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. 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.
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To determine the maximum height of the boulder and the time it takes to reach that height, we need to find the vertex of the parabolic function h(t) = -16t² + 184t + 20.
The vertex of a quadratic function in the form h(t) = at² + bt + c can be found using the formula t = -b / (2a).
In this case, a = -16 and b = 184. Plug these values into the formula to calculate the time it takes for the boulder to reach its maximum height:
t = -184 / (2 * -16)
t = -184 / -32
t = 5.75
So it takes the boulder 5.75 seconds to reach its maximum height.
To find the maximum height, substitute this time back into the function h(t):
h(t) = -16(5.75)² + 184(5.75) + 20
h(t) = -16(33.0625) + 1058 + 20
h(t) = -528.2 + 1058 + 20
h(t) = 549.8
Therefore, the boulder reaches a maximum height of approximately 549.8 feet.
for any quadratic of the form
y = ax^2 + bx + c
the x of the vertex is -b/(2a)
so for yours ...
x of the vertex is -184/(-32) = 5.75
so it took 5.75 seconds,
plug that into your original function to find the maximum height.