Chapter 9.3, Problem 52E

To determine

To find: The coordinates of the explosion.

Answer to Problem 52E

The coordinates of the explosion is $\left(3300,-2750\right)$ .

Explanation of Solution

Given information:

The location of all three stations are $\left(3300,0\right),\left(3300,1100\right)$ and $\left(-3300,0\right)$ .

Calculation:

Since the second station is 1100 feet directly north of the first station and the second station detects the explosion one second after the first one does, the x-coordinate of the explosion must be 3300. The y-coordinate must also be negative since the first station detects the explosion first.

Use the data from the first and third station to write the equation of the hyperbola of possible points of the explosion as shown below.

  $\begin{array}{c}2c=6600\\ c=3300\end{array}$

and,

  $\begin{array}{c}c+a-\left(c-a\right)=4\cdot 1100\\ 2a=4400\\ a=2200\\ a^2=4840000\end{array}$

Find the $b^2$ as follows.

  $\begin{array}{c}{b}^2=c^2-a^2\\ ={\left(3300\right)}^2-{\left(2200\right)}^2\\ =6050000\end{array}$

Write the hyperbolic equation as follows.

  $\frac{x^2}{4840000}-\frac{y^2}{6050000}=1$

Use the known x-coordinate of the explosion and the hyperbola to find the negative y-coordinate of the explosion as follows.

  $\begin{array}{c}\frac{{\left(3300\right)}^2}{4840000}-\frac{y^2}{6050000}=1\\ \frac{y^2}{6050000}=\frac{{\left(3300\right)}^2}{4840000}-1\\ y^2=1.25\cdot 6050000\\ y=2750\end{array}$

Therefore, the coordinates of the explosion is $\left(3300,-2750\right)$ .