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(f) = -16₽ + 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)
To find the maximum height of the boulder, we need to find the vertex of the quadratic function represented by the height equation.
The equation for the height of the boulder is given by:
h(t) = -16t^2 + 184t + 20
The vertex of a quadratic function in the form ax^2 + bx + c is given by x = -b/2a. In this case, a = -16 and b = 184, so the formula for the x-coordinate of the vertex is:
t = -(184) / (2(-16))
t = -184 / -32
t = 5.75
To find the maximum height, substitute t = 5.75 into the height equation:
h(5.75) = -16(5.75)^2 + 184(5.75) + 20
h(5.75) = -16(33.0625) + 184(5.75) + 20
h(5.75) = -529 + 1058 + 20
h(5.75) = 549
The boulder's maximum height is 549 feet.
To find how long it takes for the boulder to reach its maximum height, we can simply use the value of t we found for the x-coordinate of the vertex, which is 5.75 seconds.
Therefore, it takes the boulder 5.75 seconds to reach its maximum height.