Question

A toy rocket is shot vertically into the air from a 9 foot tall launching pad with an initial velocity of 72 feet per second. Suppose the height of the rocket in feet t seconds after being launched can be modeled by the function h(t)=-16t^2+v0t+h0, where v0 is this initial velocity of the rocket and h0 is the initial height of the rocket. How long will it take for the rocket to reach its maximum height? what is the maximum height

Answer 1

The height of the rocket at time t is given by the function:

h(t) = -16t^2 + v0t + h0

where v0 is the initial velocity of the rocket and h0 is the initial height of the rocket.

To find the time it takes for the rocket to reach its maximum height, we need to find the value of t that maximizes the function h(t). We can do this by finding the vertex of the parabolic function h(t).

The vertex of a parabola of the form y = ax^2 + bx + c is given by the coordinates (-b/2a, c - b^2/4a). In our case, the function is h(t) = -16t^2 + 72t + 9, so a = -16, b = 72, and c = 9.

The time it takes for the rocket to reach its maximum height is the x-coordinate of the vertex, which is given by:

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

So it takes the rocket 2.25 seconds to reach its maximum height.

To find the maximum height, we substitute t = 2.25 into the function h(t):

h(2.25) = -16(2.25)^2 + 72(2.25) + 9 = 81

So the maximum height of the rocket is 81 feet