If a cuboid of dimensions `32 cm xx 12 cm xx 9 cm` is cut into two cubes of same size, what will be the ratio of the surface area of the cuboid to the total surface area of the two cubes?

Options

A. ` 32:39`
B. `37:48`
C. `24:35`
D. `65:72`

Answer

To solve the problem, we need to find the ratio of the surface area of a cuboid to the total surface area of two cubes created from that cuboid. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the Volume of the Cuboid
The dimensions of the cuboid are given as:
- Length (L) = 32 cm
- Breadth (B) = 12 cm
- Height (H) = 9 cm

The volume (V) of the cuboid is calculated using the formula:
\[ V = L \times B \times H \]

Substituting the values:
\[ V = 32 \times 12 \times 9 \]
\[ V = 3456 \, \text{cm}^3 \]

### Step 2: Determine the Volume of One Cube
Since the cuboid is cut into two cubes of the same size, the volume of one cube (V_cube) is half the volume of the cuboid:
\[ V_{\text{cube}} = \frac{V}{2} = \frac{3456}{2} = 1728 \, \text{cm}^3 \]

### Step 3: Calculate the Side Length of One Cube
The volume of a cube is given by:
\[ V_{\text{cube}} = A^3 \]
where A is the side length of the cube. We can find A by taking the cube root of the volume:
\[ A^3 = 1728 \]
\[ A = \sqrt[3]{1728} = 12 \, \text{cm} \]

### Step 4: Calculate the Surface Area of the Cuboid
The surface area (SA) of the cuboid is calculated using the formula:
\[ SA = 2(LB + BH + HL) \]

Substituting the values:
\[ SA = 2(32 \times 12 + 12 \times 9 + 9 \times 32) \]
Calculating each term:
- \( 32 \times 12 = 384 \)
- \( 12 \times 9 = 108 \)
- \( 9 \times 32 = 288 \)

Now, summing these:
\[ SA = 2(384 + 108 + 288) = 2(780) = 1560 \, \text{cm}^2 \]

### Step 5: Calculate the Surface Area of One Cube
The surface area of one cube is given by:
\[ SA_{\text{cube}} = 6A^2 \]

Substituting the value of A:
\[ SA_{\text{cube}} = 6 \times (12^2) = 6 \times 144 = 864 \, \text{cm}^2 \]

### Step 6: Calculate the Total Surface Area of Two Cubes
Since there are two cubes, the total surface area of the two cubes is:
\[ SA_{\text{total cubes}} = 2 \times SA_{\text{cube}} = 2 \times 864 = 1728 \, \text{cm}^2 \]

### Step 7: Calculate the Ratio of Surface Areas
Now, we can find the ratio of the surface area of the cuboid to the total surface area of the two cubes:
\[ \text{Ratio} = \frac{SA_{\text{cuboid}}}{SA_{\text{total cubes}}} = \frac{1560}{1728} \]

### Step 8: Simplify the Ratio
To simplify the ratio, we can divide both numbers by their greatest common divisor (GCD). The GCD of 1560 and 1728 is 72:
\[ \frac{1560 \div 72}{1728 \div 72} = \frac{65}{72} \]

Thus, the final answer is:
\[ \text{Ratio} = 65 : 72 \]