2016 AMC 10A Problem 15

Problem 15 of 25IntermediateGeometry

Seven cookies of radius

1

inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?

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Solution

The circle of cookie dough has a radius of

3

inches since it is the same as the diameter plus the radius of a cookie. The area of the cookie dough is

3^2\pi = 9\pi,

and the cookies have an area of

7 \cdot 1^2\pi = 7\pi.

The area of the leftover scrap is therefore

9\pi - 7\pi = 2\pi.

This means that its radius is

\sqrt{2}.

Thus, the correct answer is A .

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