Question: As part of the opening ceremonies for a new skyscraper, a sponge ball is shot off the roof's edge straight into the air, with spectators watching safely from a distance. The spectator who retrieves the sponge ball will win a cash prize when the streets below the tower are reopened. The following position function describing the height of the sponge ball during its flight:

\[s\left( t \right)=-16{{t}^{2}}+128t+1344\]

(a) From this position function, identify the initial velocity.

(b) From this position function, identify the height of the office tower.

(c) What is the height of the sponge ball 3 seconds after it leaves the roof?

(d) What is the height of the sponge ball 6 seconds after it leaves the roof?

(e) When does the ball hit the ground? [COMMENTS & HINTS: What is the height of the sponge ball above the ground when it reaches the ground? Ignore any extraneous or unreasonable answers.]

(f) When will the sponge ball reach its maximum height above the ground?

(g) At what height does the ball reach maximum altitude?

(h) Confirm that the maximum is indeed a maximum two different ways. [COMMENTS & HINTS: Think analytic geometry and second derivatives.]

(i) When will the sponge ball pass the edge of the roofline on its descent to the ground? [COMMENTS & HINTS: Think initial height.]

(j) What is the velocity of the sponge ball 2 seconds after it leaves the roof?

(k) What is the velocity of the sponge ball 8 seconds after it leaves the roof?

(l) What is the velocity of the sponge ball 12 seconds after it leaves the roof?

(m) What is the velocity of the sponge ball when it reaches the ground?