Bill throws a ball upward from the top of an 84 foot building with an initial velocity of 128ft/sec. a. at what time will the ball reach maximum height? b. what is the maximum height? c. how long will it take the ball to hit the ground?

Somewhere in your text or background material you're given a position function, which gives the position, as a function of time, of an object falling at or near the surface of the earth. This function will include a term which allows for the initial velocity of the object (a linear term), and another for its initial position (a constant).

The first derivative of this position function gives the velocity of the object at any time t. Determine the deriv of the position function (the velocity function), and then determine the time t at which velocity = 0. That'll be the moment it reaches its max height (i.e. the answer to (a)). Plug that time back into your position function to arrive at its max height in feet, and (b) is done.

For (c), the ball hits the ground at the moment t for which its position is -84'.

Give those a whirl and see where you go.

Sorry, clarification needed. If your position formula wants you to consider the ball's initial position (at the top of the bldg) as zero, then the ball strikes the ground at the moment that its position is - (neg) 84.

Equivalently, you could consider the top-of-the-bldg start position as being +84. If so, then you want to use the position equation to determine the time at which its position is zero.

It'll be same time t either way.