Chapter 5.5, Problem 48PPS

To determine

To find: the time at which the population becomes zero.

Answer to Problem 48PPS

  $t=\pm 5,\pm 4i$ .

Explanation of Solution

Given:

The population of the species of animals is represented by $P\left(t\right)=-t^4+9t^2+400$ .

Calculation:

As the given equation is $P\left(t\right)=-t^4+9t^2+400$ .

The objective is to find the time when the population becomes zero.

Therefore, according to question $P\left(t\right)=0$ .

  $\begin{array}{l}0=-t^4+9t^2+400\{P\left(t\right)=0\}\\ \Rightarrow t^4-9t^2-400=0\left\{\text{multiply both sides by 1}\right\}\\ t^4-25t^2+16t^2-400=0\left\{\text{express in simplified form}\right\}\\ t^2\left(t^2-25\right)+16\left(t^2-25\right)=0\left\{\text{taking }\left(t^2-25\right)\text{as common factor}\right\}\\ \left(t^2-25\right)\left(t^2+16\right)=0\left\{\text{use distributive property of product}\right\}\end{array}$

Either $\left(t^2-25\right)=0$ or $\left(t^2+16\right)=0$

when $\left(t^2-25\right)=0$ , then $t^2=25\Rightarrow t=\pm \sqrt{25}=\pm 5$

when $\left(t^2+16\right)=0$ , then $t^2=-16\Rightarrow t=\pm \sqrt{-1\times 16}=\pm \sqrt{i^2\times 4^2}=\pm 4i$ :

Hence, the time at which the population becomes zero is $t=\pm 5,\pm 4i$ .