Question

A car traveling north at 40mi/h and a truck traveling east at 30mi/h leave an intersection at the same time. At what rate will the distance between them be changing 3 hours later?

Answer 1

By looking at a picture, we see that the distance between them is the hypotenuse of the right triangle formed by the original intersection and their positions at time t.

The question requires us to find the equation of that distance in terms of t and then take it's derivative dt. Using this derivative, we can find the rate of change of their distance at time t=3.

Answer 2

The position of the northbound ar at time t is 40t while the eastbound car is 30t. So the distance between them at time t is\[d(t)=\sqrt{(40t)^{2} + (30t)^{2}}=\sqrt{1600t^{2} + 900t^{2}}=\sqrt{2500t^{2}}=50t\]

Answer 3

From here, simply take the derivative and evaulate at t=3.

Of course, you won't really need to evaulate at t=3 since the derivative is a constant.