Question

A baseball diamond is a square whose sides are 90

feet long. Suppose a player running from second base to third base has a speed of 30 feet per second at the instant he is 20 feet from third base. At what rate is the player’s distance from home plate changing at that instant?

Answer 1

Answer: dy/dt = 37.42 fps

Explanation:

First, let's define a few variables.

t = time

x = Distance the player is from third base = 20 feet

dx/dt = Velocity the player is running towards third base = -30 fps (since he is running towards third base, the x value will decrease with time, hence the negative value)

y = Distance between the player and home plate (We will find this value below)

dy/dt = Rate of change between the player's distance from home plate (This is the answer to our problem)

Now, let's use the Pythagorean theorem to relate the x and y terms.

y2 = x2 + 902

Use x = 20 feet to find y in this instant.

y2 = 202 + 902

y = 92.2 feet

Now, let's use implicit differentiation to take the derivative of this expression.

2y*dy/dt = 2x*dx/dt + 902

And let's plug in x, y, and dx/dt to solve for dy/dt.

2(92.2)*dy/dt = 2*20*-30 + 902

dy/dt = 37.42 fps