Question

A fountain on a lake sprays water in a parabolic arch modeled by the equation y = -0.3x2 + 3x. A beam of light modeled by the equation
-2x + 5.5y = 19.5 passes through the fountain to create a rainbow effect. If the beam cuts the water spray at points A and B, such that point B is at a higher level than point A, what distance from the ground level is point A?
A) 1.66 units
B) 4.15 units
C) 5.44 units
D) 6.14 units
E) 7.13 units

Answer 1

Y=-.3x^2+3x 

....

-2x+5.5y=19.5

5.5y=19.5+2x

y=(19.5+2x)/5.5

When the light touches the parabola y=y so

(19.5+2x)/5.5=-.3x^2+3x  multiply both sides by 5.5

19.5+2x=-1.65x^2+16.5x  add 1.65x^2 to both sides

1.65x^2+2x+19.5=16.5x  subtract 16.5x from both sides

1.65x^2-14.5x+19.5=0

Using the quadratic equation for simplicity...

x=(14.5±√81.55)/3.3  we only need the lower point, or the smaller value of x

x≈1.657423  since:

y=(19.5+2x)/5.5

y≈4.15 units