Question

A and B start from the same point and travel west and north, respectively. A travels 3 miles per hour faster than B. At the end of two hours, they are 30 miles apart. Find their distances.

Answer 1

Answer:

Step-by-step explanation:

I believe you're actually looking for their respective rates.  If A travels directly west and B travels directly north, and the distance between them is 30 miles, what we have is a right triangle situation.  The hypotenuse of the triangle is 30.  If the distance an object travels at a certain rate for given time is d = rt, then for B, our formula for distance is d = 2r (since the time each traveled is 2 hours).  A traveled 3 miles per hour faster, so the formula for distance for A is d = (r + 3)2.  Again, each traveled for 2 hours, so t = 2.  Distributing we get that d = 2r + 6.  Now that we have each expression for A and B, we use them in Pythagorean's Theorem to find the only unknown we have which is r.  That's why I said in the beginning that I believe what you're actually looking for is the rate that each traveled.  Our equation is:

$30^2=(2r+6)^2+(2r)^2$ and

$900=4r^2+24r+36+4r^2$ and

$900=8r^2+24r+36$

Putting everything on one side and setting the quadratic equal to 0 to factor, we get

$8r^2+24r-864=0$

You can factor out an 8 to make the numbers a bit smaller:

$8(r^2+3r-108)=0$

Factor to get

8(r - 9)(r + 12)=0

That means, by the Zero Product Property, that 8 = 0, r - 9 = 0, or r + 12 = 0.  We all know that 8 doesn't = 0, so forget that one!! If r - 9 = 0, then r = 9.  If r + 12 = 0. then r = -12.  We all know that rate cannot be negative (velocity can, but we are not using vector math here), so we discount rate as -12.  That means that r = 9.  B's rate is 9 then and A's rate is 12.  There you go!