6.8E: Fitting Exponential Models to Data (Exercises)
64. What is the carrying capacity for a population modeled by the logistic equation \(P(t)=\frac{250,000}{1+499 e^{-0.45 t}} ?\) What is the initial population for the model?
65. The population of a culture of bacteria is modeled by the logistic equation \(P(t)=\frac{14,250}{1+29 e^{-0.62 t}},\) where \(t\) is in days. To the nearest tenth, how many days will it take the culture to reach \(75 \%\) of its carrying capacity?
For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.
66 .
| x | f(x) |
|---|---|
| 1 | 409.4 |
| 2 | 260.7 |
| 3 | 170.4 |
| 4 | 110.6 |
| 5 | 74 |
| 6 | 44.7 |
| 7 | 32.4 |
| 8 | 19.5 |
| 9 | 12.7 |
| 10 | 8.1 |
67 .
| x | f(x) |
| 0.15 | 36.21 |
| 0.25 | 28.88 |
| 0.5 | 24.39 |
| 0.75 | 18.28 |
| 1 | 16.5 |
| 1.5 | 12.99 |
| 2 | 9.91 |
| 2.25 | 8.57 |
| 2.75 | 7.23 |
| 3 | 5.99 |
| 3.5 | 4.81 |
68 .
| x | f(x) |
| 0 | 9 |
| 2 | 22.6 |
| 4 | 44.2 |
| 5 | 62.1 |
| 7 | 96.9 |
| 8 | 113.4 |
| 10 | 133.4 |
| 11 | 137.6 |
| 15 | 148.4 |
| 17 | 149.3 |