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| Math Central | Quandaries & Queries |
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Question from Jean, a student: "Two great circles lying in planes that are perpendicular to each other are drawn on a wooden sphere of radius "a". Part of the sphere is then shaved off in such a way that each cross section of the remaining solid that is perpendicular to the common diameter of the two great circles is a square whose vertices lie on these circles. Find the volume of this solid." I don't understand the geometry of the problem. |
It's easy if you know how to integrate. First, for a picture of half the object, look at our web page:
mathcentral.uregina.ca/QQ/database/QQ.09.06/danielle1.html
This shape can be used for domes of public buildings, almost like the Saskatchewan Legislative Building,
http://en.wikipedia.org/wiki/Saskatchewan_Legislative_Building
where I say "almost" because they used perpendicular ellipses rather than circles for the dome.
For the volume I would I would take the common diameter to be the x-axis -- just rotate the first figure clockwise so that one of the circles becomes the equator. I recommend that you find the volume of the right half, then multiply that number by 2 for your final answer. Thus you integrate the area of the square cross section times dx, from x = 0 to x = a. For each x you know the diagonal of the square to be y = 2*sqrt(a^2 - x^2). From that you can get the side of the square and then the area of the cross section.
Chris
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