13.147. A sheet of metal in the form of a rectangle, with the ratio of length to width being $2: 1$. From this sheet, an open-top box is made by cutting out a square with a side of 3 cm from each corner and folding up the resulting flaps. Determine the dimensions of the metal sheet if the volume of the box turned out to be $168 \mathrm{~cm}^{3}$.
Solution.

Let $2a$ cm be the length of the sheet, and $a$ cm be its width. The length of the box is $(2a-6)$ cm, its width is $(a-6)$ cm, and its height is 3 cm. The volume of the box is $3(a-6)(2a-6)=168$, from which we find $a=10$ cm is the width of the sheet, and 20 cm is its length.

Answer: $10 \times 20$ cm.