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

Revision as of 19:24, 27 January 2024

1 Problem
2 Solution
3 Solution 2
4 Video Solution 1 (super clear!) by Power Solve
5 Video Solution 2 by Math-X (First fully understand the problem!!!)
6 Video Solution 3 by SpreadTheMathLove
7 Video Solution by NiuniuMaths (Easy to understand!)
8 Video Solution 2 by OmegaLearn.org
9 Video Solution by CosineMethod [🔥Fast and Easy🔥]
10 See Also

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 angle $BOC$ , 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

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 $\pi + \frac{x}{360}(5 \pi)$ .

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

~MrThinker

Solution 2

Notice 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 $3x = 108, \boxed{A}$ .

~MaxyMoosy

Video Solution 1 (super clear!) by Power Solve

https://youtu.be/TlTN7EQcFvE

Video Solution 2 by Math-X (First fully understand the problem!!!)

https://youtu.be/4Y8GUzNEAJQ

Video Solution 3 by SpreadTheMathLove

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

~Math-X

Video Solution by NiuniuMaths (Easy to understand!)

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

~NiuniuMaths

Video Solution 2 by OmegaLearn.org

https://youtu.be/b_pfNdmLp8A

Video Solution by CosineMethod [🔥Fast and Easy🔥]

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

@@ Line 38: / Line 38: @@
 Notice 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.
-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 = 3pi</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>.
+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>.
 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}</math>.

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