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.