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Q 85.

Expert-verifiedFound in: Page 810

Book edition
6th

Author(s)
Sullivan

Pages
1200 pages

ISBN
9780321795465

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 p_{2}?

(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, p_{2} is 2162.

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

The current population:

${p}_{0}=2000$

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

${p}_{n}=1.03{p}_{n-1}+20$

Using the recursively defined formula for p_{2 }:

${p}_{1}=1.03{p}_{1-1}+20\phantom{\rule{0ex}{0ex}}\Rightarrow {p}_{1}=1.03\left(2000\right)+20\phantom{\rule{0ex}{0ex}}\Rightarrow {p}_{1}=2080\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{p}_{2}=1.03{p}_{2-1}+20\phantom{\rule{0ex}{0ex}}\Rightarrow {p}_{2}=1.03\left(2080\right)+20\phantom{\rule{0ex}{0ex}}\Rightarrow {p}_{2}=2162.4\phantom{\rule{0ex}{0ex}}$

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

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