Answer
The number of miles is $5.5\text{ miles}$.
Work Step by Step
As per the question, $ P\left( x \right)=14.7{{e}^{-0.21x}}$
Substitute the values of $ P\left( x \right)$ in the above equation and simplify:
$4.6=14.7{{e}^{-0.21x}}$
And divide both sides of the equation by $14.7$ and simplify as follows:
$\begin{align}
& \frac{4.6}{14.7}=\frac{14.7}{14.7}{{e}^{-0.21x}} \\
& 0.31={{e}^{-0.21x}}
\end{align}$
Write the above exponential form in logarithmic form using:
If ${{b}^{y}}=x $, then ${{\log }_{b}}x=y $
So, $0.31={{e}^{-0.21x}}$ is equivalent to ${{\log }_{e}}0.31=-0.21x $.
Solving the above expression as follows:
$\begin{align}
& {{\log }_{e}}0.31=-0.21x \\
& -1.17=-0.21x
\end{align}$
Change the sides of the equation and divide both sides by $-0.21$,
$\begin{align}
& \left( \frac{-0.21}{-0.21} \right)x=\frac{-1.17}{-0.21} \\
& x=5.5
\end{align}$
Thus, the peak of Mt. Everest is $5.5\text{ miles}$ above sea level.
