5.8E: Exponential and Logarithmic Models (Exercises)
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For the following exercises, use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about \(17 \%\) each hour.
54. To the nearest minute, what is the half-life of the drug?
55. Write an exponential model representing the amount of the drug remaining in the patient's system after \(t\) hours. Then use the formula to find the amount of the drug that would remain in the patient's system after 24 hours. Round to the nearest hundredth of a gram.
For the following exercises, use this scenario: A soup with an internal temperature of \(350^{\circ}\) Fahrenheit was taken off the stove to cool in a \(71^{\circ} \mathrm{F}\) room. After fifteen minutes, the internal temperature of the soup was \(175^{\circ} \mathrm{F}\).\
56. Use Newton's Law of Cooling to write a formula that models this situation.
57. How many minutes will it take the soup to cool to \(85^{\circ} \mathrm{F} ?\)
For the following exercises, use this scenario: The equation \(N(t)=\frac{1200}{1+199 e^{-0.625 t}}\) models the number of people in a school who have heard a rumor after \(t\) days.
58. How many people started the rumor?
59. To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?
60. What is the carrying capacity?
For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.
| x | f(x) |
|---|---|
| 1 | 3.05 |
| 2 | 4.42 |
| 3 | 6.4 |
| 4 | 9.28 |
| 5 | 13.46 |
| 6 | 19.52 |
| 7 | 28.3 |
| 8 | 41.04 |
| 9 | 59.5 |
| 10 | 86.28 |
62 .
| x | f(x) |
|---|---|
| 0.5 | 18.05 |
| 1 | 17 |
| 3 | 15.33 |
| 5 | 14.55 |
| 7 | 14.04 |
| 10 | 13.5 |
| 12 | 13.22 |
| 13 | 13.1 |
| 15 | 12.88 |
| 17 | 12.69 |
| 20 | 12.45 |
63. Find a formula for an exponential equation that goes through the points (-2,100) and (0,4). Then express the formula as an equivalent equation with base \(e\).