College Algebra, Chapter 5, 5.4, Section 5.4, Problem 82

The transparency of the water is determined by measuring the intensity of light at various depths in a lake. Suppose that $I = 10e^{-0.008x}$ represents the intensity of light at depth x in a certain lake.

a.) Determine the intensity l at a depth of 30ft.

b.) On what depth will the light intensity drops to I =5?



a.) if $x = 30 ft$, then


$
\begin{equation}
\begin{aligned}

I =& 10 e^{-0.008(30)}
\\
\\
I =& 10e^{-0.24}
\\
\\
I =& 7.866

\end{aligned}
\end{equation}
$


b.) if $I = 5$, then


$
\begin{equation}
\begin{aligned}

5 =& 10e^{-0.008 x}
\\
\\
\frac{5}{10} =& e^{-0.008x}
\\
\\
\ln \left( \frac{1}{2} \right) =& -0.008x
\\
\\
x =& \frac{\displaystyle \ln \left( \frac{1}{2} \right) }{-0.008}
\\
\\
x =& 86.64 \, ft

\end{aligned}
\end{equation}
$