A rectangular box with a square base is to have a volume of 1000 cubic centimeters. Find the dimensions of the box that minimize the total surface area.
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Let the dimensions of the box be $x$, $x$, and $y$, where $x$ is the length of a side of the square base and $y$ is the height. Then the volume of the box is $V = x^2 y = 1000$, so $xy = 1000/x = \frac{1000}{x}$. The surface area of the box is
\[A = 2x^2 + 4xy.\]To minimize $A$, we must minimize the sum $x^2 + 2xy = x^2 + \frac{2000}{x}$. By AM-GM,
\begin{align*}
x^2 + \frac{2000}{x} &= \left( x^2 + \frac{1000}{x} \right) + \frac{1000}{x} \\
&\ge 2 \sqrt{x^2 \cdot \frac{1000}{x}} + \frac{1000}{x} \\
&= 20x + \frac{1000}{x}.
\end{align*}Equality occurs when $x^2 = \frac{1000}{x},$ which simplifies to
\[x^3 = 1000,\]so $x = 10$. Then $y = \frac{1000}{x} = 100$, so the dimensions are $\boxed{10, 10, 100}.$
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