Three cubes of metal, whose edges are 3, 4, 5 cm respectively, are melted and formed into a single cube. The edge of the new cube is equal to :

Question

Accepted Answer

To determine the edge of the new cube formed by melting three smaller cubes with edges of 3 cm, 4 cm, and 5 cm, we need to follow these steps:

1. **Calculate the Volume of Each Cube:**
   - The volume of a cube is given by the formula $ V = a^3 $, where $ a $ is the edge length of the cube.
   - For the first cube with an edge of 3 cm: 
     $$
     V_1 = 3^3 = 27 \, \text{cm}^3
     $$
   - For the second cube with an edge of 4 cm:
     $$
     V_2 = 4^3 = 64 \, \text{cm}^3
     $$
   - For the third cube with an edge of 5 cm:
     $$
     V_3 = 5^3 = 125 \, \text{cm}^3
     $$

2. **Calculate the Total Volume:**
   - The total volume of the metal after melting all three cubes is the sum of their individual volumes:
     $$
     V_{\text{total}} = V_1 + V_2 + V_3 = 27 + 64 + 125 = 216 \, \text{cm}^3
     $$

3. **Determine the Edge of the New Cube:**
   - Let the edge of the new cube be $ a $. The volume of the new cube is given by $ a^3 $.
   - Since the total volume of the melted metal is 216 cm³, we have:
     $$
     a^3 = 216
     $$
   - Solving for $ a $, we find:
     $$
     a = \sqrt[3]{216} = 6 \, \text{cm}
     $$

Therefore, the edge of the new cube is 6 cm, which corresponds to Option 1.

**Explanation of Why Other Options Are Incorrect:**

- **Option 2 (4 cm):** If the edge of the new cube were 4 cm, the volume would be $ 4^3 = 64 \, \text{cm}^3 $, which is far less than the total volume of 216 cm³. Therefore, this option is incorrect.

- **Option 3 (5 cm):** If the edge of the new cube were 5 cm, the volume would be $ 5^3 = 125 \, \text{cm}^3 $, which is also less than the total volume of 216 cm³. Thus, this option is incorrect.

- **Option 4 (7 cm):** If the edge of the new cube were 7 cm, the volume would be $ 7^3 = 343 \, \text{cm}^3 $, which exceeds the total volume of 216 cm³. Hence, this option is incorrect.

In conclusion, the only option that satisfies the condition of the total volume being 216 cm³ is Option 1, with an edge length of 6 cm.