**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.

**Answer**
- 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 \(k=\frac{\ln\left(\frac{1}{2}\right)}{t}\).

**Answer**
- 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 \(100\)?

**Answer**
- \(20\) 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.

**Answer**
- 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?

**Answer**
- \(71.92\) 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?

**Answer**
- \(5\) 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 \(y=1\)?

**Answer**
- 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 \(9\%\)?

**Answer**
- \(8.618\%\)

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?

**Answer**
- $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 \(33°\).

**Answer**
- \(9\)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?

**Answer**
- \(239,179\) 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?

**Answer**
- \(P'(t)=43e^{0.01604t}\). 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.

**Answer**
- 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?

**Answer**
- \(P'(t)=2.259e^{0.06407t}\). 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?

## Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.