Homework 6 – Solutions
Chapter 8A
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.
6. Suppose you had a magic bank account in which your balance doubled each day. If
you started with just $1, you’d be a millionaire in less than a month.
Makes sense. After only twenty days, the balance would be $220 = $1,048,576.
Linear or Exponential? State whether the growth (or decay) is linear or exponential,
and answer the associated question.
9. The population of Meadow View is increasing at a rate of 325 people per year. If the
population is 2500 today, what will it be in four years?
Linear growth. In four years, the population will be 2500 + 4(325) = 3800.
13. The price of computer memory is decreasing at a rate of 14% per year. If a memory
chip costs $50 today, what will it cost in three years?
Exponential decay. In three years, the price will be $50(1 − 0.14)3 = $31.80.
14. The value of your car is decreasing by 10% per year. If the car is worth $12,000 today,
what will it be worth in two years?
Exponential decay. In three years, the price will be $12,000(1 − 0.10)2 = $9720.
Chessboard Parable. Use the chessboard presented in the text. Assume that each grain
of wheat weighs 1/7000 pound.
20. According to the U.S. Department of Agriculture, the current world harvest of all
wheat, rice, and corn is less than 2 billion tons per year. How does this total compare
to the weight of the wheat on the chessboard? (Hint: 1 ton = 2000 pounds.)
The total weight of the wheat on the chess board is
1 pound
1 ton
×
≈ 1.3 × 1012 tons,
(264 − 1) grains ×
7000 grains 2000 pounds
which is
1.3×1012
2×109
= 650 times the current world harvest.
Bacteria in a Bottle Parable. Use the bacteria parable presented in the text.
27. Knee Deep in Bacteria. The total surface area of the Earth is about 5.1 × 1014 m2 .
Assume that the bacteria continued their doublings for two hours (as discussed in the
text), at which point they were distributed uniformly over the Earth’s surface. How
deep would the bacterial layer be? Would it be knee-deep, more than knee-deep, or
less than knee-deep?
As discussed in the text, the volume of the colony of 2120 bacteria would occupy a
volume of 1.3 × 1015 m3 . Dividing by the surface area of the Earth, we have:
1.3 × 1015 m3 ÷5.1 × 1014 m2 ≈ 2.5 m,
which is more than 8 feet.
30. Doubling Time vs. Initial Amount.
a. Would you rather start with one penny ($0.01) and double your wealth every day
or start with one dime ($0.10) and double your wealth every five days (assuming
you want to get rich)? Explain.
You should take the penny. After five days, the penny would have grown to
25 = 32¢, whereas the dime has grown to only 20¢.
b. Would you rather start with one penny ($0.01) and double your wealth every day
or start with $1000 and double your wealth every two days (assuming you want to
get rich in the long run)? Explain.
You should take the penny. The penny will grow to 2n ¢ in n days while the
$1000 = 100,000¢ will grow to 100,000 × 2n/2 ¢ in n days, so the penny will be
worth more than $1000 after 34 days.
c. Which is more important in determining how fast exponential growth occurs: the
doubling time or the initial amount? Explain.
Doubling time is much more important in the long run than the initial amount.
Chapter 8B
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.
11. A toxic chemical decays with a half-life of 10 years, so half of it will be gone 10 years
from now and all the rest will be gone 20 years from now.
Does not make sense. While it’s true that half will be gone in ten years, over the
next ten years, over the next ten years, only half of what remains will decay.
Logarithms. Refer the Brief Review on p. 486. Determine whether each statement is true
or false without doing any calculations. Explain your reasoning.
13. 100.928 is between 1 and 10.
True. 1 = 100 < 100.928 < 101 = 10.
17. log10 π is between 3 and 4.
False. 100 = 1 < π < 10 = 101 , so 0 < log10 π < 1.
24. Using the approximation log10 5 = 0.699, find each of the following without a calculator.
a. log10 50
b. log10 5000
c. log10 0.05
d. log10 25
e. log10 0.20
f. log10 0.04
a. log10 (5 × 10) = 1.699
b. log10 (5 × 1000) = 3.699 c. log10 (5 ÷ 100) = −1.301
d. log10 (52 ) = 1.398
e. log10 (5−1 ) = −0.669
f. log10 (5−2 ) = −1.398
Doubling Time. Each exercise gives a doubling time for an exponentially growing quantity.
Answer the questions that follow.
32. The number of cells in a tumor doubles every 6 months. If the tumor begins with a
single cell, how many cells will there be after 6 years? after 10 years?
After 6 years, there will be 1 × 2
10
1 × 2 /0.5 = 1,048,576 cells.
6/0.5
= 4096 cells; while, after 10 years, there will be
World Population. In 2009 world population was 6.8 billion. Use the given doubling time
to predict the population in 2019, 2059, and 2109.
34. Assume a doubling time of 60 years.
10
2019 is 10 years later than 2009, so the population would be 6.8 × 2 /60 = 7.6 billion;
50
in 2059, the population will be 6.8×2 /60 = 12.1 billion; and, in 2109, the population
100
will be 6.8 × 2 /60 = 21.6 billion.
Doubling Time Formula. Use the approximate doubling time formula (rule of 70). Discuss whether the formula is valid for the case described.
37. The Consumer Price Index is increasing at a rate of 4% per year. What is its doubling
time? By what factor will prices increase in 3 years?
Using the rule of 70, Tdouble ≈ 70/4 = 17.5. A increase of 4% per year means that
over three years prices will increase by a factor of (1.04)3 = 1.125, which is very close
3
to the factor of 2 /17.5 = 1.126 given by the approximate doubling time formula.
Half-Life. Each exercise gives a half-life for an exponentially decaying quantity. Answer
the questions that follow.
43. The half-life of a drug in the bloodstream is 18 hours. What fraction of the original
drug dose remains in 24 hours? in 48 hours?
24/18
= 40% of the drug remains, while, after 48 hours, only
After 24 hours, 12
48/18
1
= 16% of the drug remains.
2
48. Radium-226 is a metal with a half-life of 1600 years. If you start with 1 kilogram of
radium-266, how much will remain after 1000 years? after 10,000 years?
1000/1600
After 1000 years, 1 kg × 12
= 0.65 kg of radium will remain, and, after 10,000
10000/1600
years, 1 kg × 12
= 0.013 kg of radium will remain.
Half-Life Formula. Use the approximate half-life formula (rule of 70). Discuss whether
the formula is valid for the case described.
50. A clean-up project is reducing the concentration of a pollutant in the water supply,
with a 3.5% decrease per week. What is the half-life of the concentration of pollutant?
What fraction of the original amount of the pollutant remain when the project ends
after 1 year (52 weeks)?
Using the rule of 70, Thalf ≈ 70/3.5 = 20. A decrease of 3.5% per year means that
after 52 weeks (1 − 0.035)52 = 15.7% of the pollutant will remain, which is very close
52
to the factor of 2 /20 = 16.5% given by the approximate half-life formula.
Exact Formula. Compare the doubling times found with the approximate and exact doubling time formulas. Then use the exact doubling time formula to answer the given
question.
53. Inflation is causing prices to rise at a rate of 12% per year. For an item that costs
$500 today, what will the price be in 4 years?
Using the rule 70, the approximate doubling time is Tdouble ≈ 70/12 ≈ 5.8 years, while
10 2
the exact doubling time is log log
(1+0.12) = 6.11 years. The price of a $500 item today
will be $500(2)
4/6.11
10
≈ $787.12 (or, with a more exact doubling time, $786.76).
58. Nuclear Weapons. Thermonuclear weapons use tritium for their nuclear reactions.
Tritium is a radioactive form of hydrogen (containing 1 proton and 2 neutrons) with a
half-life of about 12 years. Suppose a nuclear weapon contains 1 kilogram of tritium.
How much will remain in 50 years? Use your answer to explain why thermonuclear
weapons require regular maintenance.
50/12
After 50 years, there will be 1 kg × 21
= 0.056 kg of tritium remaining. If the
weapons are stored, and not maintained, the potency for which they are designed
will diminish fairly quickly over time.