Question

Toy rocket is launched from the top of a cliff with an initial vertical velocity of 64 ft/s.

The height h (in feet) of the rocket as a function of time t (in seconds) is modeled by the
equation:
h(t)=–16t^2+64t+80

Answer 1

The maximum height of the booster can be found by finding the vertex of the equation for height as a function of time. The maximum velocity occurs at the time when the velocity equals 0, which is also the time of maximum height.

Step-by-step explanation:

To find the maximum height and the maximum velocity of the booster, we need to analyze the given equation for height as a function of time. From the equation h(t) = -16t^2 + 64t + 80, we can determine the maximum height by finding the vertex of the parabolic equation. The maximum height occurs at the vertex, which is located at the t-coordinate of -b/2a. In this case, a = -16 and b = 64, so the t-coordinate is t = -64/-32 = 2 seconds.

Substituting t = 2 into the equation, we can find the maximum height: h(2) = -16(2)^2 + 64(2) + 80 = -64 + 128 + 80 = 144 feet. Therefore, the booster reaches a maximum height of 144 feet.

We can find the maximum velocity by taking the derivative of the equation for h(t), which gives us the equation for velocity v(t) = -32t + 64. The maximum velocity occurs at the time when the velocity equals 0, which is also the time of maximum height. Substituting t = 2 into the equation for velocity, we get v(2) = -32(2) + 64 = 0 feet per second. Therefore, when the booster reaches its maximum height of 144 feet, it momentarily stops and has a velocity of 0 feet per second.