Step 1: Understanding the Concept:
This problem deals with the conversion of solids.
When objects are melted and recast, the physical material is preserved, meaning the total volume remains the same.
The task is to find the volumes of the original objects, add them to get the volume of the new cube, find its side length, and finally calculate its surface area.
Volume conservation is the key principle here.
Step 2: Key Formula or Approach:
1. Volume of Cube \( = s^3 \).
2. Volume of Cuboid \( = l \times b \times h \).
3. Total Volume \( V_{new} = V_{cube} + V_{cuboid} \).
4. Side of new cube \( a = \sqrt[3]{V_{new}} \).
5. Total Surface Area (\( TSA \)) \( = 6a^2 \).
Step 2: Detailed Explanation:
Find the volume of the 9 cm metallic cube:
\[ V_1 = 9 \times 9 \times 9 = 729 \text{ cm}^3. \]
Find the volume of the cuboid with dimensions 5 cm, 13 cm, 31 cm:
\[ V_2 = 5 \times 13 \times 31 = 65 \times 31 = 2015 \text{ cm}^3. \]
The total volume of the resulting new single cube is:
\[ V_{total} = 729 + 2015 = 2744 \text{ cm}^3. \]
Let the side of the new cube be \( a \). Then \( a^3 = 2744 \).
To find \( a \), we look for a number whose cube is 2744.
We know \( 10^3 = 1000 \) and \( 20^3 = 8000 \). The number ends in 4, so the cube root should end in 4.
Testing 14: \( 14 \times 14 \times 14 = 196 \times 14 = 2744 \).
Thus, the side length of the new cube \( a = 14 \) cm.
Now, calculate the Total Surface Area of this new cube:
\[ TSA = 6 \times a^2 = 6 \times (14)^2 \]
\[ TSA = 6 \times 196 \]
\[ TSA = 1176 \text{ cm}^2. \]
Step 3: Final Answer:
The total surface area of the new cube is 1176 cm\(^2\).
This corresponds to Option (D).