A solid metallic cuboid of dimensions `18cmxx36cmxx72cm` is melted and recast into 8 cubes of the same volume. What is the ratio of the total surface area of the cuboid to the sum of the lateral surface areas of all 8 cubes?

Options

A. `2:3`
B. `7:12`
C. `4:7`
D. `7:8`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is given by the formula:
\[ V = l \times b \times h \]
Where:
- \( l = 18 \, \text{cm} \)
- \( b = 36 \, \text{cm} \)
- \( h = 72 \, \text{cm} \)

Substituting the values:
\[ V = 18 \times 36 \times 72 \]

### Step 2: Calculate the Volume of One Cube
Since the cuboid is melted and recast into 8 cubes of the same volume, the volume of one cube \( V_{\text{cube}} \) is:
\[ V_{\text{cube}} = \frac{V}{8} \]

### Step 3: Find the Side Length of One Cube
Let the side length of the cube be \( a \). The volume of one cube is also given by:
\[ V_{\text{cube}} = a^3 \]
Setting the two expressions for \( V_{\text{cube}} \) equal gives:
\[ a^3 = \frac{V}{8} \]

### Step 4: Calculate the Total Surface Area of the Cuboid
The total surface area \( A_{\text{cuboid}} \) of the cuboid is given by:
\[ A_{\text{cuboid}} = 2(lb + bh + lh) \]
Substituting the values:
\[ A_{\text{cuboid}} = 2(18 \times 36 + 36 \times 72 + 72 \times 18) \]

### Step 5: Calculate the Lateral Surface Area of One Cube
The lateral surface area \( A_{\text{lateral}} \) of one cube is given by:
\[ A_{\text{lateral}} = 4a^2 \]
For 8 cubes, the total lateral surface area is:
\[ A_{\text{lateral, total}} = 8 \times 4a^2 = 32a^2 \]

### Step 6: Calculate the Ratio of Total Surface Area of the Cuboid to the Total Lateral Surface Area of the Cubes
The required ratio \( R \) is:
\[ R = \frac{A_{\text{cuboid}}}{A_{\text{lateral, total}}} \]
Substituting the expressions we derived:
\[ R = \frac{2(18 \times 36 + 36 \times 72 + 72 \times 18)}{32a^2} \]

### Step 7: Substitute \( a^2 \)
From \( a^3 = \frac{V}{8} \), we can find \( a^2 \):
1. Calculate \( V = 18 \times 36 \times 72 \).
2. Find \( a = \sqrt[3]{\frac{V}{8}} \).
3. Substitute \( a^2 \) back into the ratio.

### Step 8: Simplify the Ratio
After substituting and simplifying, we will arrive at the final ratio.

### Final Answer
After calculating, we find that the ratio of the total surface area of the cuboid to the sum of the lateral surface areas of all 8 cubes is:
\[ \text{Ratio} = \frac{7}{8} \]