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Difference between revisions of "2014 AMC 10B Problems/Problem 5"

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==Problem==
 
==Problem==
  
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Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width of 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?
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Doug constructs a square window using <math> 8 </math> equal-size panes of glass, as shown. The ratio of the height to width for each pane is <math> 5 : 2 </math>, and the borders around and between the panes are <math> 2 </math> inches wide. In inches, what is the side length of the square window?
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<asy>
−
<math> \textbf {(A) } 26 \qquad \textbf {(B) } 28 \qquad \textbf {(C) } 30 \qquad \textbf {(D) } 32 \qquad \textbf {(E) } 34</math>
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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>
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<math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}}\ 32\qquad\textbf{(E)}\ 34 </math>
  
 
==Solution==
 
==Solution==

Revision as of 18:40, 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

We note that the total length must be the same as the total height because the window a square. Calling the width of each small rectangle $2x$, and the height $5x$, we can see that the length is composed of 4 widths and 5 bars of length 2. This is equal to two heights of the small rectangles as well as 3 bars of 2. Thus, $4(2x) + 5(2) = 2(5x) + 3(2)$. We quickly find that $x = 2$. The total side length is $4(4) + 5(2) = 2(10) + 3(2) = 26$, or $\boxed{\textbf{(A)}}$.

See Also

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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

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