Courtney is standing on a mountain above a lake, and she tosses a rock with a velocity of 48ft/sec. Here is a table of values that represents the height of the rock, in feet.
0 140
1 160
2 140
3 90
4 10.
What function is this?
What is the maximum height of the rock?
The rock changes directions after how mamy seonds in the air>
The function representing the height of the rock can be modeled using a quadratic equation in the form of h(t) = at^2 + bt + c, where h is the height in feet and t is the time in seconds.
Using the data points provided:
When t = 0, h(0) = a(0)^2 + b(0) + c = c = 140
When t = 1, h(1) = a(1)^2 + b(1) + c = a + b + 140 = 160
When t = 2, h(2) = a(2)^2 + b(2) + c = 4a + 2b + 140 = 140
When t = 3, h(3) = a(3)^2 + b(3) + c = 9a + 3b + 140 = 90
When t = 4, h(4) = a(4)^2 + b(4) + c = 16a + 4b + 140 = 10
Solving these equations simultaneously, we get a = -25, b = 70, c = 140.
Therefore, the equation representing the height of the rock is h(t) = -25t^2 + 70t + 140.
To find the maximum height of the rock, we first need to find the vertex of the parabola. The x-coordinate of the vertex for a quadratic equation in the form of ax^2 + bx + c is given by -b / 2a.
For this equation, the vertex occurs at t = -70 / 2(-25) = 1.4 seconds. Plugging this value back into the equation h(t), we find the maximum height to be h(1.4) = 150 feet.
The rock changes directions when its height reaches its maximum point, which occurs at t = 1.4 seconds. Therefore, the rock changes directions after being in the air for 1.4 seconds.