# Need help with this word problem using hyperbolas and need the final answer in (x,y)

John was in the lead, Ed was 1.5 miles behind and Jeff was 2 miles behind John. Then they heard an explosion. John heard it first and Ed heard it a second later and Jeff heard it 1.5 seconds after John. First quad, find (x,y) of explosion. Using hyperbolas, and please explain. I need help finding the speed that the sound traveled and i'm getting stuck. We're not using the actual speed of sound so I think we may need to use a system of equations or something. I tried setting up 1 sec + j sec = distance but I can't find the distance.

Thank You!

• Welcome to stackexchange. "Needing help" is fine, but asking us to do the whole problem for you is not. Please edit the question to show us what you tried and where you are stuck. – Ethan Bolker Apr 19 at 1:34
• Sorry about that, i have showed what I tried – No me gusta matematicas Apr 19 at 1:38
• @Nomegustamatematicas is there any more information given? It would be impossible to determine an $(x,y)$ coordinate if the problem is only talking about one dimension. – rb612 Apr 19 at 1:40
• If you’re not given the speed of sound, you’re either expected to look it up for yourself or use a coordinate system in which the speed of sound equals $1$. – amd Apr 19 at 1:52
• @rb612 Certainly. A hyperbola is the locus of points with a constant difference in distance to a pair of fixed points. This problem involves finding the intersection of a pair of hyperbolas. – amd Apr 19 at 5:38

## 1 Answer

This becomes tricky given that you don't have knowledge of where the explosion happened relative to the 3 people. This means you have to examine 2 cases. You're right though, that you have to set up a system of equations.

Imagine John, Ed, and Jeff are on the number line. John is at mile $$2$$, Ed is at mile $$0.5$$, and Jeff is just starting, at mile $$0$$. This keeps the consistency of how far they are apart.

Now, let's let $$x$$ be the location of the explosion, and let $$y$$ be the velocity of the explosion. We have to set up some systems of equations.

We need to first figure out distance. We know that John heard the explosion first, so this means that it had to have happened closest to John, so at least at or after mile marker 1.25 (or else it would be closer to Ed). So since we don't know where this happens, we don't know if x is greater than or smaller than 2 (since it could've happened in front of John or close behind and he still would've heard it first!) But we need distance to be positive, so we have to add an absolute value sign.

So the distance John is away would be $$|x-2|$$, the distance Ed would be away is $$x-0.5$$ (since we know it happens in front of Ed), and the distance Jeff would be away is just $$x$$. Now you have that velocity is $$y$$.

So, since velocity is the distance over the time, you have $$\frac{|2-x|}{y}$$ is the time it takes for John to hear it, $$\frac{|x-0.5|}{y}$$ for Ed, and $$\frac{x}{y}$$ for Jeff.

Now you know that the time John heard it, plus 1 second, is the time Ed heard it. Similarly, time John heard plus 1.5 seconds is the time Jef heard it. Can you set up a system of equations and solve from here?