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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。


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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。

众所周知,在具有半径的圆中,与$ R $角度相反的对向弧$ \ $ THETA具有长度$ \ frac {\ theta} {360} \ cdot 2 \ pi r $

在此使用它,圆圈A的弧度具有长度$ \压裂{45} {360} \ cdot2 \ PI {R_1} = \压裂{R_1 \ PI} {4} $。圆弧B具有长度$ \ frac {30} {360} \ cdot 2 \ pi {r_2} = \ frac {r_2 \ pi} {6} $。我们知道它们是平等的,所以$ \压裂{R_1 \ PI} {4} = \压裂{R_2 \ PI} {6} $,我们通过并简化得到$ \压裂{R_1} {R_2} = \压裂{2} {3} $。由于所有圆圈彼此相似,因此面积的比率只是半径比率的平方,所以我们的答案是$ \ boxed {\ text {(A)} \ 4/9} $

解决方案2

$ C_1 $$ C_2 $是圆的周长$ A $$ B $分别。

一个的长度$ 45 ^ {\保监会} $的圆的弧$ A $$ \压裂{C_1} {\压裂{360} {45}} = \压裂{C_1} {8} $,和的长度$ 30 ^ {\保监会} $的圆的弧$ B $$ \压裂{C_2} {\压裂{360} {30}} = \压裂{C_2} {12} $。我们知道$ 45 ^ {\保监会} $圆上圆弧$ A $的长度等于$ 30 ^ {\保监会} $圆弧的长度$ B $,所以$ \压裂{C_1} {8} = \压裂{C_2} {12} $。操纵方程,我们得到$ \压裂{C_1} {C_2} = \压裂{8} {12} = \压裂{2} {3} $。因为面积的比率等于圆周平方的比率,我们的答案是$ \ frac {2 ^ 2} {3 ^ 2} = \ boxed {\ text {(A)} \ 4/9} $

解决方案3

圆弧$ A $$ \压裂{45} {30} = \压裂{3} {2} $那个圈子的$ B $

圆的周长$ A $$ \压裂{2} {3} $该圆的$ B $$ B $是较大的圆圈)。

圆的半径$ A $$ \压裂{2} {3} $那个圈子的$ B $

圆的面积$ A $$ {\ left(\ frac {2} {3} \ right)} ^ 2 = \ boxed {\ text {(A)} \ 4/9} $该圆的$ B $

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