Chapter 9.5, Problem 35PPS

To determine

To Find: The equation for the horizontal hyperbola on which the lightning is located.

Answer to Problem 35PPS

  $\frac{x^2}{2722500}-\frac{y^2}{1277500}=1$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Distance between Luisa and Karl }=4000feet\\ \text{Time difference in hearing the thunder sound }=3\sec ond\end{array}$

Formula Used:

  $\begin{array}{l}{c}^2=a^2+b^2\\ \text{Hyperbola Standard equation: }\frac{x^2}{a^2}-\frac{y^{{}^2}}{b^2}=1\end{array}$

Calculation:

  $\begin{array}{l}\text{Let Luisa and Karl are at the foci and centre is at the origin and distance between }2\text{foci is }2c=4000feet:\\ \text{Standard equation is:}\\ \Rightarrow \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ \Rightarrow 2c=4000\\ \Rightarrow c=2000\\ \to \text{Speed of sound }=1100feet/\sec ond\\ \to \text{Time taken to hear the sound}=3\sec ond\\ \to \text{Distance travelled by sound}=3\left(1100\right)=3300ft=\text{it is equal to fixed distance }2a\\ \Rightarrow 2a=3300\\ \Rightarrow a=1650\\ \to \text{using equation }{c}^2=a^2+b^2\text{ to solve for }{b}^2\\ \Rightarrow b^2=c^2-a^2\\ \Rightarrow b^2={2000}^2-{1650}^2=1277500\\ \text{Now the equation is :}\\ \Rightarrow \frac{x^2}{{1650}^2}-\frac{y^2}{1277500}=1\\ \Rightarrow \frac{x^2}{2722500}-\frac{y^2}{1277500}=1\end{array}$

Conclusion:

Thus the equation of the hyperbola is $\frac{x^2}{2722500}-\frac{y^2}{1277500}=1$ .