A catapult launches a boulder with an upward velocity of 112 ft/s. The height of the boulder, h, in feet after t seconds is given by the function mc019-1.jpg. 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.
To find the time it takes for the boulder to reach its maximum height, we need to find the vertex of the function. The formula for the height of the boulder in terms of time is given by:
h(t) = -16t^2 + 112t
To find the vertex, we can use the formula t = -b / (2a), where a = -16 and b = 112.
t = -112 / (2*(-16))
t = -112 / (-32)
t = 3.5
So, it takes the boulder 3.5 seconds to reach its maximum height.
To find the maximum height, we substitute the value of t into the function:
h(3.5) = -16(3.5)^2 + 112(3.5)
h(3.5) = -16(12.25) + 392
h(3.5) = -196 + 392
h(3.5) = 196
Therefore, the boulder's maximum height is 196 feet.