Calculus Question 3

Question

A baseball diamond is a square with side 90ft. 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?

 

(b) At what rate is his distance from third base increasing at the same moment?

Answer 1

I drew a diagram with the coordinate system set up so that the home plate is at the origin. If you draw the diagram, the distance to the second base will vary with x as you run from the origin to the first base. The distance to second base can be written: 

 

R=√(x²+90²)                                                           (1)

 

This is a funcion of x (y isn't changing as you run to 1st base). 

 

The rate of change with respect to time is the time derivative of R. Note that x is changing with time, so apply the chain rule. 

 

(dR/dt)=(d/dx)√(x²+90²)  •(dx/dt)                    (2)

 

First we must differentiate R with respect to x, 

 

1/(2√(x²+90²) •2x = x/√(x²+90²)

 

Use this in equation (2) to obtain:

 

dR/dt = x/√(x²+90²) •(dx/dt)                                    (3)

 

The speed of the runner is given as dx/dt = 24 ft/s

 

Substitute the speed into equation (3) and then substitute the halfway point (x=45) to evaluate the derivative of R at x=45. I get about 10.73 ft/s.