Expert: Socrates - 1/21/2009

Question

a baseball diamond is a square with each side 90 feer in length a player runs from second base to third base at a rate of 18ft/sec. at what rate is the players distancce from firt base a changing when his dis. from third base d is 22.5?

Answer
Unfortunately , I can't draw a diagram for you ,and you really need to see one to fully understand this . Anyway, here is a solution:

After t seconds , the runner's distance to first base will be

d(t) = ( (18t)^2 + 90^2 )^1/2   you need a picture for this :(

simplify

d(t) = (18)(t^2 + 25 )^1/2

d'(t) will tell us how fast the runner is moving away from first base after t seconds

d'(t) = (18t)(t^2 + 25 )^-1/2


When the runner's distance to third base is 22.5 ft , the runner has traveled 90 - 22.5 = 67.5 ft . It takes the runner 67.5/18 = 3.75 seconds to travel 67.5 ft.

Set t = 3.75 in  d'(t) = (18t)(t^2 + 25 )^-1/2  and get

d'(3.75) = 10.8 ft/sec

The runner is moving away from first base at 10.8 ft/sec


This problem can also be solved using implicit differentiation. Express the runner's distance from first base as a function of his distance to third base. Then , considering distances as functions of t , take the derivative of both sides and then solve. If you need to see it done this way , let me know.