Difference between revisions of "2004 AMC 10A Problems/Problem 19"

(New page: ==Problem== A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete...)
 
 
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A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
 
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
  
{{image}}
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<asy>
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size(250);defaultpen(linewidth(0.8));
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draw(ellipse(origin, 3, 1));
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fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white);
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draw((3,0)--(3,16)^^(-3,0)--(-3,16));
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draw((0, 15)--(3, 12)^^(0, 16)--(3, 13));
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filldraw(ellipse((0, 16), 3, 1), white, black);
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draw((-3,11)--(3, 5)^^(-3,10)--(3, 4));
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draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89));
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draw((0,-1)--(0,15), dashed);
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draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4));
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draw((-7,0)--(-5,0)^^(-7,16)--(-5,16));
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draw((3,-3)--(-3,-3), Arrows(6));
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draw((-6,0)--(-6,16), Arrows(6));
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draw((-2,9)--(-1,9), Arrows(3));
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label("$3$", (-1.375,9.05), dir(260), fontsize(7));
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label("$A$", (0,15), N);
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label("$B$", (0,-1), NE);
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label("$30$", (0, -3), S);
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label("$80$", (-6, 8), W);</asy>
  
 
<math> \mathrm{(A) \ } 120 \qquad \mathrm{(B) \ } 180 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 360 \qquad \mathrm{(E) \ } 480  </math>
 
<math> \mathrm{(A) \ } 120 \qquad \mathrm{(B) \ } 180 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 360 \qquad \mathrm{(E) \ } 480  </math>
  
 
==Solution==
 
==Solution==
{{solution}}
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The cylinder can be "unwrapped" into a rectangle, and we see that the stripe is a parallelogram with base <math>3</math> and height <math>80</math>. Thus, we get <math>3\times80=240\Rightarrow\boxed{\mathrm{(C)}\ 240}</math>
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==Video Solution==
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https://youtu.be/PYTi6qZUAPw
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Education, the Study of Everything
  
We can't have a solution if we can't have an image, which is [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=43&year=2004 here.]
 
  
 
==See also==
 
==See also==
 +
{{AMC10 box|year=2004|ab=A|num-b=18|num-a=20}}
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[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 14:19, 21 April 2021

Problem

A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

[asy]  size(250);defaultpen(linewidth(0.8)); draw(ellipse(origin, 3, 1)); fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white); draw((3,0)--(3,16)^^(-3,0)--(-3,16)); draw((0, 15)--(3, 12)^^(0, 16)--(3, 13)); filldraw(ellipse((0, 16), 3, 1), white, black); draw((-3,11)--(3, 5)^^(-3,10)--(3, 4)); draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89)); draw((0,-1)--(0,15), dashed); draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4)); draw((-7,0)--(-5,0)^^(-7,16)--(-5,16)); draw((3,-3)--(-3,-3), Arrows(6)); draw((-6,0)--(-6,16), Arrows(6)); draw((-2,9)--(-1,9), Arrows(3)); label("$3$", (-1.375,9.05), dir(260), fontsize(7)); label("$A$", (0,15), N); label("$B$", (0,-1), NE); label("$30$", (0, -3), S); label("$80$", (-6, 8), W);[/asy]

$\mathrm{(A) \ } 120 \qquad \mathrm{(B) \ } 180 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 360 \qquad \mathrm{(E) \ } 480$

Solution

The cylinder can be "unwrapped" into a rectangle, and we see that the stripe is a parallelogram with base $3$ and height $80$. Thus, we get $3\times80=240\Rightarrow\boxed{\mathrm{(C)}\ 240}$

Video Solution

https://youtu.be/PYTi6qZUAPw

Education, the Study of Everything


See also

2004 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
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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