Question

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 of t equals negative 16t squared plus 122t plus 10 . 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.
(1 point)

Answer 1

To find the time it takes for the boulder to reach its maximum height, we need to find the vertex of the quadratic function h(t) = -16t^2 + 122t + 10.

The t-coordinate of the vertex of a quadratic function in the form h(t) = at^2 + bt + c is given by t = -b/2a.

In this case, a = -16 and b = 122. Plugging these values into the formula gives us:

t = -122 / (2 * -16)
t = -122 / -32
t = 3.8125

Therefore, it takes approximately 3.81 seconds for the boulder to reach its maximum height.

To find the maximum height, we need to substitute this value of t back into the function h(t) and calculate the corresponding height:

h(3.8125) = -16(3.8125)^2 + 122(3.8125) + 10
h(3.8125) = -16(14.515625) + 465.625 + 10
h(3.8125) = -232.25 + 465.625 + 10
h(3.8125) = 243.375

Therefore, the boulder's maximum height is approximately 243.38 feet.