A rectangle has the same perimeter and area as a rhombus, one of whose angles is $30^{\circ}$. What is the ratio of the sides of the rectangle?
Let the sides of the rectangle be $a$ and $b$, and the side of the rhombus be $c$. The perimeter of the rectangle is $2(a+b)$; its area is $a b$. The perimeter of the rhombus is $4 c$; its area is $c^{2} \sin 30^{\circ}=\frac{c^{2}}{2}$. From the equality of the perimeters and areas, we can write the following system of equations:

$$
a+b=2 c, \quad a b=\frac{c^{2}}{2}, \quad \text { hence } \quad a b=\frac{1}{2}\left(\frac{a+b}{2}\right)^{2}
$$

Perform the operations, we get $a^{2}-6 a b+b^{2}=0$. Divide the equation by $b^{2} \neq 0$:

$$
\left(\frac{a}{b}\right)^{2}-6\left(\frac{a}{b}\right)+1=0
$$

From this, $\frac{a}{b}=3+2 \sqrt{2}$, and $3-2 \sqrt{2}$. The ratio of the sides of the rectangle is $3+2 \sqrt{2} \approx 5.83$ (or $\frac{1}{3+2 \sqrt{2}}=3-2 \sqrt{2} \approx 0.17$).