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Difference between revisions of "2004 AMC 10A Problems/Problem 19"
(Undo revision 259560 by Ljdestroyer (talk))
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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?
  
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[[Image:2004 AMC 10A problem 19.png]]
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<asy>
 +
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));
 +
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));
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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>
  
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==Solution==
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==Solution 1==
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The cylinder can be "unwrapped" into a rectangle, and we see that the stripe is a parallelogram with base 3 and height 80. <math>3\cdot 80=240\Rightarrow \boxed{\mathrm{(C) \ }}</math>
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The cylinder can be "unwrapped" into a rectangle, and we see that there are two stripes which is a parallelogram with base <math>3</math> and height <math>40</math>, each. Thus, we get <math>3\times40\times2=240\Rightarrow\boxed{\mathrm{(C)}\ 240}</math>
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==Video Solution==
 +
https://youtu.be/PYTi6qZUAPw
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 +
Education, the Study of Everything
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==See also==
 
==See also==
Line 13: Line 38:
  
 
[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 16:32, 23 September 2025

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 1

The cylinder can be "unwrapped" into a rectangle, and we see that there are two stripes which is a parallelogram with base $3$ and height $40$, each. Thus, we get $3\times40\times2=240\Rightarrow\boxed{\mathrm{(C)}\ 240}$

Video Solution

https://youtu.be/PYTi6qZUAPw

Education, the Study of Everything


See also

2004 AMC 10A (Problems • Answer Key • Resources)
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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