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1. the arch of a bridge spanning a canal is in the shape of the top half of an ellipse whose major axis is horizontal. the base of the archis 16m wide, and the heighest part of the arch is 4m above the canal. determine if a flat-topped barge 6m wide and 3.6m high can safely pass under the bridge.
2. (Original post by ronnie)
the arch of a bridge spanning a canal is in the shape of the top half of an ellipse whose major axis is horizontal. the base of the archis 16m wide, and the heighest part of the arch is 4m above the canal. determine if a flat-topped barge 6m wide and 3.6m high can safely pass under the bridge.
The equation of the ellipse is (with the origin in the middle of the major axis)

x^2/8^2 + y^2/4^2 = 1

as the ellipse goes through (8,0) and (0,4).

When y = 3.6 then

x^2/64 = 1 - 3.6^2/16 = 0.19

x^2 = 12.16

x = 3.487...

So the width under the bridge at height 3.6 is 2 x 3.487 = 6.974... which is comfortably enough to accommodate the width of 6.
3. (Original post by RichE)
The equation of the ellipse is (with the origin in the middle of the major axis)

x^2/8^2 + y^2/4^2 = 1

as the ellipse goes through (8,0) and (0,4).

When y = 3.6 then

x^2/64 = 1 - 3.6^2/16 = 0.19

x^2 = 12.16

x = 3.487...

So the width under the bridge at height 3.6 is 2 x 3.487 = 6.974... which is comfortably enough to accommodate the width of 6.
Yup that's right!

Updated: August 11, 2005
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