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?

Answer 1

check the picture below, so, what is from the A coordinates, what's the y-coordinate then.

well, we can get that off either equation, if we just know A coordinates, so let's see what "x" is at those A and B points.


`\bf \begin{cases} y=-0.3x^2+3x\\ ----------\\ -2x+5.5y=19.5\\\\ y=\cfrac{19.5+2x}{5.5} \end{cases}\implies -0.3x^2+3x=\cfrac{19.5+2x}{5.5} \\\\\\ -1.65x^2+16.5x=19.5+2x\implies 0=1.65x^2-14.5x+19.5 \\\\\\ \textit{using the quadratic formula}\quad x=\cfrac{14.5\pm√(210.25-4(1.65)(19.5))}{2(1.65)} \\\\\\ x\approx \begin{cases} 7.13\\ 1.66 \end{cases}\textit{ so those are A and B, who is who?, dunno, let's check}`


`\bf y=-0.3x^2+3x\qquad x=7.13\qquad y=-0.3(7.13)^2+3(7.13) \\\\\\ y\approx 6.14\\\\ -------------------------------\\\\ y=-0.3x^2+3x\qquad x=1.66\qquad y=-0.3(1.66)^2+3(1.66) \\\\\\ y\approx 4.15`

so, as you can see, clearly 6.14 is higher, and thus is B.

thus A is 4.15, so the y-coordinate for point A is 4.15, and that's the distance from the beam hitting the fountain at A, to the ground.