.A toy rocket is shot vertically into the air from a launching pad 5 feel above the ground with an initial velocity of 72 feet per second. The height h, in

feet, of the rocket above the ground as t seconds after launch is given by the function h(t)=-16t^2+72t+5. How long will it take the rocket to reach its maximum height? What is the maximum height?

To find the time it takes for the rocket to reach its maximum height, we need to determine the vertex of the quadratic function h(t) = -16t^2 + 72t + 5. The vertex represents the maximum point of the graph.

The vertex of a quadratic function with the form f(t) = ax^2 + bx + c is given by the formula t = -b/2a. In this case, a = -16 and b = 72, so the time it takes for the rocket to reach its maximum height is:

t = -b/2a
t = -72 / (2*(-16))
t = -72 / (-32)
t = 2.25 seconds

The maximum height of the rocket can be found by substituting the value of t into the function h(t):

h(2.25) = -16(2.25)^2 + 72(2.25) + 5
h(2.25) = -81 + 162 + 5
h(2.25) = 86 feet

Therefore, it will take the rocket 2.25 seconds to reach its maximum height, and the maximum height is 86 feet.

To find the time it takes for the rocket to reach its maximum height, we need to determine the vertex of the quadratic function h(t) = -16t^2 + 72t + 5. The vertex form of a quadratic function is h(t) = a(t - h)^2 + k, where (h, k) is the vertex of the parabola.

In the given function, a = -16, b = 72, and c = 5. The formula to find the x-coordinate of the vertex is given by t = -b / (2a).

Substituting the values, we find:
t = -72 / (2 * -16)
t = -72 / -32
t = 2.25

The rocket takes 2.25 seconds to reach its maximum height.

To find the maximum height, substitute the value of t back into the equation h(t):
h(2.25) = -16(2.25)^2 + 72(2.25) + 5
h(2.25) = -16(5.0625) + 162 + 5
h(2.25) = -81 + 162 + 5
h(2.25) = 86

The maximum height the rocket reaches is 86 feet.

the max height is at the vertex of the parabola. You know that for at^2+bt+c, the vertex is at t = -b/2a.

Then use that to find h of the vertex.