Chapter 7.5, Problem 43PS

Solution

a.

To state: A recursive rule for the number of fish at the beginning of the nth year and the number of fish at the beginning of the 5th year if a lake initially contains 5000 fish.

The resultant fish are 3524.

Given information:

A lake initially contains 5000 fish. Each year the population declines 20% due to fishing and other causes, and the lake is restocked with 500 fish.

Explanation:

Consider the given information,

A lake initially contains 5000 fish. It means $a_1=5000$ .

Each year the population declines 20% due to fishing and other causes, and the lake is restocked with 500 fish.

  $\begin{array}{c}\left[\begin{array}{l}\text{Fish}\text{at}\text{the}\\ \text{start}\text{of}\text{year}n\end{array}\right]=\left(0.8\right)\left[\begin{array}{l}\text{Fish}\text{at}\text{the}\\ \text{start}\text{of}\text{year}\left(n-1\right)\end{array}\right]+\left[\begin{array}{l}\text{New}\text{fish}\\ \text{added}\end{array}\right]\\ a_n=0.8a_{n-1}+500\end{array}$

Thus, the recursive rule is $a_n=0.8a_{n-1}+500$ .

The second term is:

  $\begin{array}{c}{a}_2=0.8a_{2-1}+500\\ =0.8a_1+500\\ =0.8\left(5000\right)+500\\ =4500\end{array}$

The third term is:

  $\begin{array}{c}{a}_3=0.8a_{3-1}+500\\ =0.8a_1+500\\ =0.8\left(4500\right)+500\\ =4100\end{array}$

The fourth term is:

  $\begin{array}{c}{a}_4=0.8a_{4-1}+500\\ =0.8a_3+500\\ =0.8\left(4100\right)+500\\ =3780\end{array}$

The fifth term is:

  $\begin{array}{c}{a}_5=0.8a_{5-1}+500\\ =0.8a_4+500\\ =0.8\left(3780\right)+500\\ =3524\end{array}$

Therefore, at the beginning of 5^th year there will be 3524 fish in the lake.

b.

To state: What happens to the population of fish in the lake over time.

The fish in the lake will disappear over time.

Given information:

A lake initially contains 5000 fish.

Explanation:

A lake initially contains 5000 fish.

Each year the population declines 20% due to fishing and other causes, and the lake is restocked with 500 fish.

The lake contains 5000 fish in it. After a year, it reduced to 4500 fish. Then after two years the number of fish reduced to 4100. Then again it reduced to 3780, 3524 and so on.

This pattern will continue for a long time. It can be seen that the number of fish in the lake is decreasing and over time the fish in lake will disappear.