2006 AMC 12A Problem 12

Problem 12 of 25IntermediateAlgebra

A number of linked rings, each

1

cm thick, are hanging on a peg. The top ring has an outside diameter of

20

cm. The outside diameter of each of the other rings is

1

cm less than that of the ring above it. The bottom ring has an outside diameter of

3

cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

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Solution

The top ring spans

20

cm. Each ring below overlaps the ring above by

2

cm (twice the

1

-cm thickness), so it adds its outside diameter minus

2.

The lower rings have outside diameters

19, 18, \ldots, 3,

contributing

17, 16, \ldots, 1.

Thus the total distance is

\begin{gathered} 20 + (17 + 16 + \cdots + 1) \\ = 20 + \frac{17 \cdot 18}{2} \\ = 20 + 153 \\ = 173 \text{ cm}. \end{gathered}

Thus, the correct answer is B.