SOLUTION: A parabolic arch spans a stream 200 feet wide. How high above the stream must the arch be to give a minimum clearance of 40 feet over a channel in the center that is 120 feet wide?
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-> SOLUTION: A parabolic arch spans a stream 200 feet wide. How high above the stream must the arch be to give a minimum clearance of 40 feet over a channel in the center that is 120 feet wide?
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Question 1177813: A parabolic arch spans a stream 200 feet wide. How high above the stream must the arch be to give a minimum clearance of 40 feet over a channel in the center that is 120 feet wide?
I have tried setting up various equations, but none have worked so far. Thank you for any help! Answer by greenestamps(13200) (Show Source):
Let the origin of the coordinate system be the center of the stream at the surface; then the vertex of the parabola is at (0,k). The equation is then of the form
where, because the parabola opens downward, we know a is some negative constant.
We can determine a and k using the information in the problem:
(1) The arch is 200 feet wide. This means 100 feet either side of the center of the stream the height of the arch is zero: [1]
(2) We need a minimum vertical clearance of 40 feet everywhere in a channel 120 feet wide in the center of the stream. That means at 60 feet either side of the center of the stream the height must be 40: [2]
Solve the pair of equations [1] and [2].
Subtract the second equation from the first to eliminate k.
Substitute that value in [1] to find k, which is the height of the arch we are to find.
ANSWER: The height of the arch above the stream must be 62.5 feet.
CHECK: The equation of the parabola is .
A graph, showing a width of 200 feet and a height of 40 feet 60 feet either side of the center of the arch.
