Question
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.
Answers
GPT 3.5
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.
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.