Question

A solid metallic cube having surface area 864 cm\(^2\) is melted to form 64 small solid cubes of the same size. The sum of the surface areas of all the small cubes so formed (in cm\(^2\)) is:

• A. 3200
• B. 3328
• C. 3456
• D. 3584

Hint:

To solve cube-melting problems: 1. Use original surface area to find side and then volume. 2. Divide volume among smaller cubes to get new side length. 3. Use the formula \( 6a^2 \) for surface area.

Answer 1

The Correct Option is C

Let side of original cube be \( a \). We know surface area of cube = \( 6a^2 = 864 \Rightarrow a^2 = 144 \Rightarrow a = 12 \text{ cm} \) Volume of original cube: \[ = a^3 = 12^3 = 1728 \text{ cm}^3 \] This is melted into 64 smaller cubes. So: \[ \text{Volume of one small cube} = \frac{1728}{64} = 27 \Rightarrow \text{side} = \sqrt[3]{27} = 3 \text{ cm} \] Surface area of one small cube: \[ = 6 \times 3^2 = 6 \times 9 = 54 \text{ cm}^2 \] So, total surface area = \( 54 \times 64 = 3456 \text{ cm}^2 \)