Im giving 15 points
on a movie set, an archway is modeled by the equation y = -0.5x^2 + 3x, where y is the height in feet and x is the horizontal distance in feet. a laser is directed at the archway at an angle modeled by the equation -0.5x + 2.42y = 7.65 such that the beam crosses the archway at points a and b. at what height from the ground are the points a and b?

Check the picture below.

so, the distance from the ground to points A and B are those two green lines, namely the y-coordinate of their intersection, or solution.



anyhow, if we run the quadratic formula on those coefficients, we'll end up with x ≈ 4.01023060106514842416, and x ≈ 1.57654625843898380724.

now, what's the y-coordinate for each?  well, we can just plug that on say the first equation.

The points of intersection are the points where the (x, y) values satisfy both equations. Since we're interested in the y-values, we can use the second equation to substitute for x in the first equation.

From the second equation,
  x - 4.84y = -15.30 . . . . . multiply the second equation by -2
  x = 4.84y -15.30 . . . . . . add 4.84y to get an expression for x

Now, we can substitute this expression for x into the first equation.
  y = -0.5(4.84y -15.30)² +3(4.84y -15.30)
  y = -0.5(23.4256y² -148.104y +234.09) + 14.52y -45.90
Subtracting the right side from the whole equation we get
  11.7128y² -87.572y +162.945 = 0

This equation may be best solved using the quadratic formula. For
  ax²+bx+c = 0
it tells you the solutions are
  x = (-b ±√(b²-4ad))/(2a)

For our equation, with a=11.7128, b=-87.572, c=162.945, the solutions are
  y = (87.572 ±√((-87.572)²-4(11.7128)(162.945)))/(2·11.7128)
  y = (87.572 ±√34.6864)/23.4256
  y ≈ 3.487 ft, 3.990 ft

Points A and B are 3.487 ft and 3.990 ft above the ground.

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The numbers might be easier to work with if you solve the second equation for y, then equate the two y expressions. This gives you a quadratic in x. Once you find the solutions for x, you then need to put them back into the equation for y to find the y-values of points A and B. The method used here gives you the answer directly, even if the numbers are a little longer.

Of course, a graphing calculator can help you avoid all the algebra. It will give you the points of intersection without much trouble.