A right cylinder has height h and radius r. It is sliced by a plane that is tangent to one circular base at A and intersects the other at diameter BC. What is the volume of slice ABCD?
Note that BO=CO=DO=r, AD=h, BC is perpendicular to DO, and AD is perpendicular to DO.
Consider two right circular cylinders intersecting at right angles. Imagine the common volume and imagine an in-sphere. Any cross section of the common volume along axis of either cylinder is then a square with an inscribed circle. The ratio of the square to the circle is the same as the common volume to the in-sphere and, for a radius of one, solves to 16/3.
Inspection of the common volume reveals it may be viewed as four nesting segments, each formed of two cylindrical slices connected at their bases. Each slice then is 2/3. Both radius and height in this case are one. Since the volume of the slice is some constant times the product of radius and height, the volume of the general case is 2/3 that product.
Cee
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Posted by CeeAnne
on 2005-12-07 14:31:19 |
An Indirect Reasoning
