2016 AMC 10A Problem 15

Question

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?

\sqrt{2}

1.5

\sqrt{\pi}

\sqrt{2\pi}

\pi

Answer: A

Concepts:circle area tangent circles

Difficulty rating: 1140

Small hint:

The large circle has radius

3

Big hint:

Subtract the total area of the seven unit cookies

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 .