Chapter 10.4, Problem 76AYU

Lightning Strikes Suppose that two people standing 1 mile apart both see a flash of lightning. After a period of time the first person, standing at point $A$, hears the thunder. Two seconds later the second person, standing at point $B$, hears the thunder. If the person at point B is due west of the person at point $A$, and if the lightning strike is known to occur due north of the person standing at point $A$, where did the lightning strike occur?

To determine

To find: Where did the lightening strike occurred?

Answer to Problem 76AYU

$5236\text{ feet }\approx 0.99\text{ miles}$

Explanation of Solution

Given:

Two people standing 1 mile apart both see a flash of lightning. After a period of time the first person, standing at point $\text{A}$, hears the thunder. Two seconds later the second person, standing at point $\text{B}$, hears the thunder. The person at point $\text{B}$ is due west of the person at point $\text{A}$, and the lightning strike is known to occur due north of the person standing at point $\text{A}$,

Formula used:

Equation of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, $b^2=c^2-a^2$.

Calculation:

Let $\left(x,y\right)$ be location where lightening strike occurs and let it be a point north of $\text{A}$ such that time difference would be same as that for the first strike. All the points where lightening strike takes place would form a hyperbola with $\text{A}$ and $\text{B}$ as foci and $\left(x,y\right)$ on it.

Let $x\text{-axis}$ contain $\text{AB}$ and let origin be midpoint of $\text{AB}$.

Sound travels at 1100 feet per second.

So person at $\text{A}$ is 2200 feet closer to the location than $\text{B}$.

Hence difference between $\left(x,y\right)$ to $\text{A}$ and $\left(x,y\right)$ to $\text{B}$ is a constant 2200.

Let the equation of the hyperbola be,

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$; $2a=2200$, $a=1100$

Distance between $\text{A}$ and $\text{B}$ is $2c=1\text{ mile }=5280\text{ feet}$; $c=2640$.

To find $\text{b}$,

$b^2=c^2-a^2={2640}^2-{1100}^2=5759600$

Substituting $\text{a}$ and $\text{b}$, we get,

$\frac{x^2}{{1100}^2}-\frac{y^2}{5759600}=1$

Let $x=2640$

$\frac{{2640}^2}{{1100}^2}-\frac{y^2}{5759600}=1$

Solving the above equation,

We get $y=5236$

Hence, lightening strike occurred at $5236\text{ feet }\approx 0.99\text{ miles}$ due north of the person at $\text{A}$.