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POW Solution, Dec. 610
Posted:
Dec 13, 1993 10:03 AM


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Problem of the Week for December 610:
A circle is inscribed in a semicircle. The diameter of the circle and the radius of the semicircle are 12 units. What is the area of the region of the semicircle that is outside the circle?
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Correct solutions to the problem were submitted by
Dan Hirschhorn Jonathan Jacobs Jason Jenkins, Boston U. Kim Chatha, Chris Bernardi, and Tim Luff of DoverSherborn High School
Dan and Jonathan's solutions are included below, as they are quite different from one another.
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Solutions:
Since ratio of diameters from semicircle to circle is 2:1, by Fund. thm of Similarity, area of Full circle to inscribed circle is 4:1 so area of semicircle to inscribed circle is 2:1. Thus inscribed circle takes up 1/2 semicircle! Area of outside half = area of inscribed circle = 36 .  Daniel B. Hirschhorn  ISU Mathematics danh@math.ilstu.edu  (309) 4387849

Explanation, eh? Well, let's see ... Onehalf of a circle at radius 12 has area: (1/2)pi(12)^2 or 72pi The complete circle at diameter 12, radius 6 has area: pi(6)^2 or 36pi So, subtracting the smaller, inscribed circle from the larger semicircle gives us: 72pi  36pi, or _36 pi_. Now do I get full credit, or is it too late? (Honest ... I had the workthere originally, but I erased it ...) Jon (pigpen@hardy.u.washington.edu)



