Chapter 12.1, Problem 84AYU

Environmental Control The Environmental Protection Agency (EPA) determines that Maple Lake has $250$ tons of pollutant as a result of industrial waste and that $10\%$ of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in $15$ tons of new pollutant entering the lake each year. The amount of pollutant in the lake after $n$ years is given by the recursively defined sequence

$p_0=250p_n=0.9p_{n-1}+15$.

Determine the amount of pollutant in the lake after $2$ years. That is, determine $p_2$.

To determine

The amount of pollutant in the lake after $2$ years, that is $p_2$. Where, the Environmental Protection Agency (EPA) determines that Maple lake has $250$ tons of pollutant as a result of industrial waste and that $10\%$ of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in $15$ tons of new pollutant entering the lake each year. The amount of pollutant in the lake after $n$ years is given by the recursive sequence $P_0=250P_n=0.9P_{n-1}+15$

Answer to Problem 84AYU

Solution:

The amount of pollutant in the lake after $2$ years will be $231$ tons.

Explanation of Solution

Given information:

The Environmental Protection Agency (EPA) determines that Maple lake has $250$ tons of pollutant as a result of industrial waste and that $10\%$ of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in $15$ tons of new pollutant entering the lake each year. The amount of pollutant in the lake after $n$ years is given by the recursive sequence $P_0=250P_n=0.9P_{n-1}+15$

Explanation:

Consider the recursive relation, $P_0=250P_n=0.9P_{n-1}+15$.

The amount of pollutant in the lake after $2$ years, is $P_2$, find the value of $P_1$ first.

Substitute $n=1$ in $P_n=0.9P_{n-1}+15$ .

$\Rightarrow P_1=0.9P_{1-1}+15$

$\Rightarrow P_1=0.9P_0+15$

Substitute $P_0=250$ in above equation.

$\Rightarrow P_1=0.9\left(250\right)+15=240$.

Now to find $P_2$, substitute $n=2$ in $P_n=0.9P_{n-1}+15$ .

$\Rightarrow P_2=0.9P_{2-1}+15$

$\Rightarrow P_2=0.9P_1+15$

Substitute $P_1=240$ in above equation.

$\Rightarrow P_2=0.9\left(240\right)+15=231$.

Therefore, the amount of pollutant in the lake after $2$ years will be $231$ tons.