Difference between revisions of "2006 AMC 12A Problems/Problem 12"
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== Problem == | == Problem == | ||
| − | + | 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 | ||
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| + | <math>\mathrm{(A) \ } 171\qquad\mathrm{(B) \ } 173\qquad\mathrm{(C) \ } 182\qquad\mathrm{(D) \ } 188\qquad\mathrm{(E) \ } 210\qquad</math> | ||
== Solution == | == Solution == | ||
The sum of the inner [[diameter]]s of the rings is the series from 1 to 18. To this sum, we must add 2 for the top of the first ring and the bottom of the last ring. Thus, <math>\frac{18 * 19}{2} + 2 = 173 \Rightarrow B</math>. | The sum of the inner [[diameter]]s of the rings is the series from 1 to 18. To this sum, we must add 2 for the top of the first ring and the bottom of the last ring. Thus, <math>\frac{18 * 19}{2} + 2 = 173 \Rightarrow B</math>. | ||
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Alternatively, the sum of the consecutively increasing [[integer]]s from 3 to 20 is <math> \frac{1}{2}(18)(3+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>. | Alternatively, the sum of the consecutively increasing [[integer]]s from 3 to 20 is <math> \frac{1}{2}(18)(3+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 | + | == See Also == |
| − | * [[2006 AMC | + | *[[2006 AMC 10A Problems]] |
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| + | *[[2006 AMC 10A Problems/Problem 13|Previous Problem]] | ||
| − | + | *[[2006 AMC 10A Problems/Problem 15|Next Problem]] | |
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
Revision as of 09:03, 15 February 2007
Problem
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?
Solution
The sum of the inner diameters of the rings is the series from 1 to 18. To this sum, we must add 2 for the top of the first ring and the bottom of the last ring. Thus, 
.
Alternatively, the sum of the consecutively increasing integers from 3 to 20 is 
. However, the 17 intersections between the rings must also be subtracted, so we get 
.
