1969 AMC 12 第 13 题

Question

半径为

r

的圆位于半径为

R

的圆所围区域内。大圆所围面积是小圆外、大圆内区域面积的

\frac{a}{b}

倍。则

R:r

等于：

A circle with radius

r

is contained within the region bounded by a circle with radius

R.

The area bounded by the larger circle is

\frac{a}{b}

times the area of the region outside the smaller circle and inside the larger circle. Then

R:r

equals:

\sqrt a:\sqrt b

\sqrt a:\sqrt{a-b}

\sqrt b:\sqrt{a-b}

a:\sqrt{a-b}

b:\sqrt{a-b}

写出

\pi R^2=(\frac{a}{b})\pi(R^2-r^2)

Write

\pi R^2=(\frac{a}{b})\pi(R^2-r^2)

先求比值

\frac{R^2}{r^2}

，再开平方

Solve for the ratio

\frac{R^2}{r^2}

before taking square roots

面积条件为

\pi R^2=\frac ab\pi(R^2-r^2)\text{。}

因此

bR^2=aR^2-ar^2

，所以

ar^2=(a-b)R^2

。从而

\frac Rr=\sqrt{\frac a{a-b}}\text{，}

且

R:r=\sqrt a:\sqrt{a-b}

。

因此，正确答案是 B。

The area condition is

\pi R^2=\frac ab\pi(R^2-r^2).

Thus

bR^2=aR^2-ar^2,

so

ar^2=(a-b)R^2.

Therefore

\frac Rr=\sqrt{\frac a{a-b}},

and

R:r=\sqrt a:\sqrt{a-b}.

Therefore, the correct answer is B.

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