A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s.

Question

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s.

(a) At what rate is his distance from second base decreasing when he is halfway to first base? (Round your answer to one decimal place.)

 ft/s

(b) At what rate is his distance from third base increasing at the same moment? (Round your answer to one decimal place.)

 ft/s

Answer 1

Let the distance from the runner to first base be x.

Let s be the distance from second base to the runner, which is the hypotenuse of a right triangle.

1)

s² = x² + 90²

Taking the derivative of both sides of the equation:

2ss' = 2xx' + 0

s' = xx'/s

When x = halfway, 45 ft, s = √(45² + 90²) = = √(45²(1+2²)) =45√5

x' is negative because the difference is decreasing between the runner and first base.

s' = 45(-24)/(45√5) = -24√5 ft/s = --53.7 ft/s

2) Let z be the distance from 3rd base to the runner. Home base is a right angle.

The difference between z' and the answer for part 1) is that x is taken from home to the runner and

the velocity is positive because the distance from the home to the runner is increasing.

z' = 24√5 ft/s = 53.7 ft/s