SOLUTION: An engineer constructs side-by-side parabolic arches to support a bridge over a road and a river. The arch over the road has a maximum height of 6 m and a width of 16 m. The river

Question 934652: An engineer constructs side-by-side parabolic arches to support a bridge over a road and a river. The arch over the road has a maximum height of 6 m and a width of 16 m. The river arch has a maximum height of 8 m, but its width is reduced by 4 m because it intersects the arch over the road. Without this intersection, the river arch would have a width of 24 m. A support footing is used at the intersection point of the arches. The engineer sketched the arches on a coordinate system. She placed the origin at the left most point of the road.
(24, 8)
(8, 6)
a) Determine the system of equations that models the two arches.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the system of equations will be:

y1 = -6/64 * x^2 + 96/64 * x

y2 = -8/144 * x^2 + 384 / 144 * x - 3456 / 144

if you graph those equations, you will get:

$$$

the first equation is modeled by knowing that the zeroes of that eqution are at x = 0 and x = 16.

this means the factors of that equation are (x-0) * (x-16) = 0 which becomes x^2 - 16x = 0

the general equation becomes y = a * (x^2 - 16x)

you know that y = 6 when x = 8, so replace x in that equation to get a * (8^2 - 16(8)) = 6 which becomes a * (-64) = 6 which becomes -64*a = 6 which becomes a = -6/64.

your equation of y = a * (x^2 - 16x) becomes y = -6/64 * (x^2 - 16x) which becomes y = -6/64 * x^2 + 96/64 * x

that's the first equation that was graphed.

the second equation was solved for in a similar manner.

the roots were 12 and 36
the factors were (x-12) * (x-36) = 0
multiplying those factors out got x^2 - 48x + 432 = 0
the general equation became y = a * (x^2 - 48x + 432)
when x was 24, y was 8, so we got 8 = a * (24^2 - 48*24 + 432) which became 8 = 576 * a - 1152 * a + 432 * a which became 8 = -144 * a
divide both sides of that equation by -144 and you get a = -8/144.
your equation becomes y = -8/144 * x^2 + 384/144 * x - 3456/144
that's the equation you see in the graph.

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