A rocket is fired straight up from a 60 ft. platform with an initial velocity of 96 ft/sec. The height of the rocket, h(t), is found using the function h\left( t \right) = - 16{t^2} + 96t + 60 where t is the time in seconds.
Find the maximum height.
h(t) = -16t^2 + 96t + 60
v(t) = -32t + 96
at max height , v = 0
-32t + 96=0
32t = 96
t = 96/32 = 3
h(3) = -16(9) + 96(3) + 60 = 204
or
h(t) = -16(t^2 - 6t + 9-9) + 60
= -16( (t-3)^2 - 9) + 60
= -16(t-3)^2 + 144 + 60
= -16(t-3)^2 + 204
this has a vertex of (3,204)
thus a max of 204 when t - 3
3
To find the maximum height, we need to determine the vertex of the parabolic function h(t) = -16t^2 + 96t + 60. The vertex of a parabola is the highest or lowest point on the curve.
The formula for the x-coordinate of the vertex of a parabola in the form ax^2 + bx + c is given by x = -b / (2a).
In this case, a = -16 and b = 96. Plugging these values into the formula, we get:
x = -96 / (2 * -16) = -96 / (-32) = 3
So, the rocket reaches its maximum height at t = 3 seconds.
To find the maximum height, we substitute t = 3 into the equation h(t) = -16t^2 + 96t + 60:
h(3) = -16(3)^2 + 96(3) + 60 = -16(9) + 288 + 60 = -144 + 288 + 60 = 204
Therefore, the rocket reaches a maximum height of 204 feet.
To find the maximum height of the rocket, we need to determine the vertex of the quadratic function h(t) = -16t^2 + 96t + 60. The vertex of a quadratic function with the form f(t) = at^2 + bt + c is given by the formula:
t = -b / (2a)
In this case, a = -16 and b = 96. Substituting these values into the formula, we can calculate the time at which the rocket reaches its maximum height:
t = -96 / (2 * -16) = 3 seconds
Now, we substitute this value of t back into the original function to find the maximum height:
h(t) = -16t^2 + 96t + 60
h(3) = -16(3)^2 + 96(3) + 60
h(3) = -16(9) + 288 + 60
h(3) = -144 + 288 + 60
h(3) = 204 ft
Therefore, the maximum height of the rocket is 204 ft.