A toy rocket that is shot in the air with an upward velocity of 48 feet per second can be modeled by the function f(t)=-16t^2+48t , where t is the time in seconds since the rocket was shot and f(t) is the rocket’s height. What is the maximum height the rocket reaches?
16ft
36ft
48ft
or 144 ft
Well, that is a parabola. Where is the vertex?
-16 t^2 + 48 t = h
16 t^2 - 48 t = -h
t^2 - 3 t = -(1/16)h
t^2 - 3 t + 9/4 = -(1/16)h + 9/4
(t - 3/2)^2 = -(1/16)(h - 36)
ah ha, hits the vertex at the top when t = 1.5 seconds and h = 36 feet
thanks!!
You are welcome.
To find the maximum height the rocket reaches, we need to find the vertex of the quadratic function f(t)=-16t^2+48t. The vertex of a quadratic function is the highest or lowest point of the graph.
The formula for the x-coordinate of the vertex of a quadratic function in the form f(t) = at^2 + bt + c is given by x = -b/2a.
In this case, a = -16 and b = 48. Plugging these values into the formula, we get:
x = -48 / 2(-16) = -48 / -32 = 1.5
This means that the time at which the rocket reaches its maximum height is 1.5 seconds.
To find the maximum height, we substitute this time back into the original function:
f(1.5) = -16(1.5)^2 + 48(1.5) = -16(2.25) + 72 = -36 + 72 = 36
Therefore, the maximum height the rocket reaches is 36 feet.
So, the correct answer is 36ft.