A toy rocket is launched straight up from the roof of a garage with an initial velocity of 56 feet per second. The height h of the rocket in feet, at t seconds after it was launched, is described by h(t)=−16t2+56t+17. Find the maximum height of the rocket.
recall that the vertex of a parabola is at
t = -b/2a
so evaluate h(t) there.
To find the maximum height of the rocket, we need to find the vertex of the parabolic equation h(t) = -16t^2 + 56t + 17.
The vertex of a parabola in the form of h(t) = at^2 + bt + c can be found using the equation t = -b/2a and substituting it back into the equation to find h(t).
In this case, a = -16, b = 56, and c = 17.
Substituting the values into the equation t = -b/2a:
t = -(56) / (2(-16))
t = -56 / -32
t = 1.75
Now, substitute t = 1.75 back into the equation h(t) = -16t^2 + 56t + 17 to find the maximum height:
h(1.75) = -16(1.75)^2 + 56(1.75) + 17
h(1.75) = -16(3.0625) + 98 + 17
h(1.75) = -49 + 98 + 17
h(1.75) = 66
Therefore, the maximum height of the rocket is 66 feet.
To find the maximum height of the rocket, we need to determine the vertex of the quadratic function h(t) = -16t^2 + 56t + 17. The vertex of a quadratic function is the highest or lowest point on the graph of the function, depending on the direction of the quadratic.
The vertex of a quadratic function in the form h(t) = at^2 + bt + c can be found using the formula t = -b / (2a).
In the equation h(t) = -16t^2 + 56t + 17, we have a = -16 and b = 56.
t = -56 / (2 (-16))
t = -56 / (-32)
t = 1.75
Now, we can substitute t = 1.75 back into the original equation to find the maximum height:
h(1.75) = -16 (1.75)^2 + 56 (1.75) + 17
h(1.75) = -16 (3.0625) + 98 + 17
h(1.75) = -49 + 98 + 17
h(1.75) = 66
Therefore, the maximum height of the rocket is 66 feet.