A toy rocket is shot vertically from the base of a multiple level parking garage. the rockets height in feet is modeled by the equation h(T)= -16T squared+ 128 T, where T is the time in seconds if the rocket just reaches the top of the parking garage, how high is the parking garage, and how long does it take the rocket to reach this height?

Maximum height is reached when the velocity dh/dT = -32T + 128 = 0

That happens at T = 4.0 seconds.

Use the h(T) equation to calculate the height at that time.

h(max) = -16*(4^2) + 128*4 = 256 feet

To find the height of the parking garage, we need to determine the maximum height achieved by the rocket. This can be found by identifying the vertex (maximum point) of the parabolic equation h(T) = -16T^2 + 128T.

The equation is in the form of a quadratic equation: h(T) = aT^2 + bT + c, where a = -16, b = 128, and c = 0 (since there is no constant term).

The formula for finding the vertex of a quadratic equation is T = -b/2a. Substituting the values we have: T = -128/(2*(-16)).

Simplifying this equation, we get T = -128/(-32) = 4 seconds.

Therefore, the rocket takes 4 seconds to reach the maximum height.

To find the maximum height, we substitute the value of T (4 seconds) into the equation h(T) = -16T^2 + 128T:

h(4) = -16(4)^2 + 128(4) = -16(16) + 512 = -256 + 512 = 256 feet.

Hence, the parking garage is 256 feet high, and it takes the rocket 4 seconds to reach this height.