Homework 3 Solutions

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1 Homework 3 Solutions Chapter 3A Does it make sense? Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. 12. If they raise taxes by 10% every year, in a decade we ll be paying everything we earn in taxes. Does not make sense (generally). Increasing a quantity by several successive percent changes is not the same as a one-time increase that is the sum of the changes. For instance, if the current tax rate is 10%, the rate for the second year will be ( )% = 11%, and after 10 years the rate will be about 25.9%. (On the other hand, if the current rate is 40%, after 10 years the rate would be around 103.7% or more than 100%.) Review of Ratios. Compare the following pairs of numbers A and B in three ways: a. Find the ratio of A to B. b. Find the ratio of B to A. c. Complete the sentence: A is percent of B. (Refer to the Brief Review on p. 134).) 35. A = 150 and B = 400 a. 150 to 400 = 3 / 8 b. 400 to 150 = 8 / 3 c. A is 37.5 percent of B. 43. A = 100 square meters is the area of a square with sides of length 10 meters and B = 25π square meters is the area of a circle with a diameter of 10 meters. a. 100 m 2 to 25π m 2 = 4 / π 1.27 b. 25π m 2 to 100 m 2 = π / c. A is 127 percent of B. 52. Population Comparison. Between 2000 and 2010, the population of Las Vegas increased from 478,000 to 565,000. Over the same decade, the population of Dallas increased from 1,189,000 to 1,245,000. Which city had the greater absolute growth in population? Which city had the greater percentage growth in population? Las Vegas vs. Dallas absolute: 565, , 000 = 87, 000 > 1, 245, 000 1, 189, 000 = 56, 000 percentage: 565, , , 000 = 18.2% > 1, 245, 000 1, 189, 000 1, 189, 000 = 4.7%

2 75. Care in Wording. Assume that 30% of city employees in Carson City ride the bus to work. Consider the following two statements: The percentage of city employees who ride the bus to work is 10% higher in Freetown than in Carson City. The percentage of city employees who ride the bus work is 10 percentage points higher in Freetown than in Carson City. For each case, state the percentage of city employees in Freetown who ride the bus to work. Briefly explain why the two statements different meanings. In the first case, the increase is relative so 30% 1.1 = 33% of the city employees in Freetown ride the bus to work. In the second case, the increase is an absolute percentage (i.e. percentage points) so 30% + 10% = 40% of the city employees ride the bus to work. Solving Percentage Problems. Solve the following percentage problems. 80. The number of housing foreclosures in 2008 was 3.98 million, up 81% over How many foreclosures were there in 2007? 3.98 million ( ) 2.20 million foreclosures in Percentages in the News. Answer the question that follows each quote from a news source The unemployment rate has risen more than a percentage point, to 8.5% in February from 7.1% last November. What is the relative change in the unemployment rate expressed as a percentage? 8.5% 7.1% 7.1% or 19.7%. Chapter 3B Does it make sense? Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. 10. I ve seen about commercials on TV. No, is an astronomically large number. Review of Scientific Notation. In the following exercises, use the skills covered in the Brief Review on p Write each of the following numbers in scientific notation. a. 233 b. 126,547 c d e f a b c d e f

3 Using Scientific Notation. Rewrite the following statements using a number in scientific notation. 27. The diameter of a typical bacterium is about meter. The diameter of a typical bacterium is about meter. Approximation with Scientific Notation. Estimate the following quantities without using a calculator. Then find the exact result, using a calculator if necessary. Discuss whether your approximation technique worked. 30. a. 5.6 billion 200 = b. 4 trillion 260 million c , d. 3 billion 25, 000 = e = 2,041,567 f Perspective Through Estimation. Use estimation to make the following comparisons. Discuss your conclusion. 32. Could a person walk across the United States (New York to California) in a year? If not, about how long would it take? Yes. The distance from New York to California is about 2,400 miles, which means a person would only have to walk about 6.6 miles per day (a modest rate of 2 mph, for 8 hours a day, would be 16 miles per day). Order of Magnitude Estimates. Make order of magnitude estimates of the following quantities. Explain the assumptions you use in your estimates. 35. The number of times your heart beats in a day. Assuming one beat per second, we have 24 hr 60 min 60 sec 1 day 1 day hr min 1 beat = 86, 400 beats. sec Energy Comparisons. Use Table 3.1 to answer the following questions. 41. How many average candy bars would you have to eat to supply the energy needed for 6 hours of running? Since one average candy bar supplies about joules and one hour of running burns about , you would need to eat ( ) ( joules = 4 average candy bars for each hour of running. Thus, you would need to each a total of 4 6 = 24 candy bars to supply the energy needed for 6 hours of running. 44. Compare the energy released by burning 1 liter of of oil to that released by fusion of the hydrogen in 1 liter of water. One liter of water supplies enough enough hydrogen to produce joules of energy through fusion, which is ( ) ( ) 5, 833, 333 times as much as the energy released by burning a liter of oil (i.e. roughly 6 million times as much).

4 54. Interstellar Travel. The fastest spaceships launched to date are traveling away from Earth at speeds of about 50,000 kilometers per hour. How long would such a spaceship take to reach Alpha Centauri? (Hint: See Example 8.) Based on your answer, write one or two paragraphs discussing whether interstellar travel is a realistic possibility today. Since Alpha Centauri is about km from Earth, it would take km 50, 000 km / hr = hr, which is about 100,000 years... Making Numbers Understandable. Restate the following facts as indicated. 57. There are approximately 2.4 million deaths per year in the United States. Express this quantity in deaths per minutes deaths 1 yr 1 yr 365 day 1 day 24 hr 1 hr 60 min 4.57 deaths / min 60. In 2007, Walmart had profits of $12.7 billion. Express this profit in terms of dollars per minute. $ yr 1 yr 365 day 1 day 24 hr 1 hr $24, 162/ min. 60 min 63. Americans consume an estimated 7.6 billion pounds of candy (excluding chewing gum) per year. Express this quantity in terms of pounds per person per month. (Use a population of 305 million.) pounds of candy people 24.9 lbs. per person per year, or pounds of candy lbs. per person per month. people 12 months 68. Wood for Energy? A total of about 180,000 terawatts of solar power reaches Earth s surface, of which about 0.06% is used by plants for photosynthesis. Of the energy that goes to photosynthesis, about 1% ends up stored in plant matter (including wood). (Recall that 1 watt = 1 joule/s; 1 terawatt = watts.) a. Calculate the total amount of energy that becomes stored in plant matter each second. 180,000 terawatts of solar power is joules/ sec, which means that 1 sec joules/ sec = joules becomes stored in plants each second.

5 b. Suppose that power stations generated electricity by burning plant matter. If all the energy stored in plants could be converted to electricity, what average power, in terawatts, would be possible? Would it be enough to meet world electricity demand, which is of order 10 terawatts? Converting all the energy stored in plants each second to electricity would produce joules/ sec = W or about 1 terawatt, which is only about 10% of the electricity demand. c. Based on your answer to b, can you draw any conclusions about why humans depend on fossil fuels, such as oil and coal, which are the remains of plants that died long ago? Explain. The rate of growth of new plants could never keep up with demand, so we must use the energy stored in millions of years worth of fossilized plants to avoid using up all the plant life on Earth.