Difference between revisions of "2014 AMC 12B Problems/Problem 5"
Kevin38017 (talk | contribs) (→Solution) |
(→Solution) |
||
(6 intermediate revisions by 5 users not shown) | |||
Line 1: | Line 1: | ||
− | ==Problem== | + | == Problem == |
+ | 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? | ||
− | |||
<asy> | <asy> | ||
fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); | fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); | ||
Line 12: | Line 12: | ||
fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); | fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); | ||
</asy> | </asy> | ||
− | |||
− | + | <math> \textbf{(A)}\ 26\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 </math> | |
+ | == Solution == | ||
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 = | + | <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> | ||
− | ( | + | ==Video Solution 1 (Quick and Easy)== |
+ | https://youtu.be/IXxMDMLbd-E | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | == See Also == | ||
+ | {{AMC12 box|year=2014|ab=B|num-b=4|num-a=6}} | ||
+ | {{MAA Notice}} |
Latest revision as of 00:38, 15 November 2022
Problem
Doug constructs a square window using equal-size panes of glass, as shown. The ratio of the height to width for each pane is
, and the borders around and between the panes are
inches wide. In inches, what is the side length of the square window?
Solution
Let the height of the panes equal , and let the width of the panes equal
. Now notice that the total width of the borders equals
, and the total height of the borders is
. We have
Now, the total side length of the window equals
Video Solution 1 (Quick and Easy)
~Education, the Study of Everything
See Also
2014 AMC 12B (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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.