Beginning Algebra With Applications, Chapter 3, 3.2, Section 3.2, Problem 186

A telephone company estimates that the number $N$ of phone calls made per day between two cities of populations $P_1$ and $P_2$ that are $d$ miles apart is given by the equation $\displaystyle N = \frac{2.51 P_1 P_2}{d^2}$.

Estimate the population $P_1$ given that $P_2$ is $125,000$, the number of phone calls is $2,500,000$, and the distance between the cities is $50$ mi. Round to the nearest thousand.

Solving for $P_1$,


$
\begin{equation}
\begin{aligned}

N =& \frac{2.51 P_1 P_2}{d^2}
&& \text{Given equation}
\\
\\
Nd^2 =& 2.51 P_1 P_2
&& \text{Multiply both sides by } d^2
\\
\\
\frac{Nd^2}{2.51 P_2} =& P_1
&& \text{Divide by } 2.51 P_2
\\
\\
P_1 =& \frac{(2,500,000)(50)^2}{2.51(125,000)}
&& \text{Substitute } N = 2,500,000, d = 50 \text{ and } P_2 = 125,000
\\
\\
P_1 =& 19,920.31873 \approx 20,000
&&

\end{aligned}
\end{equation}
$


The estimated population $P_1$ is $20,000$.