Homework 2 – Solutions Chapter 2B Powers of 10. Evaluate the following expressions using skills covered in the Brief Review on on pp. 100. 1025 1015 c. 102 + 10−1

21. a.

b. 101 + 100 d. 102 − 101

a. 1010

b. 10 + 1 = 11

c. 100 + 0.1 = 100.1

d. 100 − 10 = 90

USCS Units. Answer the following questions involving conversions within the USCS system. 30. A small city produces 500,000 cubic feet of garbage per week. If all of this garbage were stacked neatly (in a nice vertical pile) on a 100-yard by 60-yard football field, how high would the pile be (in feet)?   3 ft 3 ft 500000 ft3 3 500000 ft ÷ 100 yd × × 60 yd × = ≈ 9.26 ft yd yd 54000 ft2 Metric Prefixes. Complete the following sentences with a number. All answers should greater than 1. 35. A square meter is

times as large as a square centimeter.

A square meter is 10000 times as large as a square centimeter. USCS-Metric Conversions. Convert the following quantities to the indicated units. 38. 160 centimeters to inches 160 cm ×

0.3937 in = 62.992 in 1 cm

52. Bicycle Power A 1-hour bike ride at a pace of 15 miles per hour consumes about 750 Calories. What is your average power output, in watts, during a 45-minute bike ride? Note that we’re looking for power output, which is a rate, so it doesn’t matter whether the bike ride was 45 minutes or 1 hour (in fact, a 45 minute would only consume 3/4 as many Calories in 3/4 as much time). So we have, 750 Cal 4184 joule 1 hr 1 min joule × × × ≈ 871.67 = 871.67 watts . s 1 hr 1 Cal 60 min 60 s

Densities. Compute the followings densities using the appropriate units. 59. New Jersey and Alaska have populations of 8.7 million and 680,000, respectively (U.S. Census Bureau, 2008). Their areas are 7417 and 571,951 square miles, respectively. Compute the population densities of both states. The population density for New Jersey is:

The population density for Alaska is:

8700000 people people ≈ 1173 . 7417 mi2 mi2

680000 people people ≈ 1.19 . 571951 mi2 mi2

Currency Conversions. Use the currency exchange rates in Table 2.1 in Unit 2A to answer the following questions. State all of the conversion factors that you use. 65. Suppose that a new fuel-efficient German car travels an average of 26 kilometers on 1 liter of gasoline. If gasoline costs 1.50 euros per liter, how much will it cost to drive 300 kilometers? 300 km ×

1L 1.50 euro $1.256 × × ≈ $21.74. 26 km 1L 1 euro

68. The monthly rent on on 80-square-meter apartment in Nice, France, is 1040 euros. The monthly rent on a 500-square-foot apartment in Santa Fe, New Mexico , is $800. In terms of price per area, which apartment is less expensive? The price in Santa Fe is:

The price in Nice is: 1040 euro × 80 m2



1m 3.28 ft

2 ×

$1.256 ≈ $1.52/ft2 1 euro

$800 = $1.60/ft2 500 ft2

So the apartment in Nice is less expensive. 71. Lake Victoria Lake Victoria is Africa’s largest lake and the second largest fresh-water lake in the world in terms of surface area. Its volume is approximately 2750 cubic kilometers and its surface area 68,800 square kilometers. a. What is the average depth of the lake (the depth of a box with the volume and surface area of the lake)? 2750 km3 ≈ 0.03997 km 68800 km2

or about 40 meters.

b. In the last six years, the water level of the lake has dropped 10 feet (from the depth computed in part a). Approximately how much water has the lake lost? 68800 km2 ×10 ft ×

0.3048 m 1 km × ≈ 210 km3 1 ft 1000 m

c. What percentage of the volume (2750 cubic kilometers) has been lost? 210 km3 ≈ 0.076 or 7.6% 2750 km3

79. Compact Fluorescent Light Bulbs You replace a 100-watt standard light bulb with a 25-watt compact fluorescent bulb that supplies the same light. Over a 10,000-hour life (typical for a compact fluorescent bulb), how much energy, in kilowatt-hours, do you save? If electricity cost 12¢ per kilowatt-hour, how much money do you save? (100 watt −25 watt) × 10000 hr ×

1 kW $0.12 × = $90. 1000 W 1 kW-hr

86. California Wind Power. California currently has wind farms capable of generating a total of 2500 megawatts (2.5 gigawatts) of power (roughly 2% of the state’s total electricity). a. Assuming that wind farms typically generate 30% of their capacity, how much energy, in kilowatt-hours, can the California wind farms generate in one year? Given that the average household uses about 10,000 kilowatt-hours of energy each year, how many households can be powered by these wind farms? The energy capacity, in kilowatt-hours, generated by the wind farms is: 2.5 × 109 W ×1 yr ×

365 day 24 hr 1 kW × × × 0.30 = 6.57 × 109 kW-hr. 1 yr 1 day 1000 W

This is enough to serve 1 home = 657, 000 homes. 10000kW-hr b. One of the great advantages of wind power is that it does not produce carbon dioxide emissions that contribute to global warming. On average, energy produced from fossil fuels generates about 1.5 pounds of carbon dioxide for every kilowatthour of energy. Suppose California did not have its wind farms and the energy were instead produced from fossil fuels. How much more carbon dioxide would be entering the atmosphere each year? If the energy produced by wind farms were instead produced by fossil fuels, there would be 6.57 × 109 kW-hr ×

6.57 × 109 kW-hr ×

1.5 lb = 9.855 × 109 lb 1kW-hr

or, about, 9,855,000 pounds of carbon dioxide entering the atmosphere each year.

Chapter 3A Review Questions. 5. Explain the difference between the terms percent (%) and percentage points. Give an example of how they can differ for the same situation. The term percent expresses a relative change while the term percentage points expresses an absolute change in something measured in percent (e.g. if an interest rate changes from 2% to 4%, it increases by 2 percentage points, which is a 100% increase).

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. 7. In many European countries, the percentage change in population has been negative in recent decades. Makes sense. If a country’s population is declining, the percent change will be negative. 10. I’ve decreased my caloric intake by 125%, which has helped me lose weight. Does not make sense. You an 125% decrease would mean a negative caloric intake. 16. My bank increased the interest rate on my savings account 100%, from 2% to 4%. Makes sense. A 100% increase is the same as doubling (i.e. double 2% is 4%). 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 p. 134).)

percent of B. (Refer to the Brief Review on

42. A = $87, 000 is the median price of a single-family home in Davenport, Iowa, in 2008 and B = $636, 000 is the median price of a single-family home in Honolulu, Hawaii, in 2008. a. $87, 000 to $636, 000 = 0.137. b. $636, 000 to $87, 000 = 7.31. c. A is 13.7

percent of B.

Percentage Change. Find the absolute change and the percentage change in the following cases. 53. The population of the United States increased from 249 million in 1990 to 308 million 2010. The absolute change is 308 − 249 = 59 million people. The relative change is 308 million − 249 million ≈ 0.237 = 23.7%. 249 million Percentages of Percentages. Describe each of the following changes in two ways: as an absolute change in terms of percentage points and as a relative change in terms of a percentage. 73. The percentage of Americans accessing the Internet increased from 66.9% in 2000 to 78.5% in 2010. The absolute change is 78.5% − 66.9% = 11.6 percentage points. The relative change is 78.5% − 66.9% ≈ 0.173 = 17.3%. 66.9%

Solving Percentage Problems. Solve the following percentage problems. 81. Your total cost for in a restaurant after the tip was $76.40. Assuming you left a 20% tip, what was the bill for food and taxes? $76.40/1.20 = $63.67. 83. Simon’s monthly take-home pay (after taxes) is $2200. If he pays 21% of his gross pay (before taxes) in tax, what is his gross pay? $2200/(1.00 − 0.21) = $2200/0.79 = $2784.81. Shifting Reference Value. State whether the following statements are true or false, and explain why. If a statement is false, state the true change. 86. You receive a pay raise of 5%, then receive a pay cut of 5%. After the two changes in pay, your salary is unchanged. False. Suppose you made $100. A pay raise of 5% gives you a new salary of $105. A subsequent cut of 5% of the new salary yields, $105 × (1.00 − 0.05) = $99.75, which is $0.25 less than you started with. 87. If the profits in your consulting business increase by 4% one year and decrease by 2% the following year, your profits are up by 2% over two years. False. As in (86), suppose your profits before the increase was, arbitrarily, $100. An increase of 4% gives you a new annual profit of $100 × 1.04 = $104. A subsequent decrease in profits by 2% results in a profit of $104 × 0.98 = $101.92, which is an increase of 1.92% not 2%. 95. Average Percentages. You are a teacher. Your first-period class, with 25 students, had a mean score of 86% on the midterm exam. Your second period class, with 30 students, had a mean score of 84% on the same exam. Does it follow that the mean score for both classes combined is 85%? Explain. No, the mean score is not 85%. Since the classes are different, we need to weigh them differently. Assume a 100-point exam, the first class scored a total of 25 × 86 = 2150 points, and the second scored a total of 30 × 84 = 2520 points. Thus the combined total scored by 55 students is 2150 + 2520 = 4670, and the mean score is 4670/55 ≈ 84.9%. Percentages in the News. Answer the question that follows each quote from a news source. 102. “. . . the median annual income for adults with bachelor’s degrees was $37,729, compared with $20,718 for those with only a high school diploma.” The annual income for adults with bachelor’s degrees is what percent greater than that for adults with a high school diploma only? Since $37, 729/$20, 718 ≈ 1.821, the average annual income for adults with a bachelor’s degree is 182.1% of (or, 82.1% greater than) the average income of adults with only a high school diploma.

107. Stock Market Losses a. The largest single-day point loss of the Dow Jones Industrial Average occurred on September 29, 2008, when the market lost 778 points and closed at 10,365. What was the percentage loss? 778 = 0.070 = 7.0%. 10365 + 778 b. The largest single-day percentage loss of the Dow Jones Industrial Average occurred on October 19, 1987, when the market closed down 22.6% at 1739. What was the point loss? Before the loss the Dow Jones was at $1739/(1.00 − 0.226) = 1739/0.774 ≈ $2247, so the point (i.e. absolute loss) was $2247 − $1739 = $508.