If three cubes of copper, each with an edge of 6 cm 8 cm and 10 cm respectively are melted to form a single cube, then the diagonal of the new cube will be

### Concept & Cube Transformation When multiple objects are melted to form a new object, the total volume is conserved. The volume of a cube with edge $a$ is $a^3$, and its space diagonal is given by: $$\text{Diagonal} = a\sqrt{3}$$ ### Step-by-Step Solution 1. **Calculate the volume of the original three cubes:** - Volume of first cube = $6^3 = 216 \text{ cm}^3$ - Volume of second cube = $8^3 = 512 \text{ cm}^3$ - Volume of third cube = $10^3 = 1000 \text{ cm}^3$ 2. **Find the total volume:** - Total Volume = $216 + 512 + 1000 = 1728 \text{ cm}^3$ 3. **Determine the edge of the new cube:** - Let the edge of the new cube be $A$. - $A^3 = 1728$ - $A = \sqrt[3]{1728} = 12 \text{ cm}$ 4. **Calculate the diagonal of the new cube:** - $\text{Diagonal} = A\sqrt{3} = 12\sqrt{3}$ - Knowing $\sqrt{3} \approx 1.732$, we get $12 \times 1.732 = 20.784 \text{ cm}$. 5. Rounding to one decimal place gives $20.8 \text{ cm}$. ### Exam Strategy & Shortcut Memorize the cubes of numbers from 1 to 15. The sum of $6^3 + 8^3 + 10^3$ equals $12^3$. This is a well-known Pythagorean-like relationship for cubes: $6, 8, 10 \rightarrow 12$. Recognizing this instantly gives you the new side $12$. Then calculate $12 \times 1.732$. ### Common Pitfall A frequent error is assuming the diagonal of the new cube is simply the sum of the diagonals of the three smaller cubes, ignoring the volume conservation principle. ### Final Answer Therefore, the correct answer is **20.8 cm**.