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Difference between revisions of "1998 AJHSME Problems/Problem 20"

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==Problem 20==
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==Problem==
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Carpet costs <math>\$21.95</math> per square yard and the padding to put under it costs <math>\$2.55</math> per square yard. Felix plans to install padding and carpet in the region shown in the diagram. What is the cost of the carpet and padding needed to exactly cover the room?
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Let <math>PQRS</math> be a square piece of paper.  <math>P</math> is folded onto <math>R</math> and then <math>Q</math> is folded onto <math>S</math>. The area of the resulting figure is 9 square inches.  Find the perimeter of square <math>PQRS</math>.
 
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<asy>
 
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draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
 
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label("$P$",(0,2),SE);
 
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label("$Q$",(2,2),SW);
 
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label("$R$",(2,0),NW);
 
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label("$S$",(0,0),NE);
 
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</asy>
 
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<math>\text{(A)}\ 9 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 36</math>
 
  
 
==Solution==
 
==Solution==
  
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After all folds are completed, the square would be <math>\frac{1}{4}</math> of the original square.
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After both folds are completed, the square would become a triangle that has an area of <math>\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}</math> of the original square.
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Since the area is <math>9</math> for <math>\frac{1}{4}</math> of the square, <math>9\times4=36</math> is the area of square <math>PQRS</math>
 
  
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<math>\sqrt{36}=6</math>
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Since the area is <math>9</math> square inches for <math>\frac{1}{4}</math> of the square, <math>9\times4=36</math> square inches is the area of square <math>PQRS</math>
  
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<math>6\times4=24=\boxed{D}</math>
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The length of the side of a square that has an area of <math>36</math> square inches is <math>\sqrt{36}=6</math> inches.
  
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Each side is <math>6</math> inches, so the total perimeter is <math>6\times4=24=\boxed{D}</math>
  
 
== See also ==
 
== See also ==
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* [[AJHSME Problems and Solutions]]
 
* [[AJHSME Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
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{{MAA Notice}}

Latest revision as of 00:00, 5 February 2024

Problem

Carpet costs $$21.95$ per square yard and the padding to put under it costs $$2.55$ per square yard. Felix plans to install padding and carpet in the region shown in the diagram. What is the cost of the carpet and padding needed to exactly cover the room?

Solution

After both folds are completed, the square would become a triangle that has an area of $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ of the original square.

Since the area is $9$ square inches for $\frac{1}{4}$ of the square, $9\times4=36$ square inches is the area of square $PQRS$

The length of the side of a square that has an area of $36$ square inches is $\sqrt{36}=6$ inches.

Each side is $6$ inches, so the total perimeter is $6\times4=24=\boxed{D}$

See also

1998 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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