A catapult launches a boulder with an upward velocity of 122 feet per second. The height of the boulder, (h), in feet after t seconds is given by the function h(t)=-16t^2+122t+10. How long does it take the boulder to reach its maximum height? What is the boulder’s maximum height?
To find the time it takes for the boulder to reach its maximum height, we need to find the vertex of the parabolic function h(t)=-16t^2+122t+10. The time to reach the maximum height is given by the formula t = -b/2a, where ax^2 + bx + c is the quadratic equation.
In this case, a = -16 and b = 122. Plugging them into the formula, we get:
t = -122 / (2 * (-16))
t = -122 / -32
t = 3.8125
So, it takes 3.8125 seconds for the boulder to reach its maximum height.
To find the maximum height, we need to plug this time back into the original equation.
h(3.8125) = -16(3.8125)^2 + 122(3.8125) + 10
h(3.8125) = -231.5 + 465.25 + 10
h(3.8125) = 243.75
Therefore, the boulder's maximum height is 243.75 feet.