How fast is the distance between the runners changing 1 second after the ball was hit?

Question

You are watching a baseball game. Runners are standing at first and second base. The moment the batter hits the ball, the runner in first base runs to second base 18 feet per second. The runner at 2nd runs to third at 20 feet per second. How fast is the distance between the runners changing 1 second after the ball was hit? Each base by the way is 90 feet apart. Assume the bases are on corners of a square.

Answer 1

So to approach this problem I decided to place the square of the baseball field on the coordinate plane, with the Origin being home base, first base being the point (90, 0), second base being the point (90, 90), and third base being the point (0, 90). If we consider the two runners, R1 and R2, as moving points starting at first base and second base respectively, we can model them as the points:

R1: (90, 18t)

R2: (90 - 20t, 90)

where t is the time (in seconds) since the ball has been hit. If we want to consider the distance between these two points at time t, we can do so using the distance formula, which gives us

D = √((90 - 20t - 90)² + (90 - 18t)²).

Expanding and simplifying, we find that

D = (724t² - 3240t + 8100)^1/2.

Now that we know how the distance between each runner is changing over time, can we see how fast the distance between them is changing at any point in time by taking the derivative of D with respect to t. Using the chain rule, we get:

dD/dt = (1/2)(1448t - 3240)(724t² - 3240t + 8100)^-1/2.

From there, we can simply let t = 1 second, which gives us dD/dt ≈ -12 ft/s.