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 . 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. (1 point) Responses Reaches a maximum height of 15.42 feet after 7.71 seconds. Reaches a maximum height of 15.42 feet after 7.71 seconds. Reaches a maximum height of 7.71 feet after 3.81 seconds. Reaches a maximum height of 7.71 feet after 3.81 seconds. Reaches a maximum height of 242.56 feet after 7.62 seconds. Reaches a maximum height of 242.56 feet after 7.62 seconds. Reaches a maximum height of 242.56 feet after 3.81 seconds.
The height of the boulder, h, can be determined by the equation h(t) = -16t^2 + vt + c, where t represents the time in seconds, v represents the initial upward velocity, and c represents the initial height.
In this case, the initial upward velocity is 122 ft/s, so v = 122 ft/s.
To find the time it takes for the boulder to reach its maximum height, we need to determine when the velocity becomes 0.
Using the equation v(t) = -32t + v, where v(t) represents the velocity at time t, we can set v(t) = 0 and solve for t:
-32t + 122 = 0
32t = 122
t = 3.81 seconds
Therefore, it takes the boulder 3.81 seconds to reach its maximum height.
To find the maximum height, we substitute this time into the height equation:
h(3.81) = -16(3.81)^2 + 122(3.81) + c
Since we do not have the initial height, c, we cannot determine the maximum height accurately. Therefore, none of the given responses are correct.