AMC12 2008 A

AMC12 2008 A · Q8

AMC12 2008 A · Q8. It mainly tests Exponents & radicals, 3D geometry (volume).

What is the volume of a cube whose surface area is twice that of a cube with volume 1?

一个立方体的表面积是体积为 1 的立方体表面积的两倍。这个立方体的体积是多少？

(A) \sqrt{2} \sqrt{2}

(B) 2 2

(D) 4 4

(E) 8 8

Answer

Correct choice: (C)

正确答案：（C）

Solution

A cube with volume $1$ has a side of length $\sqrt[3]{1}=1$ and thus a surface area of $6 \cdot 1^2=6$. A cube whose surface area is $6\cdot2=12$ has a side of length $\sqrt{\frac{12}{6}}=\sqrt{2}$ and a volume of $(\sqrt{2})^3=2\sqrt{2}\Rightarrow\mathrm{(C)}$. Alternatively, we can use the fact that the surface area of a cube is directly proportional to the square of its side length. Therefore, if the surface area of a cube increases by a factor of $2$, its side length must increase by a factor of $\sqrt{2}$, meaning the new side length of the cube is $1 * \sqrt{2} = \sqrt{2}$. So, its volume is $({\sqrt{2}})^3 = 2\sqrt{2} \Rightarrow\mathrm{(C)}$.

体积为 $1$ 的立方体边长为 $\sqrt[3]{1}=1$，因此表面积为 $6 \cdot 1^2=6$。表面积为 $6\cdot2=12$ 的立方体边长为 $\sqrt{\frac{12}{6}}=\sqrt{2}$，体积为 $(\sqrt{2})^3=2\sqrt{2}\Rightarrow\mathrm{(C)}$。或者，利用立方体表面积与边长的平方成正比：表面积增大 $2$ 倍，则边长增大 $\sqrt{2}$ 倍，新边长为 $1 * \sqrt{2} = \sqrt{2}$。因此体积为 $({\sqrt{2}})^3 = 2\sqrt{2} \Rightarrow\mathrm{(C)}$。

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