A rocket is launched and follows a parabolic arch reaching a max height of 14 km when it is 8 km away from the launch pad, which is 250 m above the ground. Determine the height of the rocket as a function of its horizontal distance from the launch pad
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Let
x = horizontal distance in meters
y = vertical distance (or height) in meters
There are three points on this graph
A = (0,250)
B = (8000,14000)
C = (16000,250)
Point A is the launch point at 250 m above the ground, which is why we have y = 250. I made x = 0 as the starting x value, though technically you can start with any x value you want. The next point, point B, is the vertex point. This point is 8 km or 8000 meters away from the launch point which explains why x = 8000. The y coordinate is 14000 meters (14 km) so y = 14000. The third point C is another 8000 meters from point B, but in the opposite direction as point A, and it will have the same height as point A because the parabola has symmetry along the vertical line that runs through the vertex.
The equation y = ax^2 + bx + c is the general form of any parabola. We need to find the values of a,b,c which will help us find the function we want.
Plug in (x,y) = (0,250) to find the value of c
y = ax^2 + bx + c
ax^2 + bx + c = y
a(0)^2 + b(0) + c = 250
0a + 0b + c = 250
0 + 0 + c = 250
c = 250
Now plug in (x,y) = (8000,14000)
y = ax^2 + bx + c
ax^2 + bx + c = y
a(8000)^2 + b(8000) + 250 = 14000
64000000a + 8000b + 250 = 14000
64000000a + 8000b + 250-250 = 14000-250
64000000a + 8000b = 13750
Solve the previous equation for b
64000000a + 8000b = 13750
8000b = 13750-64000000a
8000b = -64000000a+13750
b = (-64000000a+13750)/8000
b = (-64000000a)/8000+(13750)/8000
b = -8000a+1.71875
we'll plug this equation in later
Go back to the general form of a parabola. Plug in (x,y) = (16000,250)
y = ax^2 + bx + c
ax^2 + bx + c = y
a(16000)^2 + b(16000) + 250 = 250
256000000a + 16000b + 250 = 250
256000000a + 16000b + 250-250 = 250-250
256000000a + 16000b = 0
Now replace b with -8000a+1.71875 and solve for 'a'
256000000a + 16000b = 0
256000000a + 16000( b ) = 0
256000000a + 16000(-8000a+1.71875) = 0
256000000a - 128000000a + 27500 = 0
128,000,000a + 27500 = 0
128,000,000a = -27500
a = -27500/(128,000,000)
a = -0.00021484375
Use this value of 'a' to find b
b = -8000a+1.71875
b = -8000(-0.00021484375)+1.71875
b = 3.4375
We have enough info to go from this
y = ax^2 + bx + c
to this
y = -0.00021484375x^2 + 3.4375x + 250
The function is therefore:
f(x) = -0.00021484375x^2 + 3.4375x + 250
x is the horizontal distance and f(x) is the height.
If you were to plug in x = 0, then you should get f(x) = 250.
If you were to plug in x = 8000, then you should get f(x) = 14000.
If you were to plug in x = 16000, then you should get f(x) = 250.
These three facts will confirm that we have the proper parabolic function.