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 of t equals negative 16t squared plus 122t plus 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.
A. Reaches a maximum height of 15.42 feet after 7.71 seconds.
B. Reaches a 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.
To find the time it takes for the boulder to reach its maximum height, we first need to determine the time at which the boulder reaches its peak by using the formula for the vertex of a quadratic function: t = -b/2a. In this case, a = -16 and b = 122.
So, t = -(122)/(2*-16) = -122/-32 = 3.81 seconds
Therefore, the correct answer is:
B. Reaches a maximum height of 7.71 feet after 3.81 seconds.
Next, to find the maximum height of the boulder, 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) + 464.82 + 10
h(3.81) = -232.258 + 464.82 + 10
h(3.81) = 242.562
Rounded to the nearest hundredth, the maximum height is 242.56 feet.
Therefore, the complete answer is:
B. Reaches a maximum height of 242.56 feet after 3.81 seconds.