$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 6.8E: Exercises for Section 6.8

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

In exercises 1 - 2, answer True or False? If true, prove it. If false, find the true answer.

1) The doubling time for $$y=e^{ct}$$ is $$\dfrac{\ln 2}{\ln c}$$.

2) If you invest $$500$$, an annual rate of interest of $$3\%$$ yields more money in the first year than a $$2.5\%$$ continuous rate of interest.

True

3) If you leave a $$100°C$$ pot of tea at room temperature ($$25°C$$) and an identical pot in the refrigerator $$(5°C)$$, with $$k=0.02$$, the tea in the refrigerator reaches a drinkable temperature $$(70°C)$$ more than $$5$$ minutes before the tea at room temperature.

4) If given a half-life of t years, the constant $$k$$ for $$y=e^{kt}$$ is calculated by $$k=\frac{\ln\left(\frac{1}{2}\right)}{t}$$.

False; $$k=\dfrac{\ln 2}{t}$$

In exercises 5 - 18, use $$y=y_0e^{kt}.$$

5) If a culture of bacteria doubles in $$3$$ hours, how many hours does it take to multiply by $$10$$?

6) If bacteria increase by a factor of $$10$$ in $$10$$ hours, how many hours does it take to increase by $$100$$?

$$20$$ hours

7) How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is $$5730$$ years.

8) If a relic contains $$90\%$$ as much radiocarbon as new material, can it have come from the time of Christ (approximately $$2000$$ years ago)? Note that the half-life of radiocarbon is $$5730$$ years.

No. The relic is approximately $$871$$ years old.

9) The population of Cairo grew from $$5$$ million to $$10$$ million in $$20$$ years. Use an exponential model to find when the population was $$8$$ million.

10) The populations of New York and Los Angeles are growing at $$1\%$$ and $$1.4\%$$ a year, respectively. Starting from $$8$$ million (New York) and $$6$$ million (Los Angeles), when are the populations equal?

$$71.92$$ years

11) Suppose the value of $$1$$ in Japanese yen decreases at $$2\%$$ per year. Starting from $$1=¥250$$, when will $$1=¥1$$?

12) The effect of advertising decays exponentially. If $$40\%$$ of the population remembers a new product after $$3$$ days, how long will $$20\%$$remember it?

$$5$$ days $$6$$ hours $$27$$minutes

13) If $$y=1000$$ at $$t=3$$ and $$y=3000$$ at $$t=4$$, what was $$y_0$$ at $$t=0$$?

14) If $$y=100$$ at $$t=4$$ and $$y=10$$ at $$t=8$$, when does $$y=1$$?

At $$t = 12$$

15) If a bank offers annual interest of $$7.5\%$$ or continuous interest of $$7.25\%,$$ which has a better annual yield?

16) What continuous interest rate has the same yield as an annual rate of $$9\%$$?

$$8.618\%$$

17) If you deposit $$5000$$at $$8\%$$ annual interest, how many years can you withdraw $$500$$ (starting after the first year) without running out of money?

18) You are trying to save $$50,000$$ in $$20$$ years for college tuition for your child. If interest is a continuous $$10\%,$$ how much do you need to invest initially?

\$6766.76

19) You are cooling a turkey that was taken out of the oven with an internal temperature of $$165°F$$. After $$10$$ minutes of resting the turkey in a $$70°F$$ apartment, the temperature has reached $$155°F$$. What is the temperature of the turkey $$20$$ minutes after taking it out of the oven?

20) You are trying to thaw some vegetables that are at a temperature of $$1°F$$. To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of $$44°F$$. You check on your vegetables $$2$$ hours after putting them in the refrigerator to find that they are now $$12°F$$. Plot the resulting temperature curve and use it to determine when the vegetables reach $$33°$$.

$$9$$hours $$13$$minutes

21) You are an archeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era ($$146$$ million years to $$65$$ million years ago), and you find by radiocarbon dating that there is $$0.000001\%$$ the amount of radiocarbon. Is this bone from the Cretaceous?

22) The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of $$24,000$$ years. If $$1$$ barrel containing $$10$$ kg of plutonium-239 is sealed, how many years must pass until only $$10$$ g of plutonium-239 is left?

$$239,179$$ years

For exercises 23 - 26, use the following table, which features the world population by decade.

 Years since 1950 Population (millions) 0 2,556 10 3,039 20 3,706 30 4,453 40 5,279 50 6,083 60 6,849

Source: http:/www.factmonster.com/ipka/A0762181.html.

23) [T] The best-fit exponential curve to the data of the form $$P(t)=ae^{bt}$$ is given by $$P(t)=2686e^{0.01604t}$$. Use a graphing calculator to graph the data and the exponential curve together.

24) [T] Find and graph the derivative $$y′$$ of your equation. Where is it increasing and what is the meaning of this increase?

$$P'(t)=43e^{0.01604t}$$. The population is always increasing.

25) [T] Find and graph the second derivative of your equation. Where is it increasing and what is the meaning of this increase?

26) [T] Find the predicted date when the population reaches $$10$$ billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.

The population reaches $$10$$ billion people in $$2027$$.

For exercises 27 - 29, use the following table, which shows the population of San Francisco during the 19th century.

 Years since 1850 Population (thousands) 0 21.00 10 56.80 20 149.5 30 234.0

Source: http:/www.sfgenealogy.com/sf/history/hgpop.htm.

27) [T] The best-fit exponential curve to the data of the form $$P(t)=ae^{bt}$$ is given by $$P(t)=35.26e^{0.06407t}$$. Use a graphing calculator to graph the data and the exponential curve together.

28) [T] Find and graph the derivative $$y′$$ of your equation. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal?

$$P'(t)=2.259e^{0.06407t}$$. The population is always increasing.

29) [T] Find and graph the second derivative of your equation. Where is it increasing? What is the meaning of this increase?