Difference between revisions of "2018 AMC 10B Problems/Problem 7"
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+ | == Problem == | ||
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In the figure below, <math>N</math> congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let <math>A</math> be the combined area of the small semicircles and <math>B</math> be the area of the region inside the large semicircle but outside the semicircles. The ratio <math>A:B</math> is <math>1:18</math>. What is <math>N</math>? | In the figure below, <math>N</math> congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let <math>A</math> be the combined area of the small semicircles and <math>B</math> be the area of the region inside the large semicircle but outside the semicircles. The ratio <math>A:B</math> is <math>1:18</math>. What is <math>N</math>? | ||
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==Solution 1 (Work using Answer Choices)== | ==Solution 1 (Work using Answer Choices)== | ||
− | Use the answer choices and calculate them. The one that works is D | + | Use the answer choices and calculate them. The one that works is <math>\bold{\boxed{\text{(D)19}}}</math>. |
+ | |||
+ | ~Flame_Thrower | ||
==Solution 2 (More Algebraic Approach)== | ==Solution 2 (More Algebraic Approach)== | ||
− | Let the number of semicircles be <math>n</math> and let the radius of each semicircle | + | Let the number of semicircles be <math>n</math> and let the radius of each semicircle be <math>r</math>. To find the total area of all of the small semicircles, we have <math>n \cdot \frac{\pi \cdot r^2}{2}</math>. |
+ | |||
+ | Next, we have to find the area of the larger semicircle. The radius of the large semicircle can be deduced to be <math>n \cdot r</math>. So, the area of the larger semicircle is <math>\frac{\pi \cdot n^2 \cdot r^2}{2}</math>. | ||
+ | |||
+ | Now that we have found the area of both A and B, we can find the ratio. <math>\frac{A}{B}=\frac{1}{18}</math>, so part-to-whole ratio is <math>1:19</math>. When we divide the area of the small semicircles combined by the area of the larger semicircles, we get <math>\frac{1}{n}</math>. This is equal to <math>\frac{1}{19}</math>. By setting them equal, we find that <math>n = 19</math>. This is our answer, which corresponds to choice <math>\bold{\boxed{\text{(D)19}}}</math>. | ||
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+ | Solution by: Archimedes15 | ||
+ | |||
+ | |||
+ | Note: using WLOG you could have just set <math>r</math> as <math>1</math> in the beginning. | ||
+ | |||
+ | ==Solution 3== | ||
+ | Each small semicircle is <math>\frac{1}{N^2}</math> of the large semicircle. Since <math>N</math> small semicircles make <math>\frac{1}{19}</math> of the large one, <math>\frac{N}{N^2} = \frac1{19}</math>. Solving this, we get <math>\boxed{19}</math>. | ||
+ | |||
+ | ... Don't think this one works | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/4KT0AtJZQzU | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | ||
+ | https://youtu.be/qdyL6-Vmrrk | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
− | |||
− | + | == Video Solution == | |
+ | https://youtu.be/NsQbhYfGh1Q?t=1120 | ||
==See Also== | ==See Also== |
Latest revision as of 20:55, 24 October 2024
Contents
Problem
In the figure below, congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the semicircles. The ratio is . What is ?
Solution 1 (Work using Answer Choices)
Use the answer choices and calculate them. The one that works is .
~Flame_Thrower
Solution 2 (More Algebraic Approach)
Let the number of semicircles be and let the radius of each semicircle be . To find the total area of all of the small semicircles, we have .
Next, we have to find the area of the larger semicircle. The radius of the large semicircle can be deduced to be . So, the area of the larger semicircle is .
Now that we have found the area of both A and B, we can find the ratio. , so part-to-whole ratio is . When we divide the area of the small semicircles combined by the area of the larger semicircles, we get . This is equal to . By setting them equal, we find that . This is our answer, which corresponds to choice .
Solution by: Archimedes15
Note: using WLOG you could have just set as in the beginning.
Solution 3
Each small semicircle is of the large semicircle. Since small semicircles make of the large one, . Solving this, we get .
... Don't think this one works
Video Solution
~savannahsolver
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution
https://youtu.be/NsQbhYfGh1Q?t=1120
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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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