The height of a rocket fired vertically into the air from the ground is given by the formula h(t) = -16t (2nd power) + 384t + 4, where t is measured in seconds. How long will it take to reach its maximum height and what is the maximum height reached by the rocket
Complete the square with that equation. The maximum height occurs when the perfect squared term is zero.
This can be dome more easily with calculus, but you are apparently not studying that yet.
h(t) = -16t^2 + 384t + 4
= -16(t^2 -24t) -4 = 0
= -16(t^2 -24t + 144) +144 -4 = 0
= -16(t - 12)^2 + 140
h(max) = 140 when t = 12 s
To find the time it takes for the rocket to reach its maximum height, we need to determine the time value at the vertex of the quadratic equation. The vertex of a quadratic function in the form of f(t) = at^2 + bt + c is given by the formula t = -b / (2a).
In this case, the equation for the height of the rocket is h(t) = -16t^2 + 384t + 4, where a = -16, b = 384, and c = 4.
The time taken to reach the maximum height can be found using the formula:
t = -b / (2a)
t = -384 / (2 * -16)
t = -384 / -32
t = 12
The rocket will reach its maximum height after 12 seconds.
To find the maximum height reached by the rocket, we substitute this time value back into the equation h(t):
h(12) = -16(12)^2 + 384(12) + 4
h(12) = -16(144) + 4608 + 4
h(12) = -2304 + 4608 + 4
h(12) = 2308
Therefore, the rocket will reach a maximum height of 2308 feet.