Question

A baseball diamond is a square with sides 90 ft long. A batter is at bat, with runners

at first and second base. At the moment the ball is hit, the runner at first base runs to
second base at 25 ft/s. Simultaneously, the runner on second base runs to third base
at 15 ft/s. How fast is the distance between these two runners changing 2 s after the
ball is hit?

Answer 1

The runner on first has distance from second of 90 -25t, where t is the time in seconds since he left 1st base.

The runner on second has a distance from second of 15t, where t is the time in seconds since he left 2nd base

Then the Pythagorean theorem tells us the distance between the runners is :

d = √((90 -25t)² +(15t)²)

The rate of change of distance between the runners is the derivative of this with respect to time.

dd/dt = 2(90-25t)(-25)+2(15t)(15) / (2sqrt(90-25t)^2 + 15t^2)

Replace t with 2 and solve to get: -31

The distance between runners is decreasing at 31 ft/s two second after they leave the bases.