A catapult launches a boulder with an upward velocity of 122 feet per second. The height of the boulder, (h), in feet
after t seconds is given by the function h(t) = -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 determine the vertex of the quadratic function h(t) = -16t^2 + 122t + 10.
The vertex of a quadratic function in the form h(t) = at^2 + bt + c is given by the formula t = -b/(2a).
In this case, a = -16 and b = 122. Plugging these values into the formula, we get:
t = -122/(2*(-16))
t = -122/(-32)
t ≈ 3.8125 seconds
Therefore, it takes approximately 3.81 seconds for the boulder to reach its maximum height.
To find the maximum height of the boulder, we need to substitute this value of t back into the function h(t) = -16t^2 + 122t + 10:
h(3.8125) = -16(3.8125)^2 + 122(3.8125) + 10
h(3.8125) = -16(14.53) + 465.25 + 10
h(3.8125) = -232.48 + 465.25 + 10
h(3.8125) ≈ 242.77 feet
Therefore, the boulder reaches a maximum height of approximately 242.77 feet.