Short Answer

A pond currently has 2000 trout in it. A fish hatchery decides to add an additional 20 trout each month. In addition, it is known that the trout population is growing 3% per month. The size of the population after n months is given by the recursively defined sequence.
$p_{0} = 2000; p_{n} = 1.03 p_{n - 1} + 20$
(a) How many trout are in the pond at the end of the second month? That is, what is p₂?
(b) Using a graphing utility, determine how long it will be
before the trout population reaches 5000.

(a) The number of trout in second month, p₂ is 2162.

(b) The population after 26 months will be 5084.2433.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Write the given information.

The current population:

$p_{0} = 2000$

& Size of population after n months can be defined as:

$p_{n} = 1.03 p_{n - 1} + 20$

Step 2. Use the population size formula as given.

Using the recursively defined formula for p₂:

$p_{1} = 1.03 p_{1 - 1} + 20 \Rightarrow p_{1} = 1.03 (2000) + 20 \Rightarrow p_{1} = 2080 p_{2} = 1.03 p_{2 - 1} + 20 \Rightarrow p_{2} = 1.03 (2080) + 20 \Rightarrow p_{2} = 2162.4$

Step 3. Use the given formula for calculating the size of population after 26 months.

Using graphing calculator to compute the population size up to 26 months.

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Precalculus Enhanced with Graphing Utilities

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Short Answer

Step by Step Solution

Step 1. Write the given information.

Step 2. Use the population size formula as given.

Step 3. Use the given formula for calculating the size of population after 26 months.

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