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Answer :
Final Answer:
The rocket will reach its maximum height in 1 second, reaching a maximum height of 24 feet.
Explanation:
The height of the rocket above the ground at time t seconds after launch is given by the function h(t) = -16t² + 32t + 8. To find the time it takes for the rocket to reach its maximum height, we need to determine when the rocket's vertical velocity becomes zero. The rocket's vertical velocity is given by the derivative of the height function with respect to time: v(t) = h'(t) = -32t + 32.
Setting v(t) = 0 and solving for t:
-32t + 32 = 0
-32t = -32
t = 1 second.
So, it takes the rocket 1 second to reach its maximum height. To find the maximum height, we substitute this time back into the height function:
h(1) = -16(1)² + 32(1) + 8
h(1) = -16 + 32 + 8
h(1) = 24 feet.
Therefore, the rocket reaches its maximum height of 24 feet after 1 second.
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