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)
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.