A model rocket is launched with an initial upward velocity of

164
/
ft
s

. The rocket's height
h

(in feet) after
t

seconds is given by the following.
=
h

164
t
16
t
2

Find all values of
t

for which the rocket's height is
92

feet.

Round your answer(s) to the nearest hundredth.
(If there is more than one answer, use the "or" button.)

To find the values of t for which the rocket's height is 92 feet, we need to set the equation h = 92 and solve for t.

Using the given equation, h - 164t - 16t^2 = 92.

Rewriting the equation in standard form, we get 16t^2 + 164t - (h - 92) = 0.

Substituting h = 92 into the equation, we have 16t^2 + 164t - (92 - 92) = 0.

Simplifying the equation, we have 16t^2 + 164t = 0.

Factoring out 4t, we get 4t(4t + 41) = 0.

Setting each factor equal to zero, we have 4t = 0 and 4t + 41 = 0.

Solving for t, we find t = 0 and t = -10.25.

Thus, the values of t for which the rocket's height is 92 feet are t = 0 and t = -10.25 (rounded to the nearest hundredth).

To find all values of t for which the rocket's height is 92 feet, we can set the equation h = 92 and solve for t.

h - 164t = 16t^2

Rearranging the equation, we get:

16t^2 - 164t + 92 = 0

To solve this quadratic equation, we can use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 16, b = -164, and c = 92. Substituting these values into the quadratic formula:

t = (-(-164) ± √((-164)^2 - 4(16)(92))) / (2(16))

Simplifying further:

t = (164 ± √(26896 - 5888)) / 32

t = (164 ± √(21008)) / 32

Calculating the square root:

t = (164 ± 144.984) / 32

Now we can calculate the two values of t:

t1 = (164 + 144.984) / 32 ≈ 9.656

t2 = (164 - 144.984) / 32 ≈ 0.484

Therefore, the values of t for which the rocket's height is 92 feet are approximately t = 9.66 or t = 0.48.

First let me fix your post:

A model rocket is launched with an initial upward velocity of
164 ft/s. The rocket's height h (in feet) after t seconds is given by the following.
h = −16t^2 + 164t
Find all values of t for which the rocket's height is 92 feet.

so you want:
92 = -16t^2 + 164t
16t^2 - 164t + 92 = 0
4t^2 - 41t + 23 = 0
Use the quadratic equation to solve.