a water rocket is launched with an initial velocity of 112ft/sec. its height h, in feet T seconds is given by the formula h=112T-16T^2. when will the rocket be exactly 192 ft above the ground?

the rocket first reaches a height of 192ft after _____ and reaches 192ft a second time after____.

thank you!

192 = 112 t - 16 t^2

or
16 t^2 - 112 t + 192 = 0

t = [ 112 +/- sqrt ( 12544 - 12288)]/32
t = [112 +/- 16 ]/ 32
t = 3 seconds on the way up
and
t = 4 seconds on the way down

To find when the rocket is exactly 192 ft above the ground, we need to solve the equation h = 192.

Substituting h = 192 into the given formula h = 112T - 16T^2:

192 = 112T - 16T^2

This is a quadratic equation that we can solve by setting it equal to zero:

16T^2 - 112T + 192 = 0

Now, we can solve this quadratic equation. We can either factor it or use the quadratic formula. Let's use the quadratic formula:

T = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = 16, b = -112, and c = 192.

T = (-(-112) ± √((-112)^2 - 4*16*192)) / (2*16)
T = (112 ± √(12544 - 12288)) / 32
T = (112 ± √256) / 32

T = (112 ± 16) / 32

This gives two possible values for T:

T1 = (112 + 16) / 32 = 128 / 32 = 4
T2 = (112 - 16) / 32 = 96 / 32 = 3

Therefore, the rocket first reaches a height of 192 ft after 3 seconds and reaches 192 ft a second time after 4 seconds.

To find when the rocket will be exactly 192 feet above the ground, we need to set the height equation equal to 192 and solve for the time (T).

The given height equation is: h = 112T - 16T^2

Setting h = 192, we have:
192 = 112T - 16T^2

Now, let's rearrange this equation and solve for T.

16T^2 - 112T + 192 = 0

To further simplify the equation, we can divide through by 16:
T^2 - 7T + 12 = 0

This is a quadratic equation, which can be factored as:
(T - 3)(T - 4) = 0

This equation is satisfied when either T - 3 = 0 or T - 4 = 0.

For T - 3 = 0:
T = 3

For T - 4 = 0:
T = 4

Therefore, the rocket will be exactly 192 feet above the ground for the first time after 3 seconds, and it will reach the same height again after 4 seconds.