This question really just needs you to solve the system of equations. These solutions will be the two points when the curve (archway) and the line (laser) intersect.
Let us express the laser line in terms of y.
-.5x+2.42y=7.65
2.42y=.5x+7.65
y=(.5x+7.65)/2.42
The curve is just y=-.5x^2+3x so when it intersects the curve, y=y and x=x so we can say:
(.5x+7.65)/2.42=-.5x^2+3x
.5x+7.65=-1.21x^2+7.26x
1.21x^2-6.76x+7.65=0 using the quadratic for simplicity and solving for x:
x=(6.76±√8.6716)/2.42
x≈1.577, 4.010 (to nearest thousandth)
So the heights can found using the simple linear equation y=(.5x+7.65)/2.42 with those values of x...
y≈3.49, 3.99
