Difference between revisions of "2004 AMC 10A Problems/Problem 19"
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==Solution 2 (Complement Counting)== | ==Solution 2 (Complement Counting)== | ||
| − | Like in Solution 1 "unwrap" the lateral surface of the cylinder into a rectangle, the width of the rectangle is the circumference of the circular base which is <math>30\times\pi</math>. And the length is just the height of the cylinder, <math>80</math>. Now if we don't see that the stripe is just a parallelogram we can calculate the area of the two right triangles with legs <math>80</math> and <math>30\pi-3</math>, and subtract our result from the total area of the rectangle to get the area of the stripe. Thus the area of the stripe is | + | Like in Solution 1 "unwrap" the lateral surface of the cylinder into a rectangle, the width of the rectangle is the circumference of the circular base which is <math>30\times\pi</math>. And the length is just the height of the cylinder, <math>80</math>. Now if we don't see that the stripe is just a parallelogram we can calculate the area of the two right triangles with legs <math>80</math> and <math>30\pi-3</math>, and subtract our result from the total area of the rectangle to get the area of the stripe. Thus the area of the stripe is \[ |
| + | \text{Area of the stripe} = 80 \times 30\pi - 2 \times \frac{1}{2} \times 80 \times (30\pi - 3) = 80 \times 3 = 240 \Rightarrow \boxed{\mathrm{(C)}\ 240} | ||
| + | \] | ||
==Video Solution== | ==Video Solution== | ||
Revision as of 20:19, 13 August 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]](http://latex.artofproblemsolving.com/d/b/0/db0ee67d222e80fda19ce7927dc5efb178a81aff.png)
Solution 1
The cylinder can be "unwrapped" into a rectangle, and we see that there are two stripes which is a parallelogram with base 
and height 
, each. Thus, we get 
Solution 2 (Complement Counting)
Like in Solution 1 "unwrap" the lateral surface of the cylinder into a rectangle, the width of the rectangle is the circumference of the circular base which is 
. And the length is just the height of the cylinder, 
. Now if we don't see that the stripe is just a parallelogram we can calculate the area of the two right triangles with legs and 
, and subtract our result from the total area of the rectangle to get the area of the stripe. Thus the area of the stripe is \[
\text{Area of the stripe} = 80 \times 30\pi - 2 \times \frac{1}{2} \times 80 \times (30\pi - 3) = 80 \times 3 = 240 \Rightarrow \boxed{\mathrm{(C)}\ 240}
\]
Video Solution
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. 
