Calculus rate of change

Question

A baseball diamond has the shape of a square with sides 90 feet long. A player running from second base to third base at a speed of 28 feet per second is 25 feet from third base. At what rate is the players distance from home plate changing?

Professor is looking for us to round to two decimal places.

Some help tackling this would be greatly appreciated.

Answer 1

Consider a right triangle with sides

r(t):= distance between runner & 3rd base at time t and

s(t):= distance 3rd base to home plate. Note this is constant.

h(t):= distance between runner and home plate (h for hypotenuse). We want dh/dt

r²(t)+s²(t) = h²(t)

Differentiate both sides with respect to t:

2r(t)*dr/dt + 2s(t)*ds/dt = 2h(t)*dh/dt

Cancel the 2's and notice that ds/dt = 0.

When r(t) = 25', we know that h(t) = √25² + 90² ≈ 93.4'

We're also told that, at the same moment in time, dr/dt = -28 ft/sec (distance to 3rd base is decreasing, hence the negative sign)

Therefore 25 * (-28) = 93.4*dh/dt and dh/dt = 25*(-28)/93.4