Chapter 6.6, Problem 9E

(a)

To determine

To find: The population of rodents on January $1,1970$ .

Answer to Problem 9E

The population of rodents on January $1,1970$ is $1200$ .

Explanation of Solution

Given information:

The given function is $H=250+25\sin \left(\frac{\pi }{2}t-\frac{\pi }{4}\right)$ and the rodent population is $R=1200+300\sin \left(\frac{\pi }{2}t\right)$ .

Calculation:

Calculate the population of rodents on January $1,1970$

  $\begin{array}{c}R=1200+300\sin \left(0\right)\\ =1200\end{array}$

Therefore, the population of rodents on January $1,1970$ is $1200$ .

(b)

To determine

To find: The population of hawks on January $1,1970$

Answer to Problem 9E

The population of hawks on January $1,1970$ is $232$ .

Explanation of Solution

Given information:

The given function is $H=250+25\sin \left(\frac{\pi }{2}t-\frac{\pi }{4}\right)$ and the rodent population is $R=1200+300\sin \left(\frac{\pi }{2}t\right)$ .

Calculation:

Calculate the population of hawks on January $1,1970$

  $\begin{array}{c}H=250+25\sin \left(-\raisebox{1ex}{$pi$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)\\ =250+25\left(-0.7071\right)\\ =232\end{array}$

Therefore, the population of hawks on January $1,1970$ is $232$ .

(c)

To determine

To find: The maximum population of rodents and hawks.

Answer to Problem 9E

The maximum population of rodents is $1$ and hawks $1.5$ .

Explanation of Solution

Given information:

The given function is $H=250+25\sin \left(\frac{\pi }{2}t-\frac{\pi }{4}\right)$ and the rodent population is $R=1200+300\sin \left(\frac{\pi }{2}t\right)$ .

Calculation:

Calculate the maximum population of rodents.

  $\begin{array}{c}H-\raisebox{1ex}{$pi$}\!\left/ \!\raisebox{-1ex}{$2t$}\right.-\raisebox{1ex}{$pi$}\!\left/ \!\raisebox{-1ex}{$4$}\right.=\raisebox{1ex}{$pi$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-t=1.5\\ R-\raisebox{1ex}{$pi$}\!\left/ \!\raisebox{-1ex}{$2t$}\right.=\raisebox{1ex}{$pi$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-t=1\end{array}$

Therefore, the maximum population of rodents is $1$ and hawks $1.5$ .

(d)

To determine

To find: The date was the first maximum population of rodents achieved.

Answer to Problem 9E

The date was the first maximum population of rodents achieved is January $1,1971$ .

Explanation of Solution

Given information:

The given function is $H=250+25\sin \left(\frac{\pi }{2}t-\frac{\pi }{4}\right)$ and the rodent population is $R=1200+300\sin \left(\frac{\pi }{2}t\right)$ .

Calculation:

Calculate the function

The maximum population of rodents is when $t=1$ .

The first time it reached the maximum was on January $1,1971$ .

Therefore, the date was the first maximum population of rodents achieved is January $1,1971$ .

(e)

To determine

To find: The minimum population of hawks.

Answer to Problem 9E

The minimum population of hawks is $225$ .

Explanation of Solution

Given information:

The given function is $H=250+25\sin \left(\frac{\pi }{2}t-\frac{\pi }{4}\right)$ and the rodent population is $R=1200+300\sin \left(\frac{\pi }{2}t\right)$ .

Calculation:

Calculate the minimum population of hawks.

  $\begin{array}{c}H=250+25\left(-1\right)\\ =225\end{array}$

Therefore, the minimum population of hawks is $225$ .

(f)

To determine

To find: The population of rodents and hawks on January $1$ of the present year.

Answer to Problem 9E

The population of rodents is $1200$ and hawks $232$ on January $1$ of the present year.

Explanation of Solution

Given information:

The given function is $H=250+25\sin \left(\frac{\pi }{2}t-\frac{\pi }{4}\right)$ and the rodent population is $R=1200+300\sin \left(\frac{\pi }{2}t\right)$ .

Calculation:

Calculate the population of rodents and hawks.

  $\begin{array}{c}R=1200\\ H=232\end{array}$

Therefore, the population of rodents is $1200$ and hawks $232$ on January $1$ of the present year.