A model rocket is launched from the top of a building. The height (in meters) of the rocket above the ground is given by h( t) = -16t²+ 96t + 14, where t is the time (in seconds) since the rocket was launched. What is the rocket’s maximum height and when does this occur.
the max is on the axis of symmetry of the parabola
tmax = -b / 2a = -96 / (2 * -16) = 3 s
plug the 3 into h(t) to find the max height
To find the rocket's maximum height, we need to determine the vertex of the parabolic equation h(t) = -16t² + 96t + 14. The vertex of a parabola in the form y = ax² + bx + c can be found using the formula t = -b/2a.
Here, a = -16 and b = 96. Plugging these values into the formula, we have:
t = -96 / (2 * -16)
t = -96 / -32
t = 3
So, the maximum height occurs at t = 3 seconds.
To find the corresponding height, we substitute t = 3 into the equation h(t):
h(3) = -16(3)² + 96(3) + 14
h(3) = -16(9) + 288 + 14
h(3) = -144 + 288 + 14
h(3) = 158
Therefore, the rocket's maximum height is 158 meters and it occurs at t = 3 seconds.
To find the rocket's maximum height, we need to determine the vertex of the parabolic function representing the height of the rocket.
The equation for the height of the rocket above the ground is given as h(t) = -16t² + 96t + 14.
The vertex of a parabolic function represented by the equation y = ax² + bx + c is given by the x-coordinate formula: x = -b / (2a).
In this case, a = -16 and b = 96. Plugging these values into the formula, we get:
x = -96 / (2 * -16)
x = -96 / -32
x = 3
So the x-coordinate (time) for the vertex is t = 3 seconds.
To find the maximum height (y-coordinate) of the rocket, we substitute the value of t = 3 into the equation h(t):
h(3) = -16(3)² + 96(3) + 14
h(3) = -16(9) + 288 + 14
h(3) = -144 + 288 + 14
h(3) = 158
Therefore, the rocket's maximum height is 158 meters.
To determine when the maximum height occurs, we have already established that it happens at t = 3 seconds. So the rocket reaches its maximum height 3 seconds after it was launched.