2006 AMC 10A Problem 14

Problem 14 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 contributes its full outside diameter,

20

cm. Because the rings are

1

cm thick, each ring hangs

2

cm below the top of the ring above it, so each lower ring adds its outside diameter minus

2.

The outside diameters run

20, 19, \ldots, 3,

so the added distances are

17, 16, \ldots, 1.

The total is

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

Thus, the correct answer is B.