A parabolic arch has a span of 48 m and has a height of 20 m at a distance 16 m from the center of the span. What is the height of the parabolic arch?
Let the coordinate of the center of the span is (0, 0).
So the parabolic arch passes through the points (-24, 0), (24, 0) and (16, 20)
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So the equation of the parabolic arch is of the form,
y = a(x - 24)(x + 24)
where y is the height (in meters) at a distance of x meters from the center of the span.
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Now as the arch passes through the point (16, 20), so we have,
20 = a*(16 - 24)*(16 + 24)
i.e. 20 = a*(-8)*(40)
i.e. a = -20/320
i.e. a = -1/16
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So the equation of the parabolic arch is,
y = (-1/16)(x - 24)(x + 24)
i.e. y = (576 - x2)/16
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Now height of the arch can be obtained by putting x = 0, on the above equation.
So, by putting x = 0, we get, y = 576/16 = 36
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So height of the parabolic arch is 36 meters.