A model rocket is launched vertically upward so that its height (measured in feet) t sec after launch is given by the following.

h(t) = -16t2 + 272t + 8

(a) Find the times when the rocket is at a height of 488 ft.
t = sec (smaller value)
t = sec (larger value)

(b) How long is the rocket in flight? (Round your answer to the nearest whole number.)

48

its 44

To find the times when the rocket is at a height of 488 ft, we need to solve the equation h(t) = 488.

(a)
First, we substitute 488 for h(t) in the equation:
-16t^2 + 272t + 8 = 488.

Next, we rearrange the equation in standard quadratic form:
-16t^2 + 272t + 8 - 488 = 0.

Simplifying further:
-16t^2 + 272t - 480 = 0.

Now, we can solve this quadratic equation for t. There are multiple methods to solve quadratics, such as factoring, completing the square, or using the quadratic formula. I will use the quadratic formula here.

The quadratic formula is:
t = (-b ± √(b^2 - 4ac)) / (2a).

In this case, a = -16, b = 272, and c = -480. Substituting these values:
t = (-272 ± √(272^2 - 4(-16)(-480))) / (2(-16)).

Simplifying further:
t = (-272 ± √(73984 - 30720)) / -32.

t = (-272 ± √43264) / -32.

Now, we can calculate the values of t by evaluating the expression:
t = (-272 ± 208) / -32.

This gives two values for t:
t = (264 / -32) ≈ -8.25 seconds,
t = (480 / -32) ≈ -15 seconds.

Since time cannot be negative in this context, we discard the negative value.

Thus, the times when the rocket is at a height of 488 ft are approximately t = -8.25 seconds and t = -15 seconds.

(b) To find how long the rocket is in flight, we need to find the time when its height is zero. This is the time when it reaches the ground.

To do this, we set h(t) = 0 in the equation:
-16t^2 + 272t + 8 = 0.

Now, we solve this quadratic equation for t, as shown above. However, in this case, we are only interested in the positive value of t when the rocket reaches the ground.

After solving the equation, we find that the positive value of t is approximately t = 17 seconds.

Since the time is usually measured in whole numbers, we can round our answer to the nearest whole number.

Therefore, the rocket is in flight for approximately 17 seconds.

(a) just solve

-16t^2 + 272t + 8 = 488

(b) solve when h(t) = 0