Question

On a movie set, an archway is modeled by the equation y = -0.5x2 + 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?

Answer 1

In the question above, to find the height we have to calculate the value of y.
We are going to solve the 2 equations simultaneously. 
y = -0.5x² + 3x  .............................................(1)
-0.5x + 2.42y = 7.65  .....................................(2)

Solve using substitution method.
From the equation 1,
y = -0.5x² + 3x 

Substitute the value of y in the equation 2 with y = -0.5x² + 3x  

So we have, 
-0.5x + 2.42(-0.5x² + 3x ) = 7.65 
-0.5x - 1.21x² + 7.26x = 7.65
- 1.21x² + 6.76x = 7.65
1.21x² - 6.76x - 7.65 = 0

This is a quadratic equation. 
Solving the for x using quadratic formula.

x = (- - 6.76 +/- √( 6.76² - 4×1.21×7.65)) ÷ 2
   = (6.76 +/- √82.7236) ÷2
   = 6.76 +/-  9.095
 x = 15.86 or -2.335

But distance can not be negative, the value of x = 15.86 ft.
The height which is the value of y will be;
 
y = -0.5x² + 3x 
   = -0.5(15.86)² + 3(15.86)
   = 125.7698 + 47.58
   = 173.3498 ft

Height = 173.3498 ft