6.2 Differential Equations: Growth and Decay 367
6.2 Differential Equations: Growth and Decay
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Solve the differential equation.
a.
b.
c.
d.
e.
____ 2. Solve the differential equation.
a.
b.
c.
d.
e.
____ 3. Solve the differential equation.
a.
b.
c.
d.
e.
368 Chapter 6: Differential Equations
____ 4. Solve the differential equation .
a.
b.
c.
d.
e.
____ 5. Solve the differential equation.
a.
b.
c.
d.
e.
____ 6. Write and solve the differential equation that models the following verbal statement:
The rate of change of with respect to is proportional to .
a.
b.
c.
d.
e.
6.2 Differential Equations: Growth and Decay 369
____ 7. Find the function passing through the point with the first derivative
.
a.
b.
c.
d.
e.
____ 8. Find the function passing through the point with the first derivative
.
a.
b.
c.
d.
e.
____ 9. Write and solve the differential equation that models the following verbal statement.
Evaluate the solution at the specified value of the independent variable, rounding your answer to four
decimal places:
The rate of change of is proportional to . When , and when , = 84. What is the
value of when ?
a.
b.
c.
d.
e.
370 Chapter 6: Differential Equations
____ 10. The rate of change of N is proportional to N. When and when
. What is the value of N when ? Round your answer to three decimal places.
a. 2,129.520
b. 2,099.520
c. 2,049.520
d. 491.383
e. 262,440.000
____ 11. Find the exponential function that passes through the two given points.
Round your values of C and k to four decimal places.
a.
b.
c.
d.
e.
____ 12. The isotope
has a half-life of 5,715 years. Given an initial amount of 11 grams of
the isotope, how many grams will remain after 500 years? After 5,000 years? Round your answers to
four decimal places.
a. 7.2469 gm, 4.1988 gm
b. 6.2117 gm, 3.5989 gm
c. 10.3528 gm, 5.9982 gm
d. 4.1411 gm, 2.3993 gm
e. 12.4233 gm, 7.1979 gm
6.2 Differential Equations: Growth and Decay 371
____ 13. The half-life of the radium isotope Ra-226 is approximately 1,599 years. If the initial
quantity of the isotope is 38 g, what is the amount left after 1,000 years? Round your answer to two
decimal places.
a. 24.63 g
b. 30.60 g
c. 25.13 g
d. 11.88 g
e. 12.32 g
____ 14. The isotope has a half-life of 5,715 years. After 2,000 years, a sample of the
isotope is reduced to 1.2 grams. What was the initial size of the sample (in grams)? How much will
remain after 20,000 years (i.e., after another 18000 years)? Round your answers to four decimal
places.
a. 1.0706 , 0.0947
b. 2.4471 , 0.2164
c. 1.5294 , 0.1352
d. 2.1412 , 0.1893
e. 1.9883 , 0.1758
____ 15. The isotope has a half-life of 24,100 years. After 10,000 years, a sample of the
isotope is reduced 1.6 grams. What was the initial size of the sample (in grams)? How large was the
sample after the first 1,000 years? Round your answers to four decimal places.
a. 2.1332 , 2.0727
b. 2.7731 , 2.6945
c. 1.2799 , 1.2436
d. 1.7065 , 1.6582
e. 1.0666 , 1.0364
____ 16. The half life of the radium isotope Ra-226 is approximately 1,599 years. If the
amount left after 1,000 years is 1.8 g, what is the amount after 2000 years? Round your answer to
three decimal places.
a. 1.167 g
b. 0.939 g
c. 1.800 g
d. 0.490 g
e. 2.334 g
____ 17. The half-life of the radium isotope Ra-226 is approximately 1,599 years. What
percent of a given amount remains after 800 years? Round your answer to two decimal places.
a. 70.70 %
b. 5.71 %
c. 72.70 %
d. 25.02 %
e. 0.71 %
372 Chapter 6: Differential Equations
____ 18. The initial investment in a savings account in which interest is compounded
continuously is $813. If the time required to double the amount is years, what is the annual rate?
Round your answer to two decimal places.
a. 7.49 %
b. 7.70 %
c. 13.34 %
d. 6.29 %
e. 8.89 %
____ 19. The initial investment in a savings account in which interest is compounded
continuously is $604. If the time required to double the amount is years, what is the amount after
15 years? Round your answer to the nearest cent.
a. $1,917.58
b. $1,804.46
c. $1,907.37
d. $1,404.46
e. $8,278.18
____ 20. Find the principal that must be invested at the rate 8%, compounded monthly, so that
$1,000,000 will be available for retirement in 50 years. Round your answer to the nearest cent.
a. $250,000.00
b. $18,560.39
c. $717,324.37
d. $333,333.33
e. $21,321.23
____ 21. Find the time (in years) necessary for 1,000 to double if it is invested at a rate 6%
compounded continuously. Round your answer to two decimal places.
a. 1.16 years
b. 11.55 years
c. 1.39 years
d. 11.90 years
e. 11.58 years
____ 22. Suppose that the population (in millions) of Paraguay in 2007 was 6.7 and that the
expected continuous annual rate of change of the population is 0.024. Find the exponential growth
model for the population by letting correspond to 2000. Round your answer to four
decimal places.
a.
b.
c.
d.
e.
6.2 Differential Equations: Growth and Decay 373
____ 23. Suppose that the population (in millions) of a Egypt in 2007 is 80.3 and that expected
continuous annual rate of change of the population is 0.017. The exponential growth model for the
population by letting corresponds to 2000 is . Use the model to predict the
population of the country in 2013. Round your answer to two decimal places.
a. 83.08 million
b. 87.42 million
c. 90.45 million
d. 88.92 million
e. 81.68 million
____ 24. The number of bacteria in a culture is increasing according to the law of exponential
growth. After 5 hours there are 175 bacteria in the culture and after 10 hours there are 425 bacteria in
the culture. Answer the following questions, rounding numerical answers to four decimal places.
(i) Find the initial population.
(ii) Write an exponential growth model for the bacteria population. Let t represent time in hours.
(iii) Use the model to determine the number of bacteria after 20 hours.
(iv) After how many hours will the bacteria count be 15,000?
a. (i) 72.0588 ; (ii) ; (iii) 3,819.3668 ; (iv) 32.4162 hr
b. (i) 74.2088 ; (ii) ; (iii) 5,194.0840 ; (iv) 34.6442 hr
c. (i) 72.0588 ; (ii) ; (iii) 2,506.6327 ; (iv) 30.0817 hr
d. (i) 77.8388 ; (ii) ; (iii) 7,945.5374 ; (iv) 36.7554 hr
e. (i) 79.3988 ; (ii) ; (iii) 10,598.0009 ; (iv) 38.5348 hr
____ 25. A container of hot liquid is placed in a freezer that is kept at a constant temperature
of . The initial temperature of the liquid is . After 3 minutes, the liquid’s temperature is
. How much longer will it take for its temperature to decrease to ? Round your answer to
two decimal places.
a. 1.89 minutes
b. 2.84 minutes
c. 3.16 minutes
d. 1.26 minutes
e. 3.47 minutes
374 Chapter 6: Differential Equations
6.2 Differential Equations: Growth and Decay
Answer Section
6.2 Differential Equations: Growth and Decay 375