(a)

The height of the ball at time  is .

Initial velocity of the ball is  feet per second.

Find .

Substitute  in the function.

Find .

Substitute  in the function.

Therefore .

(b)

Rolles Theorem :

Let  be a function that satisfies the following three hypotheses.

1.   is continuous on .

2.   is differentiable on .

3. .

Then there is a number  in  such that .

 is continuous on  and differentiable on the interval .

So the function  satisfies the three hypotheses.

Therefore there is a number  in  such that .

The function .

Apply derivative on each side with respect to .

Equate .

The velocity of the ball will be  m/sec in the interval  is  sec.

(a) .

(b) The velocity of the ball will be  m/sec in the interval  is  sec.