2018 AMC 8 Problems/Problem 15: Difference between revisions

Revision as of 03:29, 16 March 2025

Problem

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?

$[asy] size(4cm); filldraw(scale(2)*unitcircle,gray,black); filldraw(shift(-1,0)*unitcircle,white,black); filldraw(shift(1,0)*unitcircle,white,black); [/asy]$

$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}$

Solution 1

Let the radius of the large circle be $R$ . Then, the radius of the smaller circles are $\frac R2$ . The areas of the circles are directly proportional to the square of the radii, so the ratio of the area of the small circle to the large one is $\frac 14$ ( $\frac {R^2}{4}$ is $\frac 14$ of $R^2$ .) This means the combined area of the 2 smaller circles is half of the larger circle, and therefore the shaded region is equal to the combined area of the 2 smaller circles, which is $\boxed{\textbf{(D) } 1}$ .

Solution 2

Let the radius of the two smaller circles be $r$ . It follows that the area of one of the smaller circles is ${\pi}r^2$ . Thus, the area of the two inner circles combined would evaluate to $2{\pi}r^2$ which is $1$ . Since the radius of the bigger circle is two times that of the smaller circles (the diameter), the radius of the larger circle in terms of $r$ would be $2r$ . The area of the larger circle would come to $(2r)^2{\pi} = 4{\pi}r^2$ .

Subtracting the area of the smaller circles from that of the larger circle (since that would be the shaded region), we have $4{\pi}r^2 - 2{\pi}r^2 = 2{\pi}r^2 = 1.$

Therefore, the area of the shaded region is $\boxed{\textbf{(D) } 1}$ .

Solution 3

The area of a small circle is $\frac{1}{2}= \pi r^2$ . Solving, we get $r = \sqrt{\frac{1}{2\pi}}$ .

The radius of the large circle is $R=2r$ . The area of the large circle is ${\pi}R^2={\pi}(2r)^2=4{\pi}r^2=4{\pi}\frac{1}{2\pi}=2$ .

Subtract the area of the small circles from the area of the large circle to get the area of the shaded region: $2-1=$ $\boxed{\textbf{(D) } 1}$ .

Solution 4 (Similar to Solution 3)

To get the area of the small circles, we can get the equation $\frac{1}{2}= \pi r^2$ . Solving for $r$ , we get $r = \sqrt{\frac{1}{2\pi}}$ . Then, we can get the radius of the big circle by doubling the small circle's radius, and that gives $2\sqrt{\frac{1}{2\pi}}$ . Because the area of a circle is $\pi r^2$ , you'll have to square the answer and multiple it by $\pi$ . After you square it, you'll get $4 \cdot \frac{1}{2\pi}$ , which equals to $\frac{4}{2\pi}$ . Since the area of a circle is $\pi r^2$ , we get $\pi \cdot \frac{4}{2\pi}$ , and that equals $\frac{4\pi}{2\pi}$ , which equals 2. Since each of the circles' area is $\frac{1}{2}$ , the combined area of the small circles is $1$ . Since $2-1=1$ , the area of the shaded region is $\boxed{\textbf{(D) } 1}$ .

Video Solution (CREATIVE ANALYSIS!!!)

https://youtu.be/tYfMj2SSVJc

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Video Solutions

https://youtu.be/-3WEf3EjGu0

https://youtu.be/-JR7R0PyU-w

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Video Solution by OmegaLearn

https://youtu.be/51K3uCzntWs?t=1474

~ pi_is_3.14

@@ Line 34: / Line 34: @@
 ==Solution 4 (Similar to Solution 3)==
 To get the area of the small circles, we can get the equation <math>\frac{1}{2}= \pi r^2</math>. Solving for <math>r</math>, we get <math>r = \sqrt{\frac{1}{2\pi}}</math>. Then, we can get the radius of the big circle by doubling the small circle's radius, and that gives
-<math>2\sqrt{\frac{1}{2\pi}}</math>. Because the area of a circle is <math>\pi r^2</math>, you'll have to square the answer and multiple it by <math>\pi</math>. After you square it, you'll get <math>4 \cdot \frac{1}{2\pi}</math>, which equals to <math>\frac{4}{2\pi}</math>. Since the area of a circle is <math>\pi r^2</math>, we get <math>\pi \cdot \frac{4}{2\pi}</math>, and that equals <math>\frac{4\pi}{2\pi}</math>, which equals 2. Since each of the circles' area is <math>\frac{1}{2}</math>, the combined area of the small circles is <math>1</math>. Since <math>2-1=1</math>, the area of the shaded region is <math>\boxed{\textbf{(D) } 1}</math>..
+<math>2\sqrt{\frac{1}{2\pi}}</math>. Because the area of a circle is <math>\pi r^2</math>, you'll have to square the answer and multiple it by <math>\pi</math>. After you square it, you'll get <math>4 \cdot \frac{1}{2\pi}</math>, which equals to <math>\frac{4}{2\pi}</math>. Since the area of a circle is <math>\pi r^2</math>, we get <math>\pi \cdot \frac{4}{2\pi}</math>, and that equals <math>\frac{4\pi}{2\pi}</math>, which equals 2. Since each of the circles' area is <math>\frac{1}{2}</math>, the combined area of the small circles is <math>1</math>. Since <math>2-1=1</math>, the area of the shaded region is <math>\boxed{\textbf{(D) } 1}</math>.
 ==Video Solution (CREATIVE ANALYSIS!!!)==

2018 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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