The path of a rocket is given by the equation s(t)=64t - 16t^2, where t is time and s is height.?What is the maximum height the rocket will reach?...

The height s is given by the
function:

f(s) = 64t -
16t^2

Where: t = time

The
height will be maximum at a point where:

f'(s) =
0

We find the corresponding value of t value as
follows.

f'(s) = 64 - 2*16t = 64 -
32t

Equating f('s) to 0

64 -
32t = 0

==> -32t = -
64

==> t = -64/-32 =
2

we get the maximum height by substituting this value of t
in equation of height:

Maximum height = f(2) = 64*2 -
16*(2^2) = 128 - 64 = 64

When the rocket hits the ground s
= 0

Therefor to find the time when the rocket hits the
ground we equate the equation for height to
0.

Thus:

s = 64 t - 16 t^2 =
0

==> 16t(4 - t) =
0

Therefore:

t = 0 and t =
4

t = 0 represents the time when the rocket is fired from
ground level, and t = 4 represents the time when the rocket again falls to the ground
level

Answer:

Maximum height
of rocket = 64 units

Time taken to hit the ground = 4
units