Question

Select the correct answer. An archway is modeled by the equation y = -2x2 + 8x. A rod is to be placed across the archway at an angle defined by the equation x − 2.23y + 10.34 = 0. If the rod is attached to the archway at points A and B, such that point B is at a higher level than point A, at what distance from the ground level is point B? A. 8 units B. 6 units C. 5 units D. 3 units

Answer 1

B. 6 units

Explanation:

Given equations:

`y = -2x^2 + 8x`

`x-2.23y + 10.34 = 0`

The points at which the rod is attached to the archway are the points of intersection of the two equations.

Rearrange the second equation to make x the subject:

`\implies x=2.23y-10.34`

Substitute this into the first equation to create a quadratic:

`y = -2(2.23y-10.34)^2 + 8(2.23y-10.34)`

`y = -2(4.9729y^2-46.1164y+106.9156) + 17.84y-82.72`

`y=-9.9458y^2+92.2328y-213.8312+17.84y-82.72`

`y=-9.9458y^2+110.0728y-296.5512`

`-9.9458y^2+109.0728y-296.5512=0`

`\boxed{\begin{minipage}{3.6 cm}\underline{Quadratic Formula}\\\\$x=(-b \pm √(b^2-4ac))/(2a)$\\\\when $ax^2+bx+c=0$ \\\end{minipage}}`

Solve the quadratic using the quadratic formula:

`\implies y=(-(109.0728) \pm √((109.0728)^2-4(-9.9458)(-296.5512)))/(2(-9.9458))`

`\implies y=(-109.0728 \pm √(99.12))/(-19.8916)`

`\implies y=(109.0728 \pm √(99.12))/(19.8916)`

`\implies y=5.983867702, \quad y=4.982851918`

As point B is at a higher level than point A, the y-value of point B is approximately 6 units.

Therefore, point B is 6 units from ground level.