I need help! Next period is my precal class and I just cannot figure out this hyperbola word problem. Please help me!

Question

Two farmers, Jack and Ben, heard an explosion. Jack heard the explosion 3 seconds before Ben heard it. If they are standing 1500 meters apart and directly horizontal to one another at the foci, write an equation of a horizontal hyperbola that contains all possible locations of the explosion. (The speed of sound is 350 meters per second.)

Answer 1

In order to solve this problem, we have to remember what a hyperbola actually is. A hyperbola is all the points where the difference of the distances to the two foci is always the same number. In this problem, Jack and Ben are standing at the foci, and since Jack heard the explosion 3 seconds before Ben, the explosion is always the same distance closer to Jack than to Ben. That difference stays the same no matter where the explosion is, which is exactly why all the possible locations make a hyperbola.

First, let put the center of the hyperbola at the origin, right in the middle between Jack and Ben. Since they are 1500 meters apart, each one is 750 meters from the center so it becomes:

2c = 1500, so c = 750

Next, we find the difference of the distances. Sound travels 350 meters every second, and it reached Ben 3 seconds after it reached Jack, so the explosion is farther from Ben by:

350 × 3 = 1050 meters

For a hyperbola, this constant difference is equal to 2a so it becomes:

2a = 1050, so a = 525

Now we need b. Remember, for a hyperbola the relationship is c² = a² + b², which is different from an ellipse where it is a² = b² + c². So, let solve for b² by subtracting a² on both sides so it becomes:

b² = c² − a² = 750² − 525² = 562,500 − 275,625 = 286,875

Since the hyperbola is horizontal, the x² term goes first and it is the positive one so it becomes:

x²/275,625 − y²/286,875 = 1

One small thing to notice is that since Jack heard the explosion first, it is closer to Jack, so the explosion can only be on the branch of the hyperbola on Jack's side. The equation itself still shows both branches, which is why your teacher asks for the hyperbola that contains all the possible locations.

So, our final answer is x²/275,625 − y²/286,875 = 1.