Question
This same rock is thrown and it follows the path modeled by the function:
h(t) = -16t^2 + 80t + 30
When does the rock reach it's maximum height?
What is the maximum height it reaches?
h(t) = -16t^2 + 80t + 30
When does the rock reach it's maximum height?
What is the maximum height it reaches?
Answers
GPT 3.5
To find the rock's maximum height, we need to find the vertex of the parabolic function h(t) = -16t^2 + 80t + 30.
The t-coordinate of the vertex of a parabola given by the function h(t) = at^2 + bt + c is given by t = -b/(2a).
In this case, a = -16 and b = 80. So, t = -80 / (2 * -16) = -80 / -32 = 2.5.
To find the maximum height, plug t = 2.5 into the function h(t):
h(2.5) = -16(2.5)^2 + 80(2.5) + 30
h(2.5) = -16(6.25) + 200 + 30
h(2.5) = -100 + 200 + 30
h(2.5) = 130
Therefore, the rock reaches its maximum height of 130 feet at t = 2.5 seconds.
The t-coordinate of the vertex of a parabola given by the function h(t) = at^2 + bt + c is given by t = -b/(2a).
In this case, a = -16 and b = 80. So, t = -80 / (2 * -16) = -80 / -32 = 2.5.
To find the maximum height, plug t = 2.5 into the function h(t):
h(2.5) = -16(2.5)^2 + 80(2.5) + 30
h(2.5) = -16(6.25) + 200 + 30
h(2.5) = -100 + 200 + 30
h(2.5) = 130
Therefore, the rock reaches its maximum height of 130 feet at t = 2.5 seconds.