A baseball player hits a line drive to center field. He runs from second to third at 20 feet/sec. How fast is the distance from the runner to home plate changing when he is halfway to third base?

Answer 1

If I recall, the length of a side of a baseball diamond is 90 feet.
Make a diagram of a right-angled triangle using the third base line as one side, the distance of the runner to home plate as the hypotenuse h, and letting the distance of the runner to third be x feet

then x^2 + 90^2 = h^2
2x dx/dt = 2h dh/dt
dx/dt = h (dh/dt) / x

we know dx/dt = -20 ft/sec
and when x= 45
h^2 = 45^2 + 90^2
h = √10125

dh/dt = √10125(-20)/45 = .....

(the negative value of dx/dt shows that the value of x is decreasing, as I defined x)