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Topic: POW Solution, Dec. 6-10
Replies: 1   Last Post: Dec 13, 1993 10:08 AM

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Annie Fetter

Posts: 524
Registered: 12/3/04
POW Solution, Dec. 6-10
Posted: Dec 13, 1993 10:03 AM
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Problem of the Week for December 6-10:

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?


Correct solutions to the problem were submitted by

Dan Hirschhorn
Jonathan Jacobs
Jason Jenkins, Boston U.
Kim Chatha, Chris Bernardi, and Tim Luff of Dover-Sherborn High School

Dan and Jonathan's solutions are included below, as they are quite
different from one another.



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) 438-7849


Explanation, eh? Well, let's see ...
One-half of a circle at radius 12 has area:
or 72pi
The complete circle at diameter 12, radius 6 has area:
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)

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