1958 AMC 12 第 18 题

Question

圆的半径

r

增加

n

后，面积变为原来的两倍。则

r

等于：

The area of a circle is doubled when its radius

r

is increased by

n.

Then

r

equals:

n(\sqrt2+1)

n(\sqrt2-1)

n

n(2-\sqrt2)

\dfrac{n\pi}{\sqrt2+1}

将面积加倍的条件写成

\pi(r+n)^2=2\pi r^2

Translate the doubled-area condition into

\pi(r+n)^2=2\pi r^2

取正平方根后，解方程

r+n=\sqrt2\,r

After taking positive square roots, solve

r+n=\sqrt2\,r

面积加倍的条件为

\pi(r+n)^2=2\pi r^2\text{。}

由于半径为正，

r+n=\sqrt2\,r

。因此

r=\frac{n}{\sqrt2-1} =n(\sqrt2+1)\text{。}

所以正确答案为 A。

The doubled-area condition is

\pi(r+n)^2=2\pi r^2.

Since the radii are positive,

r+n=\sqrt2\,r.

Thus

r=\frac{n}{\sqrt2-1} =n(\sqrt2+1).

Thus, the correct answer is A.

其他年份的第 18 题