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 n(t) = -161°+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.
(1 point)
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 242.56 feet after 7.62 seconds.
Reaches a maximum
• height of 242.56 feet after 3.81 seconds.
To find the maximum height of the boulder, we need to find the vertex of the parabolic function given by h(t) = -16t^2 + 122t + 10. The t-coordinate of the vertex is given by -b/2a, where a = -16 and b = 122.
t = -122 / 2(-16) = 3.81 seconds
Thus, it takes the boulder 3.81 seconds to reach its maximum height.
To find the maximum height, substitute t = 3.81 back into the function:
h(3.81) = -16(3.81)^2 + 122(3.81) + 10
= -16(14.5161) + 464.62 + 10
= -232.258 + 464.62 + 10
= 242.362 feet
Therefore, the boulder reaches a maximum height of approximately 242.36 feet after 3.81 seconds.
Correct answer: Reaches a maximum height of 242.56 feet after 3.81 seconds.