A toy rocket is shot vertically into the air from a launching pad 9 feet above the ground with an initial velocity of 80 feet per second. The height h, in feet, of the rocket above the ground at t seconds after launch is given by the function h(t)=−16t2+80t+9. How long will it take the rocket to reach its maximum height? What is the maximum height?

Question

A toy rocket is shot vertically into the air from a launching pad 9 feet above the ground with an initial velocity of 80 feet per second. The height h, in feet, of the rocket above the ground at t seconds after launch is given by the function h(t)=−16t2+80t+9. How long will it take the rocket to reach its maximum height? What is the maximum height?

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Part 1
The rocket reaches its maximum height at enter your response here second(s) after launch.
(Simplify your answer.)
Part 2
The maximum height reached by the object is enter your response here feet.
(Simplify your answer.)

Answer 1

To find the maximum height of the rocket, we need to find the vertex of the parabola described by the function h(t) = -16t^2 + 80t + 9. The t-coordinate of the vertex gives us the time at which the rocket reaches its maximum height, and the h-coordinate gives us the maximum height.

The t-coordinate of the vertex is given by t = -b/2a, where a = -16 and b = 80. Plugging in the values, we get:
t = -80/(2*(-16)) = -80/(-32) = 2.5 seconds.

So, the rocket reaches its maximum height at 2.5 seconds after launch.

To find the maximum height, we substitute this time value back into the function:
h(2.5) = -16(2.5)^2 + 80(2.5) + 9 = -16(6.25) + 200 + 9 = -100 + 200 + 9 = 109 feet.

So, the maximum height reached by the rocket is 109 feet.

Answer 2

To find the maximum height the rocket reaches, we need to find the vertex of the quadratic function h(t) = -16t^2 + 80t + 9. The vertex of a quadratic function in the form y = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the quadratic function.

In this case, the coefficients are:
a = -16
b = 80
c = 9

Using the formula -b/2a, we can find the time it takes for the rocket to reach its maximum height:
t = -80 / (2*(-16))
t = -80 / (-32)
t = 2.5 seconds

So, the rocket reaches its maximum height 2.5 seconds after launch.

To find the maximum height, we substitute this value of t back into the function h(t):
h(2.5) = -16(2.5)^2 + 80(2.5) + 9
h(2.5) = -16(6.25) + 200 + 9
h(2.5) = -100 + 200 + 9
h(2.5) = 109 feet

Therefore, the rocket reaches its maximum height after 2.5 seconds and reaches a maximum height of 109 feet.

Answer 3

Part 1: To find the time it takes for the rocket to reach its maximum height, we need to find the vertex of the parabolic function h(t) = -16t^2 +80t + 9. The vertex represents the maximum point of the parabola.

The formula for finding the x-coordinate of the vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by:

x = -b / (2a)

In our case, the equation is h(t) = -16t^2 +80t + 9, so a = -16 and b = 80. Plugging these values into the formula, we get:

t = -80 / (2*(-16))
t = 80 / 32
t = 2.5

Therefore, it takes the rocket 2.5 seconds to reach its maximum height.

Part 2: To find the maximum height reached by the rocket, we substitute the value of t we found (2.5) into the function h(t) = -16t^2 +80t + 9.

h(2.5) = -16(2.5)^2 +80(2.5) + 9
h(2.5) = -16(6.25) +200 + 9
h(2.5) = -100 +200 + 9
h(2.5) = 109

Therefore, the maximum height reached by the rocket is 109 feet.