Two cubes of sizes 4 cm and 6 cm are melted to form a cuboid of length 10 cm and breadth 8 cm. What is the height of the cuboid so formed?

Question

Accepted Answer

To determine the height of the cuboid formed by melting two cubes, we need to first calculate the total volume of the cubes and then use that volume to find the height of the cuboid.

1. **Calculate the 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.
   - For the first cube with a side length of 4 cm:
     $$
     V_1 = 4^3 = 64 \text{ cubic centimeters}
     $$
   - For the second cube with a side length of 6 cm:
     $$
     V_2 = 6^3 = 216 \text{ cubic centimeters}
     $$

2. **Calculate the total volume of the two cubes:**
   $$
   V_{\text{total}} = V_1 + V_2 = 64 + 216 = 280 \text{ cubic centimeters}
   $$

3. **Use the total volume to find the height of the cuboid:**
   - The volume of a cuboid is given by the formula $ V = \text{length} \times \text{breadth} \times \text{height} $.
   - We know the length and breadth of the cuboid are 10 cm and 8 cm, respectively.
   - Let $ h $ be the height of the cuboid. Then:
     $$
     280 = 10 \times 8 \times h
     $$
   - Solving for $ h $:
     $$
     280 = 80h \quad \Rightarrow \quad h = \frac{280}{80} = 3.5 \text{ cm}
     $$

Therefore, the height of the cuboid is 3.5 cm, which corresponds to Option 3.

**Explanation for why other options are incorrect:**

- **Option 1 (4.5 cm):** If the height were 4.5 cm, the volume of the cuboid would be $ 10 \times 8 \times 4.5 = 360 \text{ cubic centimeters} $, which is greater than the total volume of the cubes (280 cubic centimeters). Therefore, this option is incorrect.

- **Option 2 (3.8 cm):** If the height were 3.8 cm, the volume of the cuboid would be $ 10 \times 8 \times 3.8 = 304 \text{ cubic centimeters} $, which is also greater than the total volume of the cubes. Thus, this option is incorrect.

- **Option 4 (2.5 cm):** If the height were 2.5 cm, the volume of the cuboid would be $ 10 \times 8 \times 2.5 = 200 \text{ cubic centimeters} $, which is less than the total volume of the cubes. Hence, this option is incorrect.

In conclusion, Option 3 (3.5 cm) is the only correct answer as it accurately reflects the height of the cuboid formed by the given volumes.