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Let the sides of the rectangle be $x$ and $y$. The perimeter is given as 36 cm. Thus, $2(x + y) = 36$, which simplifies to $x + y = 18$, or $y = 18 - x$.
When the rectangle is revolved around side $x$, the side $x$ becomes the height ($h$) of the cylinder and side $y$ becomes the radius ($r$). The volume $V$ of the cylinder is given by: $$V = \pi r^2 h = \pi y^2 x$$ Substituting $y = 18 - x$: $$V(x) = \pi (18 - x)^2 x = \pi (324x - 36x^2 + x^3)$$
To maximize volume, differentiate $V$ with respect to $x$: $$\frac{dV}{dx} = \pi (324 - 72x + 3x^2)$$ Set $\frac{dV}{dx} = 0$: $$3(x^2 - 24x + 108) = 0$$ $$3(x - 6)(x - 18) = 0$$ The critical points are $x = 6$ and $x = 18$. Since $x = 18$ would result in $y = 0$ (no volume), we take $x = 6$.
Using the second derivative test: $$\frac{d^2V}{dx^2} = \pi (-72 + 6x)$$ At $x = 6$, $\frac{d^2V}{dx^2} = \pi (-72 + 36) = -36\pi < 0$. Thus, $x = 6$ is a point of local maxima. If $x = 6$, then $y = 18 - 6 = 12$.
Final Answer: The dimensions of the rectangle are 6 cm and 12 cm.
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