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

Question

03-05-2017
Mathematics

contestada

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

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irspow irspow · Answer 1 · 2017-05-03T21:15:37+02:00

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

PratikshaS PratikshaS · Answer 2 · 2022-06-23T18:21:14+02:00

Point A is at distance of 1.66 units from the ground level.

The correct answer is an option (A)

What is equation?

"It is a mathematical statement which consists of equal symbol between two algebraic expressions."

For given question,

A fountain on a lake sprays water in a parabolic arch modeled by the equation [tex]y = -0.3x^2 + 3x[/tex]

A beam of light modeled by the equation -2x + 5.5y = 19.5

We write the equation of a beam of light in terms of y.

⇒ -2x + 5.5y = 19.5

⇒ 5.5y = 19.5 +2x

⇒ [tex]y =\frac{19.5 +2x}{5.5}[/tex]

When a beam of light passes through the fountain,

[tex]\Rightarrow y=y\\\\\Rightarrow -0.3x^2 + 3x=\frac{19.5 +2x}{5.5}\\\\\Rightarrow -1.65x^2+16.5x=19.5+2x\\\\\Rightarrow 1.65x^2-14.5x+19.5=0\\\\\Rightarrow (x-7.1)(x-1.66)=0\\\\\Rightarrow x-7.1=0~~,~~x-1.66=0\\\\\Rightarrow x=7.1~~,~~x=1.66[/tex]

Point A is at lower level than point B.

This means, point A is at the minimum distance from the ground level.

So, x = 1.66 units would be the distance between the ground level and the point A.

Therefore, point A is at distance of 1.66 units from the ground level.

The correct answer is an option (A)

Learn more about equation here:

https://brainly.com/question/13170101

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