A catapult launches a boulder with an upward velocity of 148 ft/s. The height of the boulder, h, in feet after t seconds is given by the function h=-16t^2+148t+30. What is the boulder’s maximum height? How long does it take the boulder to reach its maximum height? Round to the nearest hundredth, if necessary.
To find the boulder's maximum height, we need to determine the vertex of the quadratic function h(t) = -16t^2 + 148t + 30.
The vertex form of a quadratic function is given by h(t) = a(t - h)^2 + k, where (h, k) represents the vertex.
Comparing the given function with the vertex form, we have:
a = -16
h = -b/2a, where b = 148 and a = -16
Using the formula, we can find h:
h = -148/(2*(-16))
h = -148/-32
h = 4.625
Therefore, the boulder's maximum height is 4.625 feet.
To find how long it takes for the boulder to reach its maximum height, we can use the formula h = -16t^2 + 148t + 30.
Since the boulder reaches its maximum height at the vertex, we need to find the value of t that corresponds to the vertex.
Using the formula t = -b/2a, where b = 148 and a = -16, we can calculate t:
t = -148/(2*(-16))
t = -148/-32
t = 4.625
Therefore, it takes the boulder approximately 4.625 seconds to reach its maximum height.
To find the boulder's maximum height, we need to find the vertex of the parabolic function h.
First, we can rewrite the function in vertex form:
h = -16(t - 4.625)^2 + 833.625
where 4.625 is the time it takes for the boulder to reach its maximum height and 833.625 is the maximum height in feet.
To find the time it takes for the boulder to reach its maximum height, we can use the formula:
t = -b/2a
where a = -16 and b = 148.
t = -148/(2(-16)) = 4.625 seconds
To find the maximum height, we substitute this value of t into the original function:
h = -16(4.625)^2 + 148(4.625) + 30 = 833.625 feet
Therefore, the boulder reaches a maximum height of 833.625 feet and it takes 4.63 seconds to reach this height.