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# 4.9E Exercises

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True or False? If true, prove it. If false, find the true answer.

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

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

Solution: True

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

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

Solution: False; $$\displaystyle k=\frac{ln(2)}{t}$$

For the following exercises, use $$\displaystyle y=y_0e^{kt}.$$

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

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

Solution: $$\displaystyle 20$$ 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 $$\displaystyle 5730$$ years.

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

Solution: No. The relic is approximately $$\displaystyle 871$$ years old.

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

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

Solution: $$\displaystyle 71.92$$ years

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

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

Solution: $$\displaystyle 5$$ days $$\displaystyle 6$$ hours $$\displaystyle 27$$minutes

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

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

Solution: $$\displaystyle 12$$

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

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

Solution: $$\displaystyle 8.618%$$

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

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

Solution: \$6766.76

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

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

Solution: $$\displaystyle 9$$hours $$\displaystyle 13$$minutes

21) You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era ($$\displaystyle 146$$ million years to $$\displaystyle 65$$ million years ago), and you find by radiocarbon dating that there is $$\displaystyle 0.000001%$$ 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 $$\displaystyle 24,000$$years. If $$\displaystyle 1$$ barrel containing $$\displaystyle 10kg$$ of plutonium-239 is sealed, how many years must pass until only $$\displaystyle 10g$$ of plutonium-239 is left?

Solution: $$\displaystyle 239,179$$ years

For the next set of exercises, 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

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

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

Solution: $$\displaystyle P'(t)=43e^{0.01604t}$$. 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 $$\displaystyle 10$$ billion. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.

Solution: The population reaches $$\displaystyle 10$$ billion people in $$\displaystyle 2027$$.

For the next set of exercises, 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

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

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

Solution: $$\displaystyle P'(t)=2.259e^{0.06407t}$$. 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?