Tommy and zach are starting out at the same position. tommy runs north at 3 miles per hour and zach starts to run east 2 hours later at the rate of 4 miles per hour. how long until tommy and zach are 8 miles apart?

Okay, Tommy runs in a positive y direction at 3mph, and his position can be expressed as:

T=3t

Then Zach starts running two hours later at 4mph and his position can be expressed as:

Z=4(t-2)=4t-8

The distance between Tommy and Zach can be expressed as the hypotenuse length of a right triangle with the distances for Tommy and Zach being the lengths of the sides...

d^2=T^2+Z^2

d^2=(3t)^2+(4t-8)^2

d^2=9t^2+16t^2-64t+64

d^2=25t^2-64t+64, and we want to know when they are 8 miles apart

25t^2-64t+64=8^2

25t^2-64t+64=64

25t^2-64t=0

25t(t-2.56)=0

So 2.56 hours after Tommy started running they will be 8 miles apart.

To be clear though, this is the time Tommy spent running, or the time that elapsed since Tommy started running.  Zach has only been running for 0.56 hours.  So if they are looking for the time Zach ran it is 0.56 not 2.56 hours.