SOLUTION: How many circular cookies measuring 2 inches in radius can be cut from a circle of dough measuring 6 inches in radius, assuming that dough between the cookies is not reused?

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Question 833336: How many circular cookies measuring 2 inches in radius can be cut from a circle of dough measuring 6 inches in radius, assuming that dough between the cookies is not reused?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
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The least unused space between cookies and the edge of the dough circle will be achieved when the cookies are against the edge of the dough circle.
The least unused space between cookies will be achieved when the cookie centers are forming an equilateral triangle. Three 1%2F6 of a cookie wedges will be inside that triangle. The area of that 3%2F6=1%2F2 of a circle of radius R is pi%2AR%5E2%2F2 , while the triangle, with sides measuring 2R will have a height of sqrt%283%29%2AR and an area of sqrt%283%29%2AR%5E2 .
The fraction of that area covered by the circle wedges is
%28pi%2AR%5E2%2F2%29%2F%28sqrt%283%29%2AR%5E2%29=pi%2F2sqrt%283%29= approx. 0.91 (91%).
If you place circles with radius R so that the centers form a square, the side of that square will be 2R and the area of that square will be %282R%29%5E2=4R%5E2 , while the 4 quarter circle wedges covering that square will have a total area of pi%2AR%5E2 , amounting to
pi%2AR%5E2%2F4R%5E2=pi%2F4= approx. 0.79 (79%) of the square area.