1. A rocket is launched from the top of a 200-foot tall cliff with an initial velocity of 120 feet per second.
The height, h(t), of the rocket at time i seconds after it was launched can be modeled by the equation
h(t) = -16t^2 120t+ 200

a. What was the height of the rocket after 1 second?

b. What is the maximum height of the rocket?

c. How long did it take the rocket to reach its maximum height?

d. How long does it take the rocket to hit the ground?

Answers

Answer 1

Answer:

a. 304 feet

b. 425 feet

c. 3.75 seconds

d. It will take the rocket approximately 8.9 seconds to hit the ground

Step-by-step explanation:

1. The given data of the question including the question relationships are listed as follows;

The height of the cliff from which the rocket was launched = 200-ft.

The initial velocity of the rocket, u = 120 feet per second

The equation that models the height of the rocket with respect to time 't' is h(t) = -16·t² + 120·t + 200

a. The height of the rocket after 1 second is given by plugging in t = 1 in the equation that models the height as follows;

h(1) = -16 × (1)² + 120 × (1) + 200 = 304

The height of the rocket after 1 second, h(1) = 304 feet

b. The maximum height can be found by finding the value of the maximum point of the curve of the equation for the height as follows;

At the maximum point, h'(t) = d(-16·t² + 120·t + 200)/dt = 0

∴ d(-16·t² + 120·t + 200)/dt = -32·t + 120 = 0

t = 120/32 = 3.75

At the maximum point, t = 3.75 seconds

The maximum height can also be found by finding the vertex of the parabola formed by the height equation as follows;

At the vertex (the maximum point), t = -b/(2·a) = -120/(2 × (-16)) = 3.75

At the vertex (the maximum point), t = 3.75

The height at the maximum point = h(3.75) = -16 × (3.75)² + 120 × (3.75) + 200 = 425

The height at the maximum point, h(3.75) = 425 feet

c. The time it will take the rocket to reach the maximum point is given by the value of 't' at the maximum point, the vertex, which is t = -b/(2·a) = -120/(2 × (-16)) = 3.75

Therefore;

The time it takes the rocket to reach the maximum point, [tex]t_{max}[/tex] = 3.75 seconds

d. The time it takes the rocket to hit the ground (height h(t)) is given by finding the values of 't' that satisfy the height equation where h(t) = 0 as follows;

At the ground level, h(t) = 0 = -16·t² + 120·t + 200

Therefore, we get;

-16·t² + 120·t + 200 =

t = (-120 ± √(120² - 4×(-16) × 200))/(2 × (-16))

t ≈ 8.9 or t ≈ -1.4

Therefore, the time it takes the rocket to hit the ground is given by the natural number, t ≈ 8.9 seconds.


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