Maximum height = 900 feet
Time to Max Height = 2.5 sec
Time to landing = 10 sec
Step-by-step explanation:
Ok, so we've got this quadratic formula h(t) = -16t^2 + 80t +800, and we have to find it's maximum height, the time to that height, and when it reaches the ground.
Lets start with the easiest, which is the maximum height of the rocket.
Since this is an upside down parabola, the vertex will be the maximum height, and since the quadratic is already in the ax^2 + bx +c format, we can use the expression -b/2a to find the x-coordinate of the vertex.
-b/2a = -80/(2*-16) = 80/32 = 2.5
Now armed with the x value we just have to plug that value in for the maximum height:
h(t) = -16t^2 + 80t +800
h(2.5) = -16(2.5)^2 + 80(2.5) +800
h(2.5) = 900
So the rocket will reach a max height of 900 feet.
Now for the second part of the question --- how long for the rocket to reach that height?
Wait, we already found that out; it's 2.5 seconds. Yay!
Ok, so now for the final part. When will the rocket reach the ground?
For this, we have to break out the quadratic formula, which will tell us when the parabola reaches 0 --- in essence, when the rocket touches the ground.
The quadratic formula is: [-b +/- sqrt(b^2 -4ac)] / 2a
Plugging in our values:
[-b +/- sqrt(b^2 -4ac)] / 2a
= [-80 +/- sqrt(80^2 - 4*-16*800)] / (2*-16)
= 2.5 +/- 7.5
Since we know the rocket landed at a future point in time, we add 7.5, which gives us the nice even number of 10.
So it takes 10 seconds for the rocket to land.
