A piece of tin is in the form of a rectangle having length 12cm and width 8 cm. This is used to construct a closed cube. The side of the cube is:

Correct Option (C)

The area of the rectangular piece of tin is determined by multiplying its length and width. Given the length as 12 cm and the width as 8 cm, the area of the tin is 12 cm × 8 cm = 96 cm².

When this rectangular tin is utilized to construct a closed cube, the total surface area of the cube must correspond to the area of the tin sheet. The formula for the total surface area of a closed cube with side 'a' is 6a².

Equating the area of the tin sheet to the surface area of the cube:

$$6a^2=96$$

To find the side 'a', we solve the equation:

$$a^2=\frac{96}{6}$$

$$a^2=16$$

$$a=\sqrt{16}$$

$$a=4\text{ cm}$$

Therefore, the side of the constructed cube is 4 cm.

Incorrect Options:

The side of the cube must yield a total surface area of 96 cm² to be consistent with the area of the tin sheet.

• If the side of the cube were 2 cm (Option A), its total surface area would be $6\times (2\text{ cm}{)}^2=6\times 4{\text{ cm}}^2=24{\text{ cm}}^2$, which does not match the available tin area of 96 cm².
• If the side of the cube were 3 cm (Option B), its total surface area would be $6\times (3\text{ cm}{)}^2=6\times 9{\text{ cm}}^2=54{\text{ cm}}^2$, which is not equal to 96 cm².
• If the side of the cube were 6 cm (Option D), its total surface area would be $6\times (6\text{ cm}{)}^2=6\times 36{\text{ cm}}^2=216{\text{ cm}}^2$, which significantly exceeds the available tin area of 96 cm².