Question

An object is projected directly upward from the ground. After t seconds its distance in feet above the ground is s = 144t -16t2
How long does it take for the object to reach its maximum height?
What is the maximum height the object reaches? When does it hit the ground?

Answer 1

To find the maximum height, we need to find the vertex of the parabolic function s = 144t -16t^2. The vertex has a t value of -b/2a, where a = -16 and b = 144. So:

t = -b/2a = -144/(2(-16)) = 4.5

To find the maximum height, we substitute 4.5 for t in the equation:

s = 144t -16t^2 = 144(4.5) -16(4.5)^2 = 324 feet

So the object reaches a maximum height of 324 feet.

To find when the object hits the ground, we set s = 0 and solve for t:

0 = 144t -16t^2
0 = t(144 -16t)
t = 0 or t = 9

t = 0 corresponds to when the object is first launched and t = 9 corresponds to when it hits the ground. So the object hits the ground after 9 seconds.

Therefore, the object takes 4.5 seconds to reach its maximum height, reaches a maximum height of 324 feet, and hits the ground after 9 seconds.