how do i find the answer

label Algebra
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A coordinate grid is mapped on a video game screen, with the origin in the lower-left corner. A game designer programs a helicopter to follow a path that can be modeled by a quadratic function with a vertex at (16, 20) and passing through the point (4, 25). She also programs an airplane to move along a linear path that passes through the points (0, 18) and (30, 20). Which system of equations can be used to determine whether the paths of the helicopter and airplane cross?

May 15th, 2015

1. " linear path that passes through the points (0, 18) and (30, 20)".
The equation of this line is (x-0)/(30-0) = (y-18)/(20-18), or
x/30 = (y-18)/2.

2. "path that can be modeled by a quadratic function with a vertex at (16, 20) and passing through the point (4, 25)"
I suppose that they speak about vertical parabola. The equation is y = a*x^2 + b*x + c.
The vertex has coordinates (-b/2a, c-b^2/(4a)) = (16, 20).
Parabola passing through (4, 25) means 25 = a*4^2 + b*4 + c.
There are 3 equations on a,b,c and we have to solve this system.
(I'll show the solution later because 20 min are insufficient)

3. "video game screen, with the origin in the lower-left corner" means x>=0, y>=0 and x<width of the screen, y<height of the screen.
Also, x and y may be non-integer but for screen we need the nearest integers of the exact solution.

So the system is:
x/30 = (y-18)/2,
y = a*x^2 + b*x + c  (a, b, c are constants we have to find before)
and x>=0, y>=0, x<W, y<H.

May 15th, 2015

About a, b, c.

b = -32*a, c = 20 + b^2/(4a) = 20 + 256*a,
25 = 16a - 128*a + 20 + 256*a, so 144*a = 5, a = 5/144.

Then b = -32*a = -10/9, c = 20 + 256*5/144 = 20 + 16*5/9 = (5/9)*(36 + 16) = 260/9.

Not good-looking but...

May 15th, 2015

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