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(t)=-16t^2 + 122t + 10. How long does it take the boulder to reach its maximum height? What is the boulders 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 can use the fact that the maximum height occurs at the vertex of the parabola. The x-coordinate of the vertex of a quadratic function given in the form y = ax^2 + bx + c is given by the formula x = -b/2a.
In this case, the function for the height of the boulder is h(t) = -16t^2 + 122t + 10. Comparing to the general form, we can see that a = -16 and b = 122.
To find the time it takes for the boulder to reach its maximum height, we use the formula:
t = -b / (2a)
t = -122 / (2(-16))
t = -122 / (-32)
t = 3.8125
Therefore, the boulder takes approximately 3.81 seconds to reach its maximum height.
To find the maximum height, we substitute this value of t into the function h(t):
h(3.8125) = -16(3.8125)^2 + 122(3.8125) + 10
h(3.8125) = -16(14.5276) + 465.81 + 10
h(3.8125) = -232.4396 + 465.81 + 10
h(3.8125) ≈ 243.37
Therefore, the boulder reaches a maximum height of approximately 243.37 feet.
The correct answer is:
D. Reaches a maximum height of 242.56 feet after 3.81 seconds.