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Answer :
the maximum height the rocket reaches is 234 feet.
The function h(t) = -16t^2 + 120t + 9 represents the height of the toy rocket above the ground at time t seconds after launch. To find the time it takes for the rocket to reach its maximum height, we need to determine the value of t when the height function h(t) is at its maximum.
The maximum or minimum value of a quadratic function occurs at its vertex. In this case, the vertex form of the quadratic function is h(t) = a(t - h)^2 + k, where (h, k) represents the vertex of the parabola.
Comparing this with the given function h(t) = -16t^2 + 120t + 9, we can see that a = -16, h = -b/(2a), and k = h(h(t)).
Using the formula h = -b/(2a), we can find h as follows:
h = -120/(2 * -16) = -120/-32 = 3.75
So, the rocket reaches its maximum height at t = 3.75 seconds.
To find the maximum height, we substitute t = 3.75 into the height function:
h(3.75) = -16(3.75)^2 + 120(3.75) + 9
h(3.75) = -16(14.0625) + 450 + 9
h(3.75) = -225 + 450 + 9
h(3.75) = 234
Therefore, the maximum height the rocket reaches is 234 feet.
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