Chapter 9, Problem 46CR

To determine

To find: The equation of two hyperbolas that would locate the explosion.

Answer to Problem 46CR

The two equation of the hyperbola are,

  $\frac{x^2}{\left(\frac{25}{64}\right)}-\frac{y^2}{\left(\frac{231}{64}\right)}=1\text{ and }\frac{x^2}{\left(\frac{25}{36}\right)}-\frac{y^2}{\left(\frac{119}{36}\right)}=1$

Explanation of Solution

Given information:

Two of your friends live 4 miles apart on the same east-west direction. When you hear the explosion a friend on the east hears after 6 seconds and the friend on the west hears it after 8 seconds

The sound travels at 1100 feet per second.

Calculation:

The friend in the east heard the explosion after 6 seconds

So,

  $\begin{array}{l}2a=6\cdot \left(\frac{1100}{5280}\right)\\ 2a=\frac{5}{4}\\ a=\frac{5}{8}\end{array}$

Thus, $2c=4\Rightarrow c=2$

Now,

  $\begin{array}{l}{b}^2=c^2-a^2\\ b^2=2^2-{\left(\frac{5}{8}\right)}^2\\ b^2=\frac{231}{64}\end{array}$

Thus write the equation of the hyperbola

  $\frac{x^2}{\left(\frac{25}{64}\right)}-\frac{y^2}{\left(\frac{231}{64}\right)}=1$

Now place the center at $\left(1,0\right)$ and determine the second hyperbola

The friend in the west heard the explosion after 8 seconds

So,

  $\begin{array}{l}2a=8\cdot \left(\frac{1100}{5280}\right)\\ 2a=\frac{5}{3}\\ a=\frac{5}{6}\end{array}$

Thus, $2c=4\Rightarrow c=2$

Now,

  $\begin{array}{l}{b}^2=c^2-a^2\\ b^2=2^2-{\left(\frac{5}{6}\right)}^2\\ b^2=\frac{119}{36}\end{array}$

Thus write the equation of hyperbola

  $\frac{x^2}{\left(\frac{25}{36}\right)}-\frac{y^2}{\left(\frac{119}{36}\right)}=1$

Therefore, the two equation of the hyperbola are,

  $\frac{x^2}{\left(\frac{25}{64}\right)}-\frac{y^2}{\left(\frac{231}{64}\right)}=1\text{ and }\frac{x^2}{\left(\frac{25}{36}\right)}-\frac{y^2}{\left(\frac{119}{36}\right)}=1$