Difference between revisions of "2024 AMC 10B Problems/Problem 21"
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Two straight pipes (circular cylinders), with radii <math>1</math> and <math>\frac{1}{4}</math>, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? | Two straight pipes (circular cylinders), with radii <math>1</math> and <math>\frac{1}{4}</math>, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? | ||
| − | + | <asy> | |
size(6cm); | size(6cm); | ||
draw(circle((0,1),1), linewidth(1.2)); | draw(circle((0,1),1), linewidth(1.2)); | ||
draw((-1,0)--(1.25,0), linewidth(1.2)); | draw((-1,0)--(1.25,0), linewidth(1.2)); | ||
draw(circle((1,1/4),1/4), linewidth(1.2)); | draw(circle((1,1/4),1/4), linewidth(1.2)); | ||
| − | + | </asy> | |
| − | <math>\textbf{(A)}~ | + | <math>\textbf{(A)}~\frac{1}{9} |
\qquad\textbf{(B)}~1 | \qquad\textbf{(B)}~1 | ||
| − | \qquad\textbf{(C)}~ | + | \qquad\textbf{(C)}~\frac{10}{9} |
| − | \qquad\textbf{(D)}~ | + | \qquad\textbf{(D)}~\frac{11}{9} |
| − | \qquad\textbf{(E)}~ | + | \qquad\textbf{(E)}~\frac{19}{9}</math> |
==Solution 1== | ==Solution 1== | ||
Revision as of 08:32, 14 November 2024
Problem
Two straight pipes (circular cylinders), with radii 
and 
, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
![[asy] size(6cm); draw(circle((0,1),1), linewidth(1.2)); draw((-1,0)--(1.25,0), linewidth(1.2)); draw(circle((1,1/4),1/4), linewidth(1.2)); [/asy]](http://latex.artofproblemsolving.com/4/0/9/4091eff69445abb83b260b3ccbdfa4e00b8f9182.png)
Solution 1
See also
| 2024 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
