Q. 89

Question

Leila must also determine hunting policies to sustain a

population $W_{0}$ of wolves that satisfy federal guidelines,

while maximizing the sustained elk population $E_{0}$ for

which the state can sell hunting tags. Her predator-prey

models approximate the number of elk over time as

$E (t) = \frac{E_{0} + 72 e^{- 0.006 t} \sin (πt / 4) + 8 t e^{- 0.006 t} \sin (πt / 4)}{1 + 0 \cdot 2 W_{0}}$

$(a)$ Use the Squeeze Theorem for Limits to show that the

population goes toward $\frac{E_{0}}{1 + 0 \cdot 2 W_{0}} a s t \to \infty$

$(b)$ Explain why L’Hopital’s Rule is ˆ not a good method for

calculating this limit.

Step-by-Step Solution

Verified

Answer

Part (a) Therefore the population goes towards $\frac{E_{0}}{1 + 0 \cdot 2 W_{0}} a s t \to \infty$

Part (b) The L’Hopital’s Rule is not good for calculating limits

1Part(a) Step 1. Given information

Given $E (t) = \frac{E_{0} + 72 e^{- 0.006 t} \sin (πt / 4) + 8 t e^{- 0.006 t} \sin (πt / 4)}{1 + 0 \cdot 2 W_{0}}$

2Part (a) Step 2. To show the population goes towards E 0 1 + 0 · 2 W 0   a s   t → ∞

$\lim_{t \to \infty} E (t) = \lim_{t \to \infty} \frac{E_{0} + 72 e^{- 0.006 t} \sin (πt / 4) + 8 t e^{- 0.006 t} \sin (πt / 4)}{1 + 0 \cdot 2 W_{0}} \lim_{t \to \infty} E (t) = \lim_{t \to \infty} \frac{E_{0}}{1 + 0 \cdot 2 W_{0}} + \lim_{t \to \infty} \frac{72 e^{- 0.006 t} \sin (πt / 4) + 8 t e^{- 0.006 t} \sin (πt / 4)}{1 + 0 \cdot 2 W_{0}} = \frac{E_{0}}{1 + 0 \cdot 2 W_{0}} + \frac{72 \times 0 \times \sin (\frac{π \times 0}{4}) + 8 \times \infty \times \sin (\frac{π \times 0}{4})}{1 + 0 \cdot 2 W_{0}} = \frac{E_{0}}{1 + 0 \cdot 2 W_{0}}$