Question

The height of a toy rocket launched from a 64-foot observation tower as a function of elapsed time since the launch is modeled by the equation shown below. At what approximate elapsed time(s) will the toy rocket be at a height of 90 feet?

h(t) = -16t^2 + 48t + 64
A) The rocket will never reach 90 feet in height.
B) The rocket will reach a height of 90 feet at approximately 0.71 seconds after it is launched.
C) The rocket will reach a height of 90 feet at approximately 4 seconds after it is launched.
D) The rocket will reach a height of 90 feet at approximately 0.71 seconds after it is launched and at 2.29 seconds after it is launched.

Answer 1

The rocket will reach a height of 90 feet at approximately 2.865 seconds after it is launched.

Step-by-step explanation:

The height of the toy rocket can be determined by setting the equation h(t) = -16t²+ 48t + 64 equal to 90 feet and solving for t.

-16t² + 48t + 64 = 90

-16t² + 48t - 26 = 0

Using the quadratic formula, we can solve for t:

t = (-b ± √(b² - 4ac)) / 2a

where a = -16, b = 48, c = -26.

Substituting the values into the formula, we get:

t = (-48 ± √(48² - 4(-16)(-26))) / (2(-16))

t = (-48 ± √(2304 - 832)) / (-32)

t = (-48 ± √1472) / (-32)

t ≈ (-48 ± 38.4) / (-32)

t ≈ -1.15 or t ≈ 2.865

Since we are only interested in positive time values, we can conclude that the rocket will reach a height of 90 feet at approximately 2.865 seconds after it is launched. Therefore, the correct answer is D) The rocket will reach a height of 90 feet at approximately 2.865 seconds after it is launched.

Answer 2

Solving the height equation h(t) for 90 using the quadratic formula, we find two times at which the rocket reaches that height. These times represent when the rocket is on the way up and on the way down.

Step-by-step explanation:

To determine when the toy rocket will be at a height of 90 feet, we need to solve the equation h(t) = -16t2 + 48t + 64 for h(t) = 90.

Substituting 90 for h(t), we get:

90 = -16t2 + 48t + 64

To solve for t, we can first rearrange the equation:

0 = -16t2 + 48t - 26

Now, we can use the quadratic formula to find the values of t:

t = ∛-b ± √(b2 - 4ac))/(2a)

Where a = -16, b = 48, and c = -26. Plugging in the values, we get two potential times, t1 and t2, when the rocket will reach a height of 90 feet.

These exact times can be calculated by evaluating the quadratic formula with a calculator or algebraic computation. The rocket will reach a height of 90 feet at both of these times, once on the way up and once on the way down.