generate on average . (The hardware required would be 100 times as much as is currently installed.)

For Exercises 19–20, refer to the following. How can an all-electric car get 99 mpg? The figure below shows the EPS fuel economy label for an all-electric car. The fuel economy is measured in miles per gallon gasoline equivalent (MPGe). A gallon of gasoline equivalent means the amount of alternative fuel (electricity, natural gas, or hydrogen) that contains the same energy as in a gallon of gasoline.

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The fuel economy label cites how many kilowatt- hours (kWh) of energy the vehicle uses per 100 miles— namely, 34 kWh. That number in turn is converted to MPGe via the equation

Thus, in the label shown, the number of kWh/100 mi is 34, so the .

19. The 2015 Kia Soul Electric vehicle uses a combined city/highway 32 kWh/100 mi.

20. Light-duty vehicles (passenger cars and trucks) travel about 2.66 trillion miles per year. Suppose that all of them were converted to run on electric power at 34 kWh/100 mi. How much electricity would be required annually?

21. In mid-2015, the cost of mailing a letter from the United States to the rest of the world was $1.15. How much was that in euros (€), the currency of the European Union (EU) in mid-2015, when the exchange rate was ? (For comparison, the cost then of a letter to the United States from the EU ranged from €0.51 to €1.70, depending on the country.)

22. The cost of mailing a letter from the United States to Canada in mid-2015 was USD $1.15. How much was that in Canadian dollars when the exchange rate was ? (The postage cost from Canada to the United States was CAD $1.10.)

23. In Germany, the fuel efficiency of cars is measured in liters of fuel per . A typical average in a compact station wagon is . What is that in miles per gallon (mpg)?

24. According to EPA ratings, the highest city gas mileage for non-electric 2015 cars was with the hybrid Toyota Prius c, at 53 mpg in the city. How many liters of gasoline does such a Prius use to travel 100 km in the city?

25. Consider a real locomotive that weighs 88 tons and an HO-gauge scale model of it, for which the linear scaling factor is 1/87.

26. What’s wrong in the following quotations?

27. Gasoline is sold in the United States by the U.S. gallon and in Europe by the liter (). What was the equivalent cost, in U.S. dollars per U.S. gallon, for gasoline in Germany priced in euros at €1.59 per liter, when , in December 2014?

28. Consider which uses more energy: your house— with heating, cooling, electric, and perhaps gas for cooking—or the cars in the household (say, 2). A common measure of energy is 1 BOE, one barrel of oil equivalent, the approximate energy released by burning one barrel of crude oil, for which we have

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29. Before 1933, the United States issued gold coins for general circulation, the highest denomination being $20. (The U.S. mint still produces gold coins but for collecting and investment purposes.) Suppose that you want to make gold coins available again, with the value of the coin the same as the value of the gold in it. Such a coin would be round and have a thickness of 1 mm (that is half the thickness of current $1 coins and thinner than a dime). Gold has a density of ; and the volume of a cylinder is , where is the radius and is the height (thickness). At a price of $1200 per ounce, what would be the diameter of a $20 gold coin?

18.3 Big Stuff

30. The weight of a 1-ft cube of steel is 500 lb. What is the pressure on the bottom face in

31. The express elevators to the observation deck of One World Trade Center in New York City travel a height of 1293 feet in 47 seconds, reaching a speed of 2000 ft/min.

32. As is the case for One World Trade Center, elevators in others of the world’s tallest buildings usually take less than 1 minute to get to the top. How fast, in meters per second, would the elevators in a mile-high building have to be to achieve that standard?

33. At birth, the average human female baby weighs 7 lb and is 19 in. in length (height). At 15 months of age, the average weight is 21 lb. What would you expect the height to be then? (Thanks for the idea to Shai Simonson, Stonehill College.)

34. (a) Give an estimate of the gorilla’s weight when it was half as tall.

(b) What assumptions are involved in your estimate?

(c) When the gorilla is standing, what is the pressure on its feet in pounds per square inch?

35. Suppose that King Kong is a gorilla scaled up with a linear scaling factor of 10.

36. You may want a waterbed, but waterbeds are not allowed in your building. Apart from the danger of flood if the bed should puncture or leak, the weight is an issue.

37. If you aren’t allowed to have a waterbed, how about a spa (hot tub)? Find the weight of the water in a spa that is in the shape of a cylinder 6 ft in diameter and 3.5 ft deep. (Hint: The volume of a cylinder is , where is the radius and is the height.)

38. What does the largest giant sequoia tree (named “Hyperion”) weigh? Model the tree as a (very elongated) cone. Assume that the tree is 379 ft high, has a circumference of 40 ft at the base, and that the density of the wood is . (The volume of a cone of height and radius is .)

39. A 6-ft-tall indoor holiday tree needs four strings of lights to decorate it. How many strings of lights are needed for an outdoor tree that is 30 ft high? (Thanks for the idea to Charlotte Chell, Carthage College, Kenosha, Wisconsin.)

40. According to Genesis 6:15, Noah was to build an ark: “The length of the ark shall be three hundred cubits, the breadth of it fifty cubits, and the height of it thirty cubits.” How does the volume of such an ark compare with that of a U.S. suburban house totaling , with 8-ft ceilings?

41. Goliath [of David and Goliath, as related in the Bible (I Samuel 17:4)] was “six cubits and a span.” A “span” was originally the distance from the tip of the thumb to the tip of the little finger when the hand is fully extended, about 9 in. What range of heights would this indicate for Goliath in feet and inches? In centimeters?

42. Calculate the BMI for a woman 160 cm tall who weighs 65 kg. Is she overweight according to the institute’s guidelines?

43. Suppose that weight and height are measured instead in U.S. customary units of pounds and inches. We can still calculate body weight divided by the square of height using these units. What conversion factor is necessary to convert this number to the BMI?

44. Calculate the BMI of a man who is 6 ft 2 in. tall and weighs 217 lb.

18.4 Dimension Tension

45. (a) Refer to Example 10 (page 756). What would you expect an individual Q. northropi to weigh if it had half the wingspan of an adult?

(b) If an individual weighed half as much as an adult, what would you expect its wingspan to be?

46. In the children’s story Peter Pan, Peter and Wendy can fly. We may suppose that they are 4 ft tall, so they are about 8 times as tall as a sparrow is long. What should their minimum flying speed be?

47. Icarus of Greek legend escaped from Crete with his father, Daedalus, on wings made by Daedalus and attached with wax. Against his father’s advice, Icarus flew too close to the sun; as a result, the wax melted, the wings fell off, and he plunged into the sea and drowned. What must have been his minimum cruising speed? What assumptions does your answer involve?

48. A typical ultralight aircraft (with pilot on board) weighs about 400 lb, has a wingspan of 28 ft, and has a maximum speed of 65 mph. Also, solar and human-powered aircraft, such as the Gossamer Condor and other superlightweight planes, fly more slowly than Example 10 (page 756) predicts; for instance, the Gossamer Condor is far longer than an ostrich, but it flies at only 12 mph. How are these aircraft possible?

49. Justify the claim in Example 10 on page 756 that a Q. northropi weighing 200 lb with a wingspan of 50 ft would have had the same wing loading as one weighing 100 lb with a wingspan of 36 ft.

50. Jonathan Swift’s Gulliver traveled to Lilliput, where the Lilliputians were human-shaped but only about 6 in. tall. In other words, they were geometrically similar in shape to ordinary human beings but only one-twelfth as tall. What would a Lilliputian weigh? Are Lilliputians ruled out by the size-shape and area-volume considerations in this chapter? If you think they are, what considerations do you find convincing? If not, why not?

51. The cult movie Them (1954) features enormous ants (8 m long by 3 m wide). We can investigate the feasibility of such a scaled-up insect by considering its oxygen consumption. A common ant, 1 cm long, needs 24 milliliters (mL) of oxygen per second for each cubic centimeter of its volume. Because an ant has no lungs, it absorbs oxygen through its “skin” at a rate of 6.2 mL per second per square centimeter. Suppose that the tissues of a scaled-up ant would have the same need for oxygen for each cubic centimeter, and that the ant’s skin could absorb oxygen at the same rate as a common ant.

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• As you may have noticed from stumps, as a tree grows older, its annual rings get less wide. Although the width of the ring varies somewhat from year to year with the amount of rainfall and other factors, the total area of each annual ring is roughly the same over the years, meaning that the tree adds roughly the same amount of mass each year. Call that amount ; the mass of the tree is at, where is its age in years.
• Over a large range of tree sizes and tree species, the diameter of a tree of a species is approximately proportional to the three-halves power of the height of the tree. (Different species have different constants of proportionality.) Thus, (which is shown in Exercise 52).

52. Suppose that a tree grows to 10 m in 15 years. How tall will it be (if it lives long enough) when it is 60 years old?

53. How long would it take the tree in Exercise 52 to grow to be 40 m tall?

54. Giant sequoias can reach 100 m after about 1000 years. If it could keep on growing at the same rate of its addition of mass, how long would it take a giant sequoia 100 m tall to grow to 200 m?

55. The branching of trees is similar to the branching of circulatory systems in the bodies of animals. For similar reasons, the area of the cross section of the tree at its base scales as the three-fourths power of the tree’s mass—that is, . Assume that most of the mass is in the trunk and model the tree either as a tall cylinder or an elongated cone . Show that the diameter of a tree is approximately proportional to the three-halves power of the height—that is, .

18.5 How We Grow

56. Listed below are the numbers of species of reptiles and amphibians on some Caribbean islands, together with the approximate areas of the islands. (Suggested by Florence Gordon of the New York Institute of Technology, with contributions from Kevin Mitchell and James Ryan of Hobart and William Smith Colleges, Geneva, New York. This table is adapted from Tables 15 and 16 in P. J. Darlington, Zoogeography: The Geographic Distribution of Animals, Wiley, New York, 1957, pp. 483-484.)

57. The accompanying table lists the weights and wingspans of some birds and of some fully loaded airplanes. (Idea and most of the data contributed by Florence Gordon, New York Institute of Technology.)

58. Each December, the Friends of the Rock River Philharmonic raises funds with a holiday sale of poinsettia plants. The choices available, with cost in 2014 and average number of flowers per plant, are shown in the accompanying table.

Chapter Review

59. In the photo below of a statue of Paul Bunyan in Akeley, Minn., he would appear to stand about 33 ft tall and be geometrically similar to an ordinary muscular man.

60. Some humans, such as the Bushmen of the Kalahari Desert in Africa, live in desert environments, where it is important to be able to do without water for periods of time. Would you expect such an environment to favor short people or tall ones? (Adapted from A. Zherdev, Horseflies and flying horses, Quantum, May-June 1994, 32-37, 59-60.)

61. Smaller birds and mammals generally maintain higher body temperatures than do larger ones. Explain why you would expect this to be so. (Adapted from A. Zherdev, Horseflies and flying horses, Quantum, May–June 1994, 32–37, 59–60.)

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62. According to Time (March 7, 2005), men’s brains on average are 10% larger than women’s, even though men on average are only 8% taller. (The article mainly discusses the many differences in brain structure that likely outweigh any size differences.) If the brain scales linearly with height, and men are 8% taller, what percentage larger would you expect their brains to be? (And this consideration has nothing to do with relative intelligence!)

63. A common measure of water in agriculture is the KAF, which is one thousand acre-feet. An acre-foot of water is the amount of water to cover a field of area one acre to a depth of one foot. A common measure of water usage in homes is the water billing unit, which is 748 gallons.

64. The largest animal ever to live is the blue whale. No blue whale has been weighed as a whole, but estimates are that the largest ones weigh 200 tons, are 30 m long, and have a lung volume of 5000 L. An adult human male 6 ft tall and weighing 200 lb has a lung capacity of about 6 L.

65. The U.S. Energy Security and Independence Act of 2007 required that 15 billion gallons of ethanol and other renewable fuels be blended with gasoline in 2015. In the United States, corn is the main source of ethanol. What percentage of total arable U.S. land (land that can be planted to crops) would be needed to raise that much corn ethanol? The contiguous United States (without Alaska or Hawaii) has hectares of arable land, which could yield about 400 gal/acre of ethanol.

66. Unit conversion can matter, and the number of significant digits in the conversion factor can matter. New electronic components are designed using metric units, but accompanying circuit boards are often designed using U.S. customary units. Consider a connector with 100 pins on a side with design distance (“pitch”) of 0.6 mm from one pin to the next. (Thanks for the idea to Mark Biegert, Calix, Inc., Plymouth, Minnesota.)