Chapter 8.1, Problem 123E

(a)

To determine

To write: a recursive sequence that given the population $p_n$ of trout in the lake in terms of the year $n$ , with $n=0$ corresponding to 2015.

Answer to Problem 123E

The recursive sequence that gives the population of trout in the lake in terms of year $n$ is:

  $p_n=\left(0.75\right)P_{n-1}+500$

Explanation of Solution

Given information:

A landlocked lake has been selected to be stocked in the year 2015 with 5500 trout, and to be restocked given year thereafter with 500 trout. Given year the fish population declines 25%duw to harvesting and other natural causes.

Calculation:

The recursive sequence that gives the population of trout in the lake in terms of year $n$ is:

  $\begin{array}{l}{P}_0=5500\\ p_n=\left(0.75\right)P_{n-1}+500\end{array}$

(b)

To determine

To find: the numbers of trout in the lake for $n=1,2,3$ and 4.

Answer to Problem 123E

The numbers of trout in the lake are:

4625, 3969, 3477, 3108

Explanation of Solution

Given information:

A landlocked lake has been selected to be stocked in the year 2015 with 5500 trout, and to be restocked given year thereafter with 500 trout. Given year the fish population declines 25%duw to harvesting and other natural causes.

Calculation:

To find the numbers of trout in the lake, substitute $n=1,n=2,n=3,n=4$ in the sequence one by one.

  $\begin{array}{c}{P}_1=\left(0.75\right)P_{1-1}+500\\ =\left(0.75\right)\left(5500\right)+500\\ =4625\\ P_2\approx 3969\\ P_3\approx 3477\\ P_4\approx 3108\end{array}$

Thus, the numbers of trout in the lake are:

4625, 3969, 3477, 3108

These values represent the number of trout in the lake for the years 2016 through 2019.

(c)

To determine

To find: the number of trout in the lake as time passes infinitely using graphing utility.

Answer to Problem 123E

The number of trout in the lake is found as 2000 trout.

Explanation of Solution

Given information:

A landlocked lake has been selected to be stocked in the year 2015 with 5500 trout, and to be restocked given year thereafter with 500 trout. Given year the fish population declines 25%duw to harvesting and other natural causes.

Calculation:

Using graphing utility, the number of trout in the lake is found as 2000 trout.

As time passes the population of trout decreases at a decreasing rate. Because the population is growing smaller and still decline 25%, each time 25% is taken from a smaller number, there is a smaller decline in the number of trout.