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) = –16t² + 184t +20. What is the boulder’s maximum height? How long does it take the boulder to reach its maximum height? Round to the nearest hundredth, if necessary.
(1 point)
A) Reaches a maximum height of 11.6 feet after 5.75 seconds.
B) Reaches a maximum height of 549 feet after 11.5 seconds.
C) Reaches a maximum height of 549 feet after 5.75 seconds.
D) Reaches a maximum height of 23.2 feet after 11.6 seconds.
To find the maximum height of the boulder, we need to find the vertex of the quadratic function.
The vertex of a quadratic function in the form h(t) = at^2 + bt + c is given by the coordinates (t, h) where t = -b/2a and h = h(t).
In this case, the function is h(t) = -16t^2 + 184t + 20.
a = -16 and b = 184, so t = -184 / (2 * -16) = -184 / -32 = 5.75.
To find the maximum height, we substitute t = 5.75 into the function.
h(5.75) = -16(5.75)^2 + 184(5.75) + 20 = -16(33.0625) + 1058 + 20 = -529 + 1058 + 20 = 549.
So, the boulder's maximum height is 549 feet.
To find how long it takes for the boulder to reach its maximum height, we use the value of t = 5.75.
Therefore, the correct answer is:
C) Reaches a maximum height of 549 feet after 5.75 seconds.