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 26 ft/s.

(a) At what rate is his distance from second base decreasing when he is halfway to first base?
ft/s

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

Answer 1

a) First create a diagram similar to the one above.

Your first leg will be from home to first base (a = 45).

Your second leg will be from first base to second base (b = 90).

You want to find your hypotenuse. a² + b² = c²

(45)² + (90)² = c² 2025 + 8100 = c² √10125 = √c² c≈101

Now you want to use implicit differentiation to find the rate with respect to time.

a² + b² = c²

2a+ 2b= 2c

Since the runner is moving from home to first base (a), the path from first base to second (b) doesn't change. So that means = 0ft/s

So lets see what we hav so far

a = 45

= 26ft/s

b = 90

= 0ft/s

c = 101

= ?

so we want to find .

Let's plug in everything we have from the implicit differentiation equation

2(45)(26) + 2(90)(0) = 2(101)()

2(45)(26) = 2(101)()

=

≈ 11.6 ft/s

So the distance is 11.6ft/s

b)

Due to symmetry the distance to third base is increasing at exactly the same rate, at the halfway point only