Chapter 3.2, Problem 82E

Population Change The rabbit population on a small island is observed to be given by the function

$P(t)=\text{12}0t-0.\text{4}{t}^{\text{4}}+\text{1}000$

where t is the time (in months) since observations of the island began.

1. (a) When is the maximum population attained, and what is that maximum population?
2. (b) When does the rabbit population disappear from the island?

To determine

The maximum population of rabbit and the time in months when the population of rabbit is maximum.

Answer to Problem 82E

The maximum population is 1379.54 and the time when the maximum population attends is 4.2 months.

Explanation of Solution

Given:

Rabbit population is a function of time is,

$P\left(t\right)=120t-0.4t^4+1000$

Calculation:

Rabbit population is a function of time is,

$P\left(t\right)=120t-0.4t^4+1000$ (1)

Here,

• $P(t)$ is the population as a function of time.
• t is the time in months.

In the above equation t is a independent variable and $P(t)$ is a dependent variable.

For $P\left(t\right)$ intercept, substitute 0 for t in equation (1).

$\begin{array}{c}P\left(t\right)=120t-0.4t^4+1000\\ P(0)=120\times 0-0.4\times {(}^0+1000\\ =1000\end{array}$

For t intercept, substitute 0 for $P(t)$ in equation (1).

$\begin{array}{c}P\left(t\right)=120t-0.4t^4+1000\\ 0=120t-0.4t^4+1000\end{array}$

After hit and trial method,

$t_1=-5.432$ and $x_2=8.42$.

From the property of graph that at t intercept the value of $P(t)$ is zero and at the $P(t)$ intercept the value of t is zero.

Plot these points on the $t-P(t)$ axis and connect the points by a smooth curve to complete the graph.

Figure (1)

Point $\left(4.2,1379.54\right)$ represents the peak point of graph.

From the above graph it is clear that the maximum value of population is 1379.54 and the time at which the maximum population attends is 4.2 months.

Thus, the maximum population is 1379.54 and the time when the maximum population attends is 4.2 months.

To determine

The time at which the population of rabbit is disappear from the island.

Answer to Problem 82E

Rabbit population was disappeared after 8.2 months.

Explanation of Solution

Given:

Total profit after firm produces and sells x blenders annually in dollars is,

$P\left(x\right)=8x+0.3x^2-0.0013x^3-372$

Calculation:

Rabbit population becomes zero only when the population graph cuts the time axis.

From the part (a) Figure (1) it is clear that graph cuts the time axis at two points $\left(-5.432,0\right)$ and $\left(8.42,0\right)$.

First point shows that at $t=-5.432$ population are zero but it can not be possible.

Second point shows that at $t=8.42$ population are zero.

Thus, rabbit population was disappeared after 8.2 months.