Chapter 8.5, Problem 44PS

Solution

To calculate:

Two microphones placed one mile apart capture an elk bull's bugling.

The location might be in either the first quadrant or the fourth quadrant if microphone $B$ is on the positive

Given information:

  $\begin{array}{c}c=\text{0}\text{.5mile×}\frac{\text{5280}\text{ft}\text{.}}{\text{mile}}\\ \text{=2640ft}\end{array}$

The foci are where the microphones are. The intersections of equal waves originating from the foci are in the form of a hyperbola. The opposite is also true; sounds coming from a point will similarly find their way to the foci.

Calculation:

Elk is located ${\text{d}}_1$ from microphone $B$ and microphone $A$ is located at $d_2$ away.

  $d_2-d_1=AB$ A constant and matching $\text{2 a}$

  $\begin{array}{c}{d}_2-d_1=2200\\ 2a=2200\\ a=1100\text{ft}\text{.}\end{array}$

  $\begin{array}{c}{a}^2+b^2=c^2\\ b^2=c^2-a^2\\ ={2640}^2-{1100}^2\\ =5759600\end{array}$

Find $b^2$

  $\begin{array}{r}\hfill \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ \hfill \frac{x^2}{{1100}^2}-\frac{y^2}{5759600}=1\\ \hfill \frac{x^2}{1210000}-\frac{y^2}{5759600}=1\end{array}$

Input the data into the fundamental equation with the origin.

Even with the hyperbola equation, there is still insufficient detail because the location might be in either the first or fourth quadrant if microphone $\text{B}$ is on the positive $\text{x-axis}$ .