To find the time it takes for the boulder to reach its maximum height, we need to first find the vertex of the parabolic function h(t) = -16t^2 + 122t + 10.
The vertex of a parabolic function in the form h(t) = at^2 + bt + c is located at the point t = -b/2a. In this case, a = -16 and b = 122.
t = -122/(2(-16))
t = -122/-32
t = 3.8125
Therefore, it takes approximately 3.81 seconds for the boulder to reach its maximum height.
To find the maximum height, we substitute t = 3.81 into the function h(t) = -16t^2 + 122t + 10.
h(3.81) = -16(3.81)^2 + 122(3.81) + 10
h(3.81) = -16(14.5161) + 465.82 + 10
h(3.81) = -232.258 + 465.82 + 10
h(3.81) = 243.562
Therefore, the boulder reaches a maximum height of approximately 243.56 feet.
A catapult launches a boulder with an upward velocity of 122 ft/s. The height of the boulder (h) in feet after t seconds is given by the function h = –16t² + 122t + 10. How long does it take the boulder to reach its maximum height? What is the boulder’s maximum height? Round to the nearest hundredth, if necessary.
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