Chapter 9.3, Problem 51E

To determine

To find: The equation that gives the possible places where the lightning could have occurred.

Answer to Problem 51E

The equation that gives the possible places where the lightning could have occurred is $\frac{x^2}{98010000}-\frac{y^2}{13503600}=1$ .

Explanation of Solution

Given information:

The speed of sound is 100 feet per second.

Calculation:

Convert 4 miles to feet and find the distance traveled by the sound of the thunder as follows.

  $\begin{array}{c}4\text{miles}=4\times 5280\text{feet}\\ =\text{21120}\text{feet}\\ 18\text{seconds}\times 1100\frac{\text{feet}}{\text{seconds}}=19800\text{feet}\end{array}$

Let the two people be foci. 2c is the distance between the two people.

Find c as follows.

  $\begin{array}{c}2c=21120\\ c=10560\end{array}$

A hyperbola is the set of all points in a plane, the difference of whose differences from the two foci is constant. Suppose the source of lightning was on the same axis as the two foci. Set up the equation used to find a.

Solve for a as follows.

  $\begin{array}{c}c+a-\left(c-a\right)=19800\\ 2a=19800\\ a=9900\end{array}$

Find the square of b as follows.

  $\begin{array}{c}{b}^2=c^2-a^2\\ ={\left(10560\right)}^2-{\left(9900\right)}^2\\ =13503600\end{array}$

Write the equation for the hyperbola. This hyperbola has a horizontal transverse axis.

  $\frac{x^2}{98010000}-\frac{y^2}{13503600}=1$

Therefore, the required equation is $\frac{x^2}{98010000}-\frac{y^2}{13503600}=1$ .