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Difference between revisions of "2014 AMC 12B Problems/Problem 5"

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Let the height of the panes equal <math>5x</math>, and let the width of the panes equal <math>2x</math>.  Now notice that the total width of the borders equals <math>10</math>, and the total height of the borders is <math>6</math>.  We have
 
Let the height of the panes equal <math>5x</math>, and let the width of the panes equal <math>2x</math>.  Now notice that the total width of the borders equals <math>10</math>, and the total height of the borders is <math>6</math>.  We have
 
<cmath>10 + 4(2x) = 6 + 2(5x)</cmath>
 
<cmath>10 + 4(2x) = 6 + 2(5x)</cmath>
−
<cmath>x = 4</cmath>
+
<cmath>x = 2</cmath>
 
Now, the total side length of the window equals  
 
Now, the total side length of the window equals  
 
<cmath>10+ 4(2x) = 10 + 16 = \boxed{\textbf{(A)}\ 26}</cmath>
 
<cmath>10+ 4(2x) = 10 + 16 = \boxed{\textbf{(A)}\ 26}</cmath>
  
 
(Solution by kevin38017)
 
(Solution by kevin38017)

Revision as of 19:48, 20 February 2014

Problem

Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? [asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill((6,0)--(8,0)--(8,26)--(6,26)--cycle,gray); fill((12,0)--(14,0)--(14,26)--(12,26)--cycle,gray); fill((18,0)--(20,0)--(20,26)--(18,26)--cycle,gray); fill((24,0)--(26,0)--(26,26)--(24,26)--cycle,gray); fill((0,0)--(26,0)--(26,2)--(0,2)--cycle,gray); fill((0,12)--(26,12)--(26,14)--(0,14)--cycle,gray); fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); [/asy] $\textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}}\ 32\qquad\textbf{(E)}\ 34$ (Error compiling LaTeX. Unknown error_msg)

Solution

Let the height of the panes equal $5x$, and let the width of the panes equal $2x$. Now notice that the total width of the borders equals $10$, and the total height of the borders is $6$. We have \[10 + 4(2x) = 6 + 2(5x)\] \[x = 2\] Now, the total side length of the window equals \[10+ 4(2x) = 10 + 16 = \boxed{\textbf{(A)}\ 26}\]

(Solution by kevin38017)

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