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Question

A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 30 feet per second is 21feet from third base. At what rate is the player's distance from home plate changing? (Round your answer to two decimal places.)

Answer 1

Hi Jessica,

This is a related rates problem using a right triangle. I'm going to draw a triangle with point P on the top right (Player), point T on the top left (Third), and point H directly below T (Home).

Let PT = x

Let TH = y

Let PH = D

Pythagorean Theorem states that

x^2 + y^2 = D^2

We're interested in the rate of change of the distance from the player to home plate, or dD/dt. In order to generate this term, we can implicity differentiate the Pythagorean equation above with respect to t.

d/dt ( x^2 + y^2 = D^2 )

2x*dx/dt + 2y*dy/dt = 2D*dD/dt

x*dx/dt + y*dy/dt = D*dD/dt

x is the distance from the player to third base, which the problem says is 21.

y is the distance from third base to home plate, which the problem says is 90

D, which can be determined by the Pythagorean Theorem, is ~92.42

dx/dt is the rate of change of the distance between the player and third base. The player is running towards third base at a rate of 30 feet per second. Since the length x is decreasing, dx/dt is -30.

dy/dt is the rate of change of the distance between third base and home plate. The bases themselves don't move, so this value is 0.

dD/dt is the rate of change we're looking for.

x*dx/dt + y*dy/dt = D*dD/dt

(21)(-30) + (90)(0) = (92.42)*dD/dt

-630 = 92.42*dD/dt

dD/dt = -6.82 ft/ sec

So the player's distance from home plate is decreasing by 6.82 ft/ sec.

Hope this helps!