Difference between revisions of "2012 AMC 10B Problems/Problem 2"

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== Problem ==
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A circle of radius 5 is inscribed in a rectangle as shown.  The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?
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<math> \textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\  125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200 </math>
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==Solution==
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Note that the diameter of the circle is equal to the shorter side of the rectangle. Since the radius is <math>5</math>, the diameter is <math>2\cdot 5 = 10</math>.
 
Note that the diameter of the circle is equal to the shorter side of the rectangle. Since the radius is <math>5</math>, the diameter is <math>2\cdot 5 = 10</math>.
 
Since the sides of the rectangle are in a <math>2:1</math> ratio, the longer side has length <math>2\cdot 10 = 20</math>.
 
Since the sides of the rectangle are in a <math>2:1</math> ratio, the longer side has length <math>2\cdot 10 = 20</math>.
 
Therefore the area is <math>20\cdot 10 = 200</math> or <math>\boxed{E}</math>.
 
Therefore the area is <math>20\cdot 10 = 200</math> or <math>\boxed{E}</math>.

Revision as of 18:59, 25 February 2012

Problem

A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?

$\textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\  125\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 200$

Solution

Note that the diameter of the circle is equal to the shorter side of the rectangle. Since the radius is $5$, the diameter is $2\cdot 5 = 10$. Since the sides of the rectangle are in a $2:1$ ratio, the longer side has length $2\cdot 10 = 20$. Therefore the area is $20\cdot 10 = 200$ or $\boxed{E}$.

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