Question

The trajectory of a rocket is represented by the function h(t) = -3t? + 24t + 36, where

his height in meters and t is time in seconds.
(a) What is the initial height of the rocket before it takes off?
(b) What is the height of the rocket after 2 seconds?
(c) At what time does the rocket reach its maximum height?
(d) What is the maximum height reached by the rocket?

Answer 1

The initial height of the rocket is 36 meters, after 2 seconds the rocket is at 48 meters, it reaches its maximum height of 84 meters after 4 seconds.

Step-by-step explanation:

The trajectory of a rocket is represented by the function h(t) = -3t² + 24t + 36, where h is the height in meters and t is the time in seconds.

1. The initial height of the rocket before it takes off can be found by evaluating h(0), which gives us h(0) = -3(0)² + 24(0) + 36 = 36 meters.
2. The height of the rocket after 2 seconds is found by evaluating h(2), resulting in h(2) = -3(2)² + 24(2) + 36 = 48 meters.
3. The rocket reaches its maximum height at the vertex of the parabola. Since the equation is in standard form, we can find the time to reach the maximum height by using the formula t = -b/(2a), where a and b are coefficients from the quadratic equation h(t) = at² + bt + c. Thus, t = -24/(2(-3)) = 4 seconds.
4. The maximum height reached by the rocket can be calculated by plugging the time back into the function: h(4) = -3(4)² + 24(4) + 36 = 84 meters.