A catapult launches a boater with an upward velocity of a 122 feet per second, the height of the Boulder H.In feet after 2 seconds is given by the function H times T equals negative 16T squared + a 122.T plus 10, how long does it take the Boulder to reach its maximum height?What is the boulder's maximum height
To find the time it takes for the boulder to reach its maximum height, we first need to determine the time at which the velocity becomes 0. This occurs at the peak of the motion, which is also the maximum height.
The equation for velocity is given as V(t) = -16t + 122, where t is the time in seconds.
Setting V(t) = 0 and solving for t:
0 = -16t + 122
16t = 122
t = 122/16
t = 7.625 seconds
Therefore, it takes the boulder approximately 7.625 seconds to reach its maximum height.
To find the maximum height, we substitute this time back into the height equation H(t) = -16t^2 + 122t + 10:
H(7.625) = -16(7.625)^2 + 122(7.625) + 10
H(7.625) = -16(58.515625) + 932.75 + 10
H(7.625) = -936.25 + 932.75 + 10
H(7.625) = 6.5 feet
Therefore, the boulder reaches a maximum height of 6.5 feet.