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 = –16t2 + 120t + 10. How long does it take to reach maximum height? What is the boulder’s maximum height? Round to the nearest hundredth, if necessary.
To find the boulder's maximum height, we need to find the vertex of the parabolic function h = -16t^2 + 120t + 10. The formula for the x-coordinate of the vertex of a parabola in the form y = ax^2 + bx + c is given by x = -b/2a. In this case, a = -16 and b = 120, so we have:
x = -b/2a = -120/(2*(-16)) = 3.75
Therefore, the boulder reaches its maximum height after 3.75 seconds.
To find the maximum height, we substitute t = 3.75 into the equation for h:
h = -16(3.75)^2 + 120(3.75) + 10 ≈ 168.75
Therefore, the boulder's maximum height is approximately 168.75 feet.