Afountain 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?

  4.15 units

Explanation:

1) You must find the solution to this system:

y = -0.3x² + 3x
-2x + 5.5y = 19.5

2) replace y from the first equation into the second equation:

-2x + 5.5 [-0.3x² + 3x] = 19.5

3) use distributive property: -2x - 1.65x² + 16.5x = 19.5

4) combine like terms: - 1.65x² + 14.5x = 19.5

5) transpose terms: 1.65x² - 14.5x + 19.5 = 0

6) use quadratic formula to find the solution:

 x ≈ 1.66 and x ≈ 7.13.

To find which is the lower point, you replace the two values and find which has smaller y value:

y = -0.3x² + 3x

x = 1.66 ⇒ y = -0.3(1.66)² + 3(1.66) = 4.15

x = 7.13 ⇒ y = -0.3(7.13)² + 3(7.13) = 6.14

Therefore, the answer is y = 4.15.

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