Difference between revisions of "2006 AMC 12A Problems/Problem 12"

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== Problem ==
 
== Problem ==
 
 
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A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outisde diameter of 20 cm. The outside diameter of each of the outer 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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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 outer 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?
  
 
<math> \mathrm{(A) \ } 171\qquad \mathrm{(B) \ } 173\qquad \mathrm{(C) \ } 182\qquad \mathrm{(D) \ } 188</math><math>\mathrm{(E) \ }  210</math>
 
<math> \mathrm{(A) \ } 171\qquad \mathrm{(B) \ } 173\qquad \mathrm{(C) \ } 182\qquad \mathrm{(D) \ } 188</math><math>\mathrm{(E) \ }  210</math>
  
 
== Solution ==
 
== Solution ==
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The sum of the [[consecutive]]ly increasing [[integers]]s from 3 to 20 is <math>\frac{1}{2}183+20 = 207</math>. However, the 17 [[intersection]]s between the rings must also be subtracted, so we get <math>207 – 2(17) = 173 \Rightarrow B</math>.
  
 
== See also ==
 
== See also ==
 
* [[2006 AMC 12A Problems]]
 
* [[2006 AMC 12A Problems]]
*[[2006 AMC 12A Problems/Problem 11|Previous Problem]]
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*[[2006 AMC 12A Problems/Problem 13|Next Problem]]
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{{AMC box|year=2006|n=12A|num-b=11|num-a=13}}
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]

Revision as of 18:37, 31 January 2007

Problem


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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 outer 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?

$\mathrm{(A) \ } 171\qquad \mathrm{(B) \ } 173\qquad \mathrm{(C) \ } 182\qquad \mathrm{(D) \ } 188$$\mathrm{(E) \ }  210$

Solution

The sum of the consecutively increasing integerss from 3 to 20 is $\frac{1}{2}183+20 = 207$ (Error compiling LaTeX. Unknown error_msg). However, the 17 intersections between the rings must also be subtracted, so we get $207 – 2(17) = 173 \Rightarrow B$.

See also


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Preceded by
Problem 11
AMC 12A
2006
Followed by
Problem 13