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Difference between revisions of "2024 AMC 8 Problems/Problem 18"

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m (Previously, the equation had pi + ... on the left side with 3 pi + ... on the right side. Edited it to have 3 pi on the left side and pi on the right side.)
 
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==Problem==
 
==Problem==
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Not held yet
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Three concentric circles centered at <math>O</math> have radii of <math>1</math>, <math>2</math>, and <math>3</math>. Points <math>B</math> and <math>C</math> lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angles <math>BOC</math>, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of <math>\angle{BOC}</math> in degrees?
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<asy>
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size(150);
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import graph;
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draw(circle((0,0),3));
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real radius = 3;
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real angleStart = -54;  // starting angle of the sector
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real angleEnd = 54;  // ending angle of the sector
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label("$O$",(0,0),W);
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pair O = (0, 0);
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filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, gray);
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filldraw(circle((0,0),2),gray);
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filldraw(circle((0,0),1),white);
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draw((1.763,2.427)--(0,0)--(1.763,-2.427));
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label("$B$",(1.763,2.427),NE);
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label("$C$",(1.763,-2.427),SE);
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</asy>
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<math>\textbf{(A) } 108\qquad\textbf{(B) } 120\qquad\textbf{(C) } 135\qquad\textbf{(D) } 144\qquad\textbf{(E) } 150</math>
  
 
==Solution 1==
 
==Solution 1==
−
Not held yet
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 +
Let <math>x=\angle{BOC}</math>.
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We see that the shaded region is the inner ring plus a sector <math>x^\circ</math> of the outer ring. The area of this in terms of <math>x</math> is <math>\left( 4 \pi - \pi \right)+\frac{x}{360} \left( 9 \pi - 4 \pi \right)</math>. This simplifies to <math>3 \pi + \frac{x}{360}(5 \pi)</math>.
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Also, the unshaded portion is comprised of the smallest circle plus the sector <math>(360-x)^\circ</math> of the outer ring. The area of this is <math>\pi + \frac{360-x}{360}(5 \pi)</math>.
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We are told these are equal, therefore <math>3 \pi + \frac{x}{360}(5 \pi) = \pi + \frac{360-x}{360}(5 \pi)</math>. Solving for <math>x</math> reveals <math>x=\boxed{\textbf{(A) } 108}</math>.
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~MrThinker
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~Dash11 (Edited coefficient of pi. Credit goes to MrThinker for the solution and explanation.)
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==Solution 2==
 +
 
 +
Notice that for the 3rd most outer ring of the circle, the ratio of the shaded region to non-shaded region is the ratio of <math>\angle{BOC}</math> to <math>360-\angle{BOC}</math>. With that, all we need to do is solve for the shaded region.
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 +
The inner most circle has radius <math>1</math>, and the second circle has radius 2. Therefore, the first shaded area has <math>4 \pi - \pi = 3 \pi</math> area. The circle has total area <math>9 \pi</math>, so the other shaded region must have <math>1.5 \pi</math> area, as the non-shaded and shaded area is equivalent. So for the 3rd outer ring, the total area is <math>9 \pi - 4 \pi = 5 \pi</math>, so the non-shaded part of the outer ring is <math>5 \pi - 1.5 \pi = 3.5 \pi</math>.
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Now as said before, the ratio of these two areas is the ratio of <math>\angle{BOC}</math> and <math>360 - \angle{BOC}</math>. So, <math>\frac{3.5}{1.5} = \frac{7}{3}</math>. We have <math>7x:3x</math> where <math>7x+3x = 360</math>, <math>x = 36</math>, so our answer is <math>3x = 108, \boxed{(A) 108}</math>.
 +
 
 +
~MaxyMoosy
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 +
==Solution 3 (Ruler Tool)==
 +
 
 +
The AMC 8s allow a ruler tool that you can rotate and drag. You can use the tool to make a straight segment (which we know is <math>180</math> degrees), and we let the angle of desire be <math>x</math>. We can estimate that <math>180-x</math> is just about <math>30</math> degrees short of <math>x</math> itself, so <math>x-30=180-x</math>, solving gives <math>x=105</math>, therefore the closest answer is <math>\boxed{\textbf{(A)}\,108}</math>.
 +
 
 +
*Note: This isn't the most accurate method to use on AMC 8s, this is just a quick method if time is short or you do not know how to do the problem and want to guess at it.
 +
 
 +
==Solution 4==
 +
Suppose the desired angle is some fraction <math>x</math> of the total degree measure of the circle. We now compile a list of the shaded and unshaded areas. The inner circle of radius <math>1</math> is completely unshaded, so it contributes <math>1</math> to the unshaded area. (Everything will be a multiple of <math>\pi</math>, so we omit it.) The inner annulus has area <math>2^2 - 1^2 = 3</math>, which it contributes to the shaded area. The outer annulus has a total area of <math>3^2 - 2^2 = 5</math>; the fraction <math>x</math> is shaded, so the shaded portion of the outer annulus contributes <math>5x</math> to the shaded area, while the other <math>1 - x</math> fraction is unshaded, so the unshaded portion contributes <math>5(1-x)</math> to the unshaded area. We now equate and solve. <cmath>1 + 5(1-x) = 3 + 5x</cmath> Upon solving, we find that <math>x = \frac{3}{10}</math>, so the degree measure is <math>360 \cdot \frac{3}{10} = \boxed{\textbf{(A)} 108}</math>.
 +
 
 +
~ [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi]
 +
 
 +
==Video Solution 1 by Math-X (First fully understand the problem!!!)==
 +
https://youtu.be/BaE00H2SHQM?si=lg7OGcJ7OwdDFHAn&t=4872
 +
 
 +
~Math-X
 +
 
 +
==Video Solution 2 (A Clever Explanation You’ll Get Instantly)==
 +
https://youtu.be/5ZIFnqymdDQ?si=A8VVtUidVZlXDyrN&t=2517
 +
 
 +
~hsnacademy
 +
 
 +
==Video Solution 3 (super clear!) by Power Solve==
 +
https://youtu.be/TlTN7EQcFvE
 +
 
 +
==Video Solution 4 by SpreadTheMathLove==
 +
https://www.youtube.com/watch?v=Svibu3nKB7E
 +
 
 +
==Video Solution 5 by NiuniuMaths (Easy to understand!)==
 +
https://www.youtube.com/watch?v=V-xN8Njd_Lc
 +
 
 +
~NiuniuMaths
 +
 
 +
==Video Solution 6 by OmegaLearn.org==
 +
https://youtu.be/b_pfNdmLp8A
 +
 
 +
== Video Solution 7 by CosineMethod [🔥Fast and Easy🔥]==
 +
 
 +
https://www.youtube.com/watch?v=ahVNjSlwKmA
 +
 
 +
==Video Solution 8 by Interstigation==
 +
https://youtu.be/ktzijuZtDas&t=2045
 +
 
 +
==Video Solution 9 by Dr. David==
 +
https://youtu.be/ySuCpQZtsZY
 +
 
 +
==Video Solution by WhyMath==
 +
https://youtu.be/v0Aba87nnA4
 +
 
 +
==See Also==
 +
{{AMC8 box|year=2024|num-b=17|num-a=19}}
 +
{{MAA Notice}}

Latest revision as of 13:00, 22 December 2024

Problem

Three concentric circles centered at $O$ have radii of $1$, $2$, and $3$. Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angles $BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle{BOC}$ in degrees?

[asy] size(150); import graph; draw(circle((0,0),3)); real radius = 3; real angleStart = -54; // starting angle of the sector real angleEnd = 54; // ending angle of the sector label("$O$",(0,0),W); pair O = (0, 0); filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, gray); filldraw(circle((0,0),2),gray); filldraw(circle((0,0),1),white); draw((1.763,2.427)--(0,0)--(1.763,-2.427)); label("$B$",(1.763,2.427),NE); label("$C$",(1.763,-2.427),SE); [/asy] $\textbf{(A) } 108\qquad\textbf{(B) } 120\qquad\textbf{(C) } 135\qquad\textbf{(D) } 144\qquad\textbf{(E) } 150$

Solution 1

Let $x=\angle{BOC}$.

We see that the shaded region is the inner ring plus a sector $x^\circ$ of the outer ring. The area of this in terms of $x$ is $\left( 4 \pi - \pi \right)+\frac{x}{360} \left( 9 \pi - 4 \pi \right)$. This simplifies to $3 \pi + \frac{x}{360}(5 \pi)$.

Also, the unshaded portion is comprised of the smallest circle plus the sector $(360-x)^\circ$ of the outer ring. The area of this is $\pi + \frac{360-x}{360}(5 \pi)$.

We are told these are equal, therefore $3 \pi + \frac{x}{360}(5 \pi) = \pi + \frac{360-x}{360}(5 \pi)$. Solving for $x$ reveals $x=\boxed{\textbf{(A) } 108}$.

~MrThinker

~Dash11 (Edited coefficient of pi. Credit goes to MrThinker for the solution and explanation.)

Solution 2

Notice that for the 3rd most outer ring of the circle, the ratio of the shaded region to non-shaded region is the ratio of $\angle{BOC}$ to $360-\angle{BOC}$. With that, all we need to do is solve for the shaded region.

The inner most circle has radius $1$, and the second circle has radius 2. Therefore, the first shaded area has $4 \pi - \pi = 3 \pi$ area. The circle has total area $9 \pi$, so the other shaded region must have $1.5 \pi$ area, as the non-shaded and shaded area is equivalent. So for the 3rd outer ring, the total area is $9 \pi - 4 \pi = 5 \pi$, so the non-shaded part of the outer ring is $5 \pi - 1.5 \pi = 3.5 \pi$.

Now as said before, the ratio of these two areas is the ratio of $\angle{BOC}$ and $360 - \angle{BOC}$. So, $\frac{3.5}{1.5} = \frac{7}{3}$. We have $7x:3x$ where $7x+3x = 360$, $x = 36$, so our answer is $3x = 108, \boxed{(A) 108}$.

~MaxyMoosy

Solution 3 (Ruler Tool)

The AMC 8s allow a ruler tool that you can rotate and drag. You can use the tool to make a straight segment (which we know is $180$ degrees), and we let the angle of desire be $x$. We can estimate that $180-x$ is just about $30$ degrees short of $x$ itself, so $x-30=180-x$, solving gives $x=105$, therefore the closest answer is $\boxed{\textbf{(A)}\,108}$.

  • Note: This isn't the most accurate method to use on AMC 8s, this is just a quick method if time is short or you do not know how to do the problem and want to guess at it.

Solution 4

Suppose the desired angle is some fraction $x$ of the total degree measure of the circle. We now compile a list of the shaded and unshaded areas. The inner circle of radius $1$ is completely unshaded, so it contributes $1$ to the unshaded area. (Everything will be a multiple of $\pi$, so we omit it.) The inner annulus has area $2^2 - 1^2 = 3$, which it contributes to the shaded area. The outer annulus has a total area of $3^2 - 2^2 = 5$; the fraction $x$ is shaded, so the shaded portion of the outer annulus contributes $5x$ to the shaded area, while the other $1 - x$ fraction is unshaded, so the unshaded portion contributes $5(1-x)$ to the unshaded area. We now equate and solve. \[1 + 5(1-x) = 3 + 5x\] Upon solving, we find that $x = \frac{3}{10}$, so the degree measure is $360 \cdot \frac{3}{10} = \boxed{\textbf{(A)} 108}$.

~ cxsmi

Video Solution 1 by Math-X (First fully understand the problem!!!)

https://youtu.be/BaE00H2SHQM?si=lg7OGcJ7OwdDFHAn&t=4872

~Math-X

Video Solution 2 (A Clever Explanation You’ll Get Instantly)

https://youtu.be/5ZIFnqymdDQ?si=A8VVtUidVZlXDyrN&t=2517

~hsnacademy

Video Solution 3 (super clear!) by Power Solve

https://youtu.be/TlTN7EQcFvE

Video Solution 4 by SpreadTheMathLove

https://www.youtube.com/watch?v=Svibu3nKB7E

Video Solution 5 by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=V-xN8Njd_Lc

~NiuniuMaths

Video Solution 6 by OmegaLearn.org

https://youtu.be/b_pfNdmLp8A

Video Solution 7 by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=ahVNjSlwKmA

Video Solution 8 by Interstigation

https://youtu.be/ktzijuZtDas&t=2045

Video Solution 9 by Dr. David

https://youtu.be/ySuCpQZtsZY

Video Solution by WhyMath

https://youtu.be/v0Aba87nnA4

See Also

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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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