A cuboid of dimensions 51, 85 and 102 cm is first painted by red colour then it is cut into minimum possible identical cubes. Now the total surface area of all those faces of cubes which are not red is :

Options

A. `119646 cm^2`
B. `52020 cm^2`
C. `18514 cm^2`
D. `36414 cm^2`

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript and break it down into manageable parts.

### Step 1: Identify the dimensions of the cuboid
The dimensions of the cuboid are given as:
- Length (l) = 102 cm
- Width (w) = 85 cm
- Height (h) = 51 cm

### Step 2: Find the highest common factor (HCF)
To cut the cuboid into the minimum possible identical cubes, we need to find the HCF of the dimensions 51, 85, and 102.

1. **Prime factorization**:
- 51 = 3 × 17
- 85 = 5 × 17
- 102 = 2 × 3 × 17

2. **Identify common factors**:
The common factor among all three numbers is 17.

Thus, HCF(51, 85, 102) = 17 cm.

### Step 3: Calculate the number of cubes
Now, we can find the number of identical cubes that can be formed by dividing the dimensions of the cuboid by the HCF.

- Number of cubes along length = 102 / 17 = 6
- Number of cubes along width = 85 / 17 = 5
- Number of cubes along height = 51 / 17 = 3

Total number of cubes = 6 × 5 × 3 = 90 cubes.

### Step 4: Calculate the surface area of one cube
The side length of each cube is equal to the HCF, which is 17 cm.

The surface area (SA) of one cube is given by the formula:
\[ SA = 6a^2 \]
where \( a \) is the side length of the cube.

So, substituting \( a = 17 \):
\[ SA = 6 \times (17^2) \]
\[ SA = 6 \times 289 = 1734 \, \text{cm}^2 \]

### Step 5: Calculate the total surface area of all cubes
Total surface area of all 90 cubes:
\[ \text{Total SA} = 90 \times 1734 = 156060 \, \text{cm}^2 \]

### Step 6: Calculate the surface area of the red-painted faces
Now, we need to find the surface area of the faces of the cubes that are not painted red.

To find the area of the red-painted faces, we need to calculate the surface area of the original cuboid:
\[ \text{Surface Area of Cuboid} = 2(lw + lh + wh) \]
Substituting the values:
\[ \text{Surface Area} = 2(102 \times 85 + 102 \times 51 + 85 \times 51) \]
Calculating each term:
- \( 102 \times 85 = 8670 \)
- \( 102 \times 51 = 5202 \)
- \( 85 \times 51 = 4335 \)

Now substituting back:
\[ \text{Surface Area} = 2(8670 + 5202 + 4335) \]
\[ = 2(18207) = 36414 \, \text{cm}^2 \]

### Step 7: Calculate the non-painted surface area
The non-painted surface area of the cubes is given by:
\[ \text{Non-painted SA} = \text{Total SA of cubes} - \text{Surface Area of Cuboid} \]
\[ = 156060 - 36414 = 119646 \, \text{cm}^2 \]

### Final Answer
The total surface area of all those faces of cubes which are not red is **119646 cm²**.

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