Difference between revisions of "2010 AMC 8 Problems/Problem 16"
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== Problem == | == Problem == | ||
A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? | A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? | ||
| + | <math> \textbf{(A)}\ \frac{\sqrt{\pi}}{2}\qquad\textbf{(B)}\ \sqrt{\pi}\qquad\textbf{(C)}\ \pi\qquad\textbf{(D)}\ 2\pi\qquad\textbf{(E)}\ \pi^{2} </math> | ||
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== Solution == | == Solution == | ||
| − | Let the side length of the square be <math>s</math>, and let the radius of the circle be <math>r</math>. Thus we have <math>s^2=r^2\pi</math>. Dividing each side by <math>r^2</math>, we get <math>s^2/r^2=\pi</math>. Since <math>(s/r)^2=s^2/r^2</math>, we have <math>s/r=\sqrt{\pi}\Rightarrow \boxed{B}</math> | + | Let the side length of the square be <math>s</math>, and let the radius of the circle be <math>r</math>. Thus we have <math>s^2=r^2\pi</math>. Dividing each side by <math>r^2</math>, we get <math>s^2/r^2=\pi</math>. Since <math>(s/r)^2=s^2/r^2</math>, we have <math>s/r=\sqrt{\pi}\Rightarrow \boxed{\textbf{B}\ \sqrt{\pi}}</math> |
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| + | ==See Also== | ||
| + | {{AMC8 box|year=2010|num-b=15|num-a=17}} | ||
Revision as of 16:31, 5 November 2012
Problem
A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?
Solution
Let the side length of the square be 
, and let the radius of the circle be 
. Thus we have 
. Dividing each side by 
, we get 
. Since 
, we have 
See Also
| 2010 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
