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
(10 + 0.1 ∗ 10)% = 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
p. 134).)
percent of B. (Refer to the Brief Review on
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 m2 to 25π m2 = 4/π ≈ 1.27
b. 25π m2 to 100 m2 = π/4 ≈ 0.785
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
565, 000 − 478, 000 = 87, 000
565, 000 − 478, 000
percentage:
= 18.2%
478, 000
absolute:
vs.
>
>
Dallas
1, 245, 000 − 1, 189, 000 = 56, 000
1, 245, 000 − 1, 189, 000
= 4.7%
1, 189, 000
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 2007. How
many foreclosures were there in 2007?
3.98 million ÷(1 + 0.81) ≈ 2.20 million foreclosures in 2007.
Percentages in the News. Answer the question that follows each quote from a news
source.
103. “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%
≈ 0.197 or 19.7%.
7.1%
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 1050 commercials on TV.
No, 1050 is an astronomically large number.
Review of Scientific Notation. In the following exercises, use the skills covered in the
Brief Review on p. 145.
17. Write each of the following numbers in scientific notation.
a. 233
d. 9736.23
a. 2.33 × 102
d. 9.73623 × 103
b. 126,547
e. 124.58
b. 1.26547 × 105
e. 1.2458 × 102
c. 0.11
f. 0.8642
c. 1.1 × 10−1
f. 8.642 × 10−1
Using Scientific Notation. Rewrite the following statements using a number in scientific
notation.
27. The diameter of a typical bacterium is about 0.000001 meter.
The diameter of a typical bacterium is about 1 × 10−6 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 = 2.8 × 107
b. 4 trillion ÷260 million ≈ 15385
c. 9000 × 54, 986 ≈ 4.95 ×
e. 5987 × 341 = 2,041,567
d. 3 billion ÷25, 000 = 1.2 × 105
f. 43 ÷ 765 ≈ 0.056
108
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
1 day ×
24 hr
60 min 60 sec 1 beat
×
×
×
= 86, 400 beats .
1 day
hr
min
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 1 × 106 joules and one hour of running
burns about 4×106 , you would need to eat (4×106 )÷(1×106 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 7 × 1013 joules of
energy through fusion, which is (7 × 1013 ) ÷ (1.2 × 107 ) ≈ 5, 833, 333 times as much
as the energy released by burning a liter of oil (i.e. roughly 6 million times as much).
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 4.1 × 101 3 km from Earth, it would take
4.1 × 101 3 km
= 8.2 × 108 hr,
50, 000 km/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.
2.4 × 106 deaths
1 yr
1 day
1 hr
×
×
×
≈ 4.57 deaths/min
1 yr
365 day
24 hr
60 min
60. In 2007, Walmart had profits of $12.7 billion. Express this profit in terms of dollars
per minute.
$1.27 × 1010
1 yr
1 day
1 hr
×
×
×
≈ $24, 162/ min .
yr
365 day
24 hr
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.)
7.6 × 109 pounds of candy
≈ 24.9 lbs. per person per year, or
305 × 106 people
7.6 × 109 pounds of candy
≈ 2.1 lbs. per person per month.
305 × 106 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 = 1012 watts.)
a. Calculate the total amount of energy that becomes stored in plant matter each
second.
180,000 terawatts of solar power is 1.8 × 1017 joules/sec, which means that
1 sec ×1.8 × 1017 joules/sec × 0.0006 × .01 = 1.08 × 1012 joules
becomes stored in plants each second.
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
1.08 × 1012 joules/sec = 1.08 × 1012 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.