Difference between revisions of "2002 AMC 12A Problems/Problem 7"

Latest revision as of 23:51, 25 July 2024

The following problem is from both the 2002 AMC 12A #7 and 2002 AMC 10A #7, so both problems redirect to this page.

Problem

A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?

$\textbf{(A)}\ 4/9 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 5/6 \qquad \textbf{(D)}\ 3/2 \qquad \textbf{(E)}\ 9/4$

Solutions

Solution 1

Let $r_1$ and $r_2$ be the radii of circles $A$ and $B$ , respectively.

It is well known that in a circle with radius $r$ , a subtended arc opposite an angle of $\theta$ degrees has length $\frac{\theta}{360} \cdot 2\pi r$ .

Using that here, the arc of circle A has length $\frac{45}{360}\cdot2\pi{r_1}=\frac{r_1\pi}{4}$ . The arc of circle B has length $\frac{30}{360} \cdot 2\pi{r_2}=\frac{r_2\pi}{6}$ . We know that they are equal, so $\frac{r_1\pi}{4}=\frac{r_2\pi}{6}$ , so we multiply through and simplify to get $\frac{r_1}{r_2}=\frac{2}{3}$ . As all circles are similar to one another, the ratio of the areas is just the square of the ratios of the radii, so our answer is $\boxed{\textbf{(A) } 4/9}$ .

Solution 2

The arc of circle $A$ is $\frac{45}{30}=\frac{3}{2}$ that of circle $B$ .

The circumference of circle $A$ is $\frac{2}{3}$ that of circle $B$ ( $B$ is the larger circle).

The radius of circle $A$ is $\frac{2}{3}$ that of circle $B$ .

The area of circle $A$ is ${\left(\frac{2}{3}\right)}^2=\boxed{\textbf{(A) } 4/9}$ that of circle $B$ .

Solution 3

We can use a very cheap way to find the answer based on the choices. Firstly, if a higher angle measure equals a lower angle measure, that means that the ratio of areas must be lower, cutting D and E. Then we realize that the ratio has to be with square numbers, as there are no square roots. Therefore, it is $(A)$

-dragoon

Solution 4

Written by a 6th grader, please don't mind*

Set Circle A's arc to 45 degrees. In a full circle, there is 360 degrees, so 360/45 = 8. If we set the arc to be pi degrees, then in total there will be 8 pi in the circumference of this circle, yielding 4 as the radius and 16pi as the area.

Since the arc of 30 degrees in Circle B is also pi, and there are 12 30 degree arcs in a circle, there will be 12pi in the circumference of Circle B, yielding 6 as the radius and 36pi as the area.

Now, we put 16pi/36pi as our ratio, yielding 16/36=4/9. Therefore, it is $(A)$ 4/9.

~MathKatana

Video Solution by Daily Dose of Math

https://youtu.be/BEn0GJOFbU8

~Thesmartgreekmathdude

2002 AMC 12A (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 12 Problems and Solutions

2002 AMC 10A (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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

@@ Line 14: / Line 14: @@
 It is well known that in a circle with radius <math> r </math>, a subtended arc opposite an angle of <math>\theta</math> degrees has length <math>\frac{\theta}{360} \cdot 2\pi r</math>.
-Using that here, the arc of circle A has length <math>\frac{45}{360}\cdot2\pi{r_1}=\frac{r_1\pi}{4}</math>. The arc of circle B has length <math>\frac{30}{360} \cdot 2\pi{r_2}=\frac{r_2\pi}{6}</math>. We know that they are equal, so <math>\frac{r_1\pi}{4}=\frac{r_2\pi}{6}</math>, so we multiply through and simplify to get <math>\frac{r_1}{r_2}=\frac{2}{3}</math>. As all circles are similar to one another, the ratio of the areas is just the square of the ratios of the radii, so our answer is <math>\boxed{\text{(A)}\ 4/9}</math>.
+Using that here, the arc of circle A has length <math>\frac{45}{360}\cdot2\pi{r_1}=\frac{r_1\pi}{4}</math>. The arc of circle B has length <math>\frac{30}{360} \cdot 2\pi{r_2}=\frac{r_2\pi}{6}</math>. We know that they are equal, so <math>\frac{r_1\pi}{4}=\frac{r_2\pi}{6}</math>, so we multiply through and simplify to get <math>\frac{r_1}{r_2}=\frac{2}{3}</math>. As all circles are similar to one another, the ratio of the areas is just the square of the ratios of the radii, so our answer is <math>\boxed{\textbf{(A) } 4/9}</math>.
 ===Solution 2===
@@ Line 23: / Line 23: @@
 The radius of circle <math>A</math> is <math>\frac{2}{3}</math> that of circle <math>B</math>.
-The area of circle <math>A</math> is <math>{\left(\frac{2}{3}\right)}^2=\boxed{\text{(A)}\ 4/9}</math> that of circle <math>B</math>.
+The area of circle <math>A</math> is <math>{\left(\frac{2}{3}\right)}^2=\boxed{\textbf{(A) } 4/9}</math> that of circle <math>B</math>.
+===Solution 3===
+We can use a very cheap way to find the answer based on the choices. Firstly, if a higher angle measure equals a lower angle measure, that means that the ratio of areas must be lower, cutting D and E. Then we realize that the ratio has to be with square numbers, as there are no square roots. Therefore, it is <math>(A)</math>
+-dragoon
+===Solution 4===
+*Written by a 6th grader, please don't mind*
+Set Circle A's arc to 45 degrees. In a full circle, there is 360 degrees, so 360/45 = 8. If we set the arc to be pi degrees, then in total there will be 8 pi in the circumference of this circle, yielding 4 as the radius and 16pi as the area.
+Since the arc of 30 degrees in Circle B is also pi, and there are 12 30 degree arcs in a circle, there will be 12pi in the circumference of Circle B, yielding 6 as the radius and 36pi as the area.
+Now, we put 16pi/36pi as our ratio, yielding 16/36=4/9. Therefore, it is <math>(A)</math>4/9.
+~MathKatana
+==Video Solution by Daily Dose of Math==
+https://youtu.be/BEn0GJOFbU8
+~Thesmartgreekmathdude
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