Chapter 6.3, Problem 56E

a.

To determine

To find: the number of pumas and deer in the region at the given value of $t$ .

Answer to Problem 56E

Number of pumas = 357

Number of deer = 1500

Explanation of Solution

Given information:

The equations representing the Pumas $\left(P\right)$ and deer $\left(D\right)$ after $t$ years are shown below.

  $P=500+200\sin \left[0.4\left(t-2\right)\right]$............................ (1)

  $D=1500+400\sin \left(0.4t\right)$............................ (2)

  $t=0$

Calculation:

Put the value of t in equation (1) and evaluate P as follows:

  $\begin{array}{c}P=500+200\sin \left[0.4\left(0-2\right)\right]\\ =500+200\sin \left[-0.8\right]\\ =500+200\left(-0.73\right)\\ =500-143\\ \simeq 357\end{array}$

Put the value of t in equation (2) and evaluate D as follows:

  $\begin{array}{c}D=1500+400\sin \left(0.4\left(0\right)\right)\\ =1500+400\sin \left(0\right)\\ =1500+400\left(0\right)\\ =1500\end{array}$

Thus the number of pumas and deer for the given time is 357 and 1500 respectively.

b.

To determine

To find: the number of pumas and deer in the region at the given value of $t$ .

Answer to Problem 56E

Number of pumas = 488

Number of deer = 1197

Explanation of Solution

Given information:

The equations representing the Pumas $\left(P\right)$ and deer $\left(D\right)$ after $t$ years are shown below.

  $P=500+200\sin \left[0.4\left(t-2\right)\right]$............................ (1)

  $D=1500+400\sin \left(0.4t\right)$............................ (2)

  $t=10$

Calculation:

Put the value of t in equation (1) and evaluate P as follows:

  $\begin{array}{c}P=500+200\sin \left[0.4\left(10-2\right)\right]\\ =500+200\sin \left[3.2\right]\\ =500+200\left(-0.058\right)\\ =500-11.67\\ \simeq 488\end{array}$

Put the value of t in equation (2) and evaluate D as follows:

  $\begin{array}{c}D=1500+400\sin \left(0.4\left(10\right)\right)\\ =1500+400\sin \left(4\right)\\ =1500+400\left(-0.76\right)\\ =1500-302.7\\ \simeq 1197\end{array}$

Thus the number of pumas and deer for the given time is 488 and 1197 respectively.

c.

To determine

To find: the number of pumas and deer in the region at the given value of $t$ .

Answer to Problem 56E

Number of pumas = 545

Number of deer = 1282

Explanation of Solution

Given information:

The equations representing the Pumas $\left(P\right)$ and deer $\left(D\right)$ after $t$ years are shown below.

  $P=500+200\sin \left[0.4\left(t-2\right)\right]$............................ (1)

  $D=1500+400\sin \left(0.4t\right)$............................ (2)

  $t=25$

Calculation:

Put the value of t in equation (1) and evaluate P as follows:

  $\begin{array}{c}P=500+200\sin \left[0.4\left(25-2\right)\right]\\ =500+200\sin \left[9.2\right]\\ =500+200\left(0.223\right)\\ \simeq 545\end{array}$

Put the value of t in equation (2) and evaluate D as follows:

  $\begin{array}{c}D=1500+400\sin \left(0.4\left(25\right)\right)\\ =1500+400\sin \left(10\right)\\ =1500+400\left(-0.544\right)\\ \simeq 1282\end{array}$

Thus the number of pumas and deer for the given time is 545 and 1282 respectively.