Question

A baseball diamond is a square with sides of length 90 ft. A batter hits the ball and runs toward first base with a speed of 21 ft/s.

At what rate is his distance from second base changing when he is halfway to first base?
At what rate is his distance from third base changing at the same moment?

Answer 1

https://answers.yahoo.com/question/index?qid=20100220141822AAoU9KV

Answer 2

At any time the runner is between home and first, the line from runner to first, the line from runner to second, and the line from first to second form a right triangle with the line from runner to second being the hypotenuse. Let x = distance between runner and first and y = distance between runner and second.
y^2 = x^2 + 90^2 = x^2 + 8100
2y dy/dx = 2x
dy/dx = x/y = x/(x^2 + 8100) or dy/dx = -x/(x^2 + 8100)
His distance to second is decreasing as he runs to first so use dy/dx = -x/(x^2 + 8100)
At x = 90/2 = 45,
dy/dx = -45/(45^2 + 8100)^(1/2) =
45/(10125)^(1/2) = -0.4472 ft/s

Likewise, the line from home to third, home to runner, and third to runner form a right triangle with third to runner being the hypotenuse. Let x = distance between home and runner and y = distance between runner and third.
y^2 = x^2 + 90^2
y^2 = x^2 + 90^2 = x^2 + 8100
2y dy/dx = 2x
dy/dx = x/y = x/(x^2 + 8100) or dy/dx = -x/(x^2 + 8100)
His distance to third is increasing as he runs to first so use dy/dx = x/(x^2 + 8100)
At x = 90/2 = 45,
dy/dx = -45/(45^2 + 8100)^(1/2) =
45/(10125)^(1/2) = 0.4472 ft/s

Answer 3

Refer to a solution using Mathematica 9 Home Edition.