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) = -16t^2 +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 find 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
So, it takes approximately 3.81 seconds for the boulder to reach its maximum height.
To find the maximum height of the boulder, we plug this value back into the original function:
h(3.8125) = -16(3.8125)^2 + 122(3.8125) + 10
h(3.8125) = -16(14.54297) + 465.625 + 10
h(3.8125) = -232.6875 + 475.625 + 10
h(3.8125) = 253.9375
Therefore, the boulder reaches a maximum height of approximately 253.94 feet.
The answers are,
A. Reaches a maximum height of 15.42 feet after 7.71 seconds.
B. Reaches maximum height of 7.71 feet after 3.81 seconds.
C. Reaches a maximum height of 242.56 feet after 7.62 seconds.
D. Reaches a maximum height of 242.56 feet after 3.81 seconds.
The correct answer is:
C. Reaches a maximum height of 242.56 feet after 7.62 seconds.