Difference between revisions of "2002 AMC 12A Problems/Problem 7"
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A <math>45^\circ</math> arc of circle A is equal in length to a <math>30^\circ</math> arc of circle B. What is the ratio of circle A's area and circle B's area? | A <math>45^\circ</math> arc of circle A is equal in length to a <math>30^\circ</math> arc of circle B. What is the ratio of circle A's area and circle B's area? | ||
− | <math>\ | + | <math>\textbf{(A)}\ 4/9 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 5/6 \qquad \textbf{(D)}\ 3/2 \qquad \textbf{(E)}\ 9/4</math> |
==Solutions== | ==Solutions== | ||
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Let <math>r_1</math> and <math>r_2</math> be the radii of circles <math>A</math> and<math> B</math>, respectively. | Let <math>r_1</math> and <math>r_2</math> be the radii of circles <math>A</math> and<math> B</math>, respectively. | ||
− | 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>. | + | 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{\ | + | 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>. |
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===Solution 2=== | ===Solution 2=== | ||
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The radius of circle <math>A</math> is <math>\frac{2}{3}</math> that of circle <math>B</math>. | 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{\ | + | 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>. |
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+ | ===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 | ||
==See Also== | ==See Also== |
Latest revision as of 00:51, 26 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.
Contents
Problem
A arc of circle A is equal in length to a
arc of circle B. What is the ratio of circle A's area and circle B's area?
Solutions
Solution 1
Let and
be the radii of circles
and
, respectively.
It is well known that in a circle with radius , a subtended arc opposite an angle of
degrees has length
.
Using that here, the arc of circle A has length . The arc of circle B has length
. We know that they are equal, so
, so we multiply through and simplify to get
. 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
.
Solution 2
The arc of circle is
that of circle
.
The circumference of circle is
that of circle
(
is the larger circle).
The radius of circle is
that of circle
.
The area of circle is
that of circle
.
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
-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 4/9.
~MathKatana
Video Solution by Daily Dose of Math
~Thesmartgreekmathdude
See Also
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.