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.
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.
1 answer