SOLUTION: Additional picnic pyramids are being constructed in the northeast corner of the park. Bridges are being built across the small stream to allow people to access the structures.

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Question 1182285: Additional picnic pyramids are being constructed in the northeast corner of the park. Bridges are being built across the small stream to allow people to access the structures.


The design of one bridge includes a parabolic arch, a horizontal handrail and vertical rods. The parabolic arch is defined by the equation,
y = - 0.1875x2 + 0.5625x + 0.328125
where x is the horizontal distance from one end of the bridge in metres and y is the height of the arch in metres.
The arch spans the stream and is anchored to the land 0.5 m beyond the edge of the stream on each side.
Determine the height of the handrail above the bridge surface.
Determine the width of the stream.
What is the height of the arch at the edge of the stream? Round to 2 decimal places.
Picture of bridge: https://ibb.co/31Td44H
Answer by ikleyn(52793) About Me  (Show Source):
You can put this solution on YOUR website!
.

First find the roots of the polynomial from the quadratic equation


        -0.1875x2 + 0.5625x + 0.328125 = 0


    It is EQUIVALENT to

        x^2 - 3x - 1.75 = 0


    The roots are


        x%5B1%2C2%5D = %283+%2B-+sqrt%283%5E2+%2B+4%2A1.75%29%29%2F2 = %283+%2B-+sqrt%2816%29%29%2F2 = %283+%2B-+4%29%2F2.


        x%5B1%5D = %283-4%29%2F2 = -0.5;   x%5B2%5D = %283%2B4%29%2F2 = 3.5.


So, the arch is anchored at  x%5B1%5D = -0.5  and  x%5B2%5D = 3.5.


The stream is between  x= 0  and  x= 3,

The width of the stream is  3 - 0 = 3 meters.


The arch has maximum height exactly half the way between the roots at x= 1.5.


The height of the arch is the value of the polynomial at x= 1.5


    h%5Bmax%5D = -0.1875*1.5^2 + 0.5625*1.5 + 0.328125 = 0.75 m.

Solved.