Related Rates, Base ball diamond

Question

A (square) baseball diamond has sides that are 90 feet long. A player 23 feet from third base is running at a speed of 27 feet per second. At what rate is the player's distance from home plate changing? (Round your answer to two decimal places.)

Answer 1

Turn the diamond so it looks like a square. A right triangle exists where 3rd base is a right angle between 2nd base and home. Since the runner is running toward 3rd base, the velocity dx/dt = -27ft/s. Let s be the hypotenuse of the triangle that is the distance between the runner and home plate.

s² = x² + 90²

Differentiating both sides of the equation:

2sds/dt = 2xdx/dt --> ds/dt = (x/s)dx/dt

When x = 23, s = √(8100 + 529) = 92.89

ds/dt = (23/92.89)(-27) = -6.69 ft/s --> the rate is decreasing