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# AMC10每日一题（2002年真题#06）

• 2018-08-01
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今天课窝小编为大家整理了AMC10真题练习，希望考生们认真阅读，能够对你的考试有所帮助。

## Problem

$45^\circ$ 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?

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

## 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 $\boxed{\text{(A)}\ 4/9}$.

### Solution 2

Let $c_1$ and $c_2$ be the circumference of circles $A$ and $B$, respectively.

The length of a $45^{\circ}$ arc of circle $A$ is $\frac{c_1}{\frac{360}{45}}=\frac{c_1}{8}$, and the length of a $30^{\circ}$ arc of circle $B$ is $\frac{c_2}{\frac{360}{30}}=\frac{c_2}{12}$. We know that the length of a $45^{\circ}$ arc on circle $A$ is equal to the length of a $30^{\circ}$ arc of circle $B$, so $\frac{c_1}{8}=\frac{c_2}{12}$. Manipulating the equation, we get $\frac{c_1}{c_2}=\frac{8}{12}=\frac{2}{3}$. Because the ratio of the areas is equal to the ratio of the circumferences squared, our answer is $\frac{2^2}{3^2}=\boxed{\text{(A)}\ 4/9}$

### Solution 3

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

## 问题

$45 ^ \ 保监会$圆A的弧的长度等于一$30 ^ \ 保监会$圆B的弧什么是圆A的面积和圆B的面积之比？

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

## 解决方案

### 解决方案1

$R_1$$R_2$是圆的半径$A$$B$分别。

### 解决方案2

$C_1$$C_2$是圆的周长$A$$B$分别。

### 解决方案3

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