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.
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.