A catapult launches a boulder with an upward velocity of 184 feet per second. the height of the boulder, (h), in feet after t seconds is given by the function h (t) = - 16t^2+184t+20. 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.

Given:

Upward velocity of 184 feet per second

Height of boulder:

h (t) = - 16t^2+184t+20

Since we are given the velocity, we need to differentiate the equation to reach an equation for velocity:

v (t) = - 32t + 184

now, substitute the value of velocity:

-184 feet per second = - 32t + 184

-368 = - 32t

t = 11.5 seconds to reach the maximum height

It takes 11.5 seconds for the boulder to reach the maximum height. The max height is

h (t) = - 16t^2+184t+20

where t = 11.5 s

h (t) = - 16 (11.5) ^2 + 184 (11.5) + 20

h (t) = 20 feet