Chapter 11.EA, Problem 1EA

To determine

(a)

To find:

The time in years to reduce the pollution to $40\%$ in Lake Erie.

Answer to Problem 1EA

Solution:

The time of reducing the pollution in Lake Erie to $40\%$ is $2.4$ years.

Explanation of Solution

Given:

The value of $\frac{1}{k}$ is $2.6$ and $\frac{P_L\left(t\right)}{P_L\left(0\right)}$ is equal to $40\%$.

Approach:

Use the value of $\frac{1}{k}$ and equation $t=\frac{1}{k}\ln \left(\frac{P_L\left(t\right)}{P_L\left(0\right)}\right)$.

Calculation:

$\begin{array}{rll}\frac{P_L\left(t\right)}{P_L\left(0\right)}& =& 40\%\\ & =& 0.4\end{array}$

Then the value of inverted ratio.

$\begin{array}{rll}\frac{P_L\left(0\right)}{P_L\left(t\right)}& =& \frac{1}{0.4}\\ & =& 2.5\end{array}$

Put the values of $\frac{1}{k}$ and $\frac{P_L\left(0\right)}{P_L\left(t\right)}$ in the equation of t to find the time.

$\begin{array}{rll}t& =& \frac{1}{k}\ln \left(\frac{P_L\left(t\right)}{P_L\left(0\right)}\right)\\ & =& 2.6\ln \left(2.5\right)\\ & =& 2.4\end{array}$

Therefore, the time of reducing the pollution to $40\%$ is $2.4$ years.

Hence, in $2.4$ years the pollution reduced to $40\%$.

To determine

(b)

To find:

The time of reducing the pollution to $30\%$ is $2.4$ years.

Answer to Problem 1EA

Solution:

The time to reduce the pollution of Lake to $30\%$ is $\text{3}\text{.1 years}$.

Explanation of Solution

Calculation:

The pollution is reduced to $30\%$.

$\begin{array}{rll}\frac{P_L\left(t\right)}{P_L\left(0\right)}& =& 30\%\\ & =& 0.3\end{array}$

Then the value of inverted ratio.

$\begin{array}{rll}\frac{P_L\left(0\right)}{P_L\left(t\right)}& =& \frac{1}{0.3}\\ & =& 3.33\end{array}$

Put the values of $\frac{1}{k}$ and $\frac{P_L\left(0\right)}{P_L\left(t\right)}$ in the equation of t to find the time.

$\begin{array}{rll}t& =& \frac{1}{k}\ln \left(\frac{P_L\left(t\right)}{P_L\left(0\right)}\right)\\ & =& 2.6\ln \left(3.33\right)\\ & =& 3.1\end{array}$

Therefore, the time is $\text{3}\text{.1 years}$ to reduce the pollution to $30\%$

Hence, the time to reduce the pollution of Lake to $30\%$ is $\text{3}\text{.1 years}$.