A catapult launches a boulder with an upward velocity of 120 ft/s. The height of the boulder, h, in feet after t seconds is given by the function h = -16t^2 + 120t + 10. How long does it take to reach maximum height? What is the boulder's maximum height? What is the boulder's maximum height? Round to the nearest hundredth, if necessary.

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18-05-2017
Mathematics

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A catapult launches a boulder with an upward velocity of 120 ft/s. The height of the boulder, h, in feet after t seconds is given by the function h = -16t^2 + 120t + 10. How long does it take to reach maximum height? What is the boulder's maximum height? What is the boulder's maximum height? Round to the nearest hundredth, if necessary.

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The time it takes for the boulder to reach maximum height can be determined by taking the derivative of the function and equating it to zero as in: h' = -32t + 120 = 0. This is because at the maximum point, the slope (derivative) is zero. Thus, the time it takes for the boulder to reach maximum height is 4/15 seconds or ~0.27 seconds.

To solve for the maximum height, substitute the time it takes to reach the maximum height in the given function as in: h = -16(0.27)^2 + 120(0.27) + 10. Thus, the boulder's maximum height is 41.23 ft.