# Jim, we have lift off!

## Job

An Aerospace Engineer is testing a new fuel propellant and parachute system with a rocket. He tests the rocket in a confined space where there is no air resistance. The rocket is 50 yards away from its predicted landing point. He also thinks the maximum height should be 60 yards. What equations can he use to calculate the exact parabola of the rocket? He'll need these equations to put into his computer for further testing.

## Analysis part 1

We can see that the roots are (0,0) and (50,0), the maximum value is 60, the vertex is (60,50), and the axis of symmetry is x=0. He can us vertex form to figure out the equation.

Vertex Form:

y=a(x-h)^2+k

He plugs in (0,0) and (60,50)/(h,k) to find a.

0=a(0-60)^2+50

-50=a(-60)^2

-50=3600a

-1/72=a

He can now find the equation in vertex form.

y=-1/72(x-60)^2+50

## Analysis part 2

He knows the parachute will open 20 yards away from the target. He wants to find how high it'll be at that point.

y=-1/72(10-60)^2+50

y=-1/72(-50)^2+50

y= 15.27 yards

## Analysis part 3

His boss comes in and tells him to put it in standard form. He does this by using the vertex form.

y=-1/72(x-60)^2+50

y=-1/72(x-60)(x-60)+50

y=-1/72(x^2-120x+3600)+50

y=(-1/72)x^2+(5/3)x-50+50

y=(-1/72)x^2+(5/3)x