Two cubes each of volume 64 cm$^3$ are joined end to end. Then the surface area of the resulting cuboid is :

Question

Accepted Answer

To determine the surface area of the resulting cuboid when two cubes, each with a volume of 64 cm³, are joined end to end, we first need to find the dimensions of the cubes and then the dimensions of the resulting cuboid.

1. **Volume of Each Cube:**
   - The volume of a cube is given by the formula $ V = a^3 $, where $ a $ is the side length of the cube.
   - Given that the volume of each cube is 64 cm³, we have:
     $$
     a^3 = 64
     $$
   - Solving for $ a $, we find:
     $$
     a = \sqrt[3]{64} = 4 \text{ cm}
     $$
   - Therefore, each cube has side lengths of 4 cm.

2. **Dimensions of the Resulting Cuboid:**
   - When two cubes are joined end to end, the resulting shape is a cuboid.
   - The length of the cuboid becomes $ 4 \text{ cm} + 4 \text{ cm} = 8 \text{ cm} $.
   - The width and height remain the same as the side of the cube, which is 4 cm each.
   - Thus, the dimensions of the cuboid are 8 cm (length) × 4 cm (width) × 4 cm (height).

3. **Surface Area of the Cuboid:**
   - The surface area $ S $ of a cuboid is calculated using the formula:
     $$
     S = 2(lw + lh + wh)
     $$
   - Substituting the dimensions of the cuboid:
     $$
     S = 2(8 \times 4 + 8 \times 4 + 4 \times 4)
     $$
     $$
     S = 2(32 + 32 + 16)
     $$
     $$
     S = 2 \times 80 = 160 \text{ cm}^2
     $$

4. **Justification for Option 3:**
   - The calculated surface area of the resulting cuboid is 160 cm², which matches Option 3.

5. **Explanation for Incorrectness of Other Options:**
   - **Option 1: 140 cm²** - This value is less than the calculated surface area of 160 cm². It does not account for all the surfaces of the cuboid.
   - **Option 2: 148 cm²** - This value is also less than the calculated surface area. It is incorrect as it does not match the formula-based calculation.
   - **Option 4: 176 cm²** - This value exceeds the calculated surface area. It likely includes an overestimation of the surface area.

In conclusion, the correct surface area of the resulting cuboid is 160 cm², which is why Option 3 is the correct answer.