what is the maximum height reached by the rocket

Question

after t seconds a model rocket is launched in the air from ground level and reaches a height h (t) =96t -12t². The rocket then falls back to the ground

what is the maximum height reached by the rocket?

Answer 1

h(t) = 96t -12t^2

h'(t) = 96 -24t = 0

t = 96/24 = 4 seconds

h(4) = 96(4) -12(4^2) = 384-192= 192 feet = maximum height

it hits the ground when 96-12t = 0, t = 96/12 = 8 seconds

max height is half 8 seconds, midway between lift off and splash down

seconds height

0 - 0

1 - 96-12 = 84

2 - 96(2) - 12(4) = 192-48 = 144

3 - 96(3) - 12(9) = 288 - 108 = 180

4 - 96(4) - 12(16) = 384-192 = 192 feet high

5 - 96(5) - 12(25) = 180

6 96(6) -12(36) = 144

7 96(7) -12(49) = 84

8 96(8) -12(64) = 0

general height formula is

h(t) = (1/2)gt^2 +vot + ho where vo=initial velocity, ho = initial height, g = acceleration due to gravity

g= about -9.8 m/sec^2 or -32 ft/sec^2, using -12t^2 suggests g= -24 as if there's a lot of air resistance