Question

a baseball 'diamond' is a square with sides of length 90 feet. consider a player running from second base to third base at a rate of 25 feet per second. determine the rate of change of the distance between the player and home plate when the player is 20 feet from third base. baseball diamond

Answer 1

If a baseball 'diamond' is a square with sides of length 90 feet. The rate of change of the distance between the player and home plate when the player is 20 feet from third base is 7 feet per second.

What is the rate?

Using the concept of similar triangles set up the following equation:

(D - 20) / (D - D(t)) = 20 / D

Solve for D(t).

Simplify the equation:

(D - 20) * D = 20 * (D - D(t))

Expand and rearrange the equation:

`D^2 - 20D = 20D - 20D(t)\\D^2 - 40D = -20D(t)`

Dividing both sides by -20:

`D(t) = (D^2 - 40D) / 20\\`

Solve for dD/dt.

dD(t)/dt = (2D - 40) / 20

Substitute

dD(t)/dt = (2(90) - 40) / 20

dD(t)/dt = (180 - 40) / 20

dD(t)/dt = 140 / 20

dD(t)/dt = 7 feet per second

Therefore the rate of change of the distance between the player and home plate when the player is 20 feet from third base is 7 feet per second.