If a cuboid of dimensions `32 m xx 12 cm xx 9` cm is into two cubes of same size. What will be the rati of the surface are of the cuboid to the total surface are of the two cube ?

Options

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

Answer

To solve the problem of finding the ratio of the surface area of a cuboid to the total surface area of two cubes formed from it, we will follow these steps:

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is given by the formula:
\[ V = \text{Length} \times \text{Breadth} \times \text{Height} \]

Given dimensions:
- Length = 32 cm
- Breadth = 12 cm
- Height = 9 cm

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

### Step 2: Calculate the Volume
Now, let's calculate the volume:
\[ V = 32 \times 12 = 384 \]
\[ V = 384 \times 9 = 3456 \, \text{cm}^3 \]

### Step 3: Relate the Volume of the Cuboid to the Volume of the Cubes
Since the cuboid is melted into two cubes of the same size, the volume of each cube will be:
\[ \text{Volume of one cube} = \frac{V}{2} = \frac{3456}{2} = 1728 \, \text{cm}^3 \]

### Step 4: Find the Side Length of the Cube
The volume \( V \) of a cube is given by:
\[ V = a^3 \]
where \( a \) is the side length of the cube.

Setting the volume equal to 1728 cm³:
\[ a^3 = 1728 \]
To find \( a \), we take the cube root:
\[ a = \sqrt[3]{1728} = 12 \, \text{cm} \]

### Step 5: Calculate the Total Surface Area of the Cuboid
The total surface area (TSA) of a cuboid is given by:
\[ \text{TSA} = 2(LB + BH + HL) \]

Substituting the values:
\[ \text{TSA} = 2(32 \times 12 + 12 \times 9 + 9 \times 32) \]

Calculating each term:
- \( 32 \times 12 = 384 \)
- \( 12 \times 9 = 108 \)
- \( 9 \times 32 = 288 \)

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

### Step 6: Calculate the Total Surface Area of the Two Cubes
The total surface area of one cube is given by:
\[ \text{TSA of one cube} = 6a^2 \]

Substituting \( a = 12 \, \text{cm} \):
\[ \text{TSA of one cube} = 6 \times (12)^2 = 6 \times 144 = 864 \, \text{cm}^2 \]

Since there are two cubes:
\[ \text{Total TSA of two cubes} = 2 \times 864 = 1728 \, \text{cm}^2 \]

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

### Step 8: Simplify the Ratio
To simplify the ratio:
\[ \frac{1560}{1728} = \frac{65}{72} \]

### Final Answer
The ratio of the surface area of the cuboid to the total surface area of the two cubes is:
\[ \text{Ratio} = 65 : 72 \]

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