A catapult launches a boulder with an upward velocity of 92 m/s. The height of the
boulder, h, in meters after t seconds is given by the function h = -5?+ 92 + 16. 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 time it takes to reach the maximum height, we need to first find the vertex of the parabolic function h(t) = -5t^2 + 92t + 16.
The formula for the t-coordinate of the vertex of a parabola in the form y = ax^2 + bx + c is given by t = -b/2a.
In this case, a = -5 and b = 92.
Therefore, t = -92 / (2(-5)) = -92 / (-10) = 9.2 seconds.
To find the maximum height, we substitute t = 9.2 seconds into the equation h(t):
h(9.2) = -5(9.2)^2 + 92(9.2) + 16
h(9.2) = -5(84.64) + 844.4 + 16
h(9.2) = -423.2 + 844.4 + 16
h(9.2) = 437.2
Therefore, it takes 9.2 seconds for the boulder to reach its maximum height, which is 437.2 meters.