A cuboid `(3 cm xx 4 cm xx 5 cm)` is cut into unit cubes. The ratio of the total surface area of all the unit cubes to that of the cuboid is

Options

A. `180 :3 `
B. `180:9`
C. `180:36 `
D. `180:47`

Answer

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

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is calculated using the formula:
\[ V = l \times b \times h \]
where \( l \), \( b \), and \( h \) are the length, breadth, and height of the cuboid, respectively.

For the given cuboid:
- Length \( l = 3 \, \text{cm} \)
- Breadth \( b = 4 \, \text{cm} \)
- Height \( h = 5 \, \text{cm} \)

Calculating the volume:
\[ V = 3 \times 4 \times 5 = 60 \, \text{cm}^3 \]

### Step 2: Determine the Number of Unit Cubes
Since the cuboid is cut into unit cubes (1 cm x 1 cm x 1 cm), the number of unit cubes is equal to the volume of the cuboid:
\[ \text{Number of unit cubes} = V = 60 \]

### Step 3: Calculate the Total Surface Area of the Unit Cubes
The surface area \( S \) of one unit cube is given by:
\[ S_{\text{cube}} = 6a^2 \]
where \( a \) is the side length of the cube. For a unit cube, \( a = 1 \, \text{cm} \):
\[ S_{\text{cube}} = 6 \times 1^2 = 6 \, \text{cm}^2 \]

Thus, the total surface area of all 60 unit cubes is:
\[ \text{Total Surface Area of unit cubes} = 60 \times 6 = 360 \, \text{cm}^2 \]

### Step 4: Calculate the Total Surface Area of the Cuboid
The total surface area \( S \) of a cuboid is calculated using the formula:
\[ S = 2(lb + bh + hl) \]

Substituting the values:
\[ S = 2(3 \times 4 + 4 \times 5 + 5 \times 3) \]
Calculating each term:
- \( lb = 3 \times 4 = 12 \)
- \( bh = 4 \times 5 = 20 \)
- \( hl = 5 \times 3 = 15 \)

Now substituting back:
\[ S = 2(12 + 20 + 15) = 2(47) = 94 \, \text{cm}^2 \]

### Step 5: Calculate the Ratio of Surface Areas
Now we find the ratio of the total surface area of the unit cubes to that of the cuboid:
\[ \text{Ratio} = \frac{\text{Total Surface Area of unit cubes}}{\text{Total Surface Area of cuboid}} = \frac{360}{94} \]

To simplify this ratio:
\[ \text{Ratio} = \frac{180}{47} \]

### Final Answer
Thus, the ratio of the total surface area of all the unit cubes to that of the cuboid is:
\[ \frac{180}{47} \]

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