What is the total surface area of the identical cubes of largest possible size that are cut from a cuboid of size `85 cm xx 17 cm xx 5.1 cm ` ?

Options

A. a)`26010 cm^(2) `
B. b)`21600 cm^(2) `
C. c)`26100 cm^(2) `
D. d)none of these

Answer

To find the total surface area of the identical cubes of the largest possible size that can be cut from a cuboid with dimensions 85 cm, 17 cm, and 5.1 cm, we will follow these steps:

### Step 1: Find the dimensions of the cuboid
The dimensions of the cuboid are given as:
- Length (L) = 85 cm
- Breadth (B) = 17 cm
- Height (H) = 5.1 cm

### Step 2: Find the largest possible cube size
To determine the largest possible cube that can be cut from the cuboid, we need to find the greatest common factor (GCF) of the three dimensions.

1. Convert the dimensions into a simpler form:
- 85 cm = 1.7 x 10
- 17 cm = 1.7 x 1
- 5.1 cm = 1.7 x 3

2. The common factor among these dimensions is 1.7 cm.

Thus, the side length of the largest possible cube is **1.7 cm**.

### Step 3: Calculate the volume of the cuboid
The volume of the cuboid can be calculated using the formula:
\[ \text{Volume of cuboid} = L \times B \times H \]
\[ \text{Volume of cuboid} = 85 \times 17 \times 5.1 \]

Calculating this gives:
\[ \text{Volume of cuboid} = 85 \times 17 = 1445 \]
\[ 1445 \times 5.1 = 73695 \, \text{cm}^3 \]

### Step 4: Calculate the volume of one cube
The volume of one cube can be calculated using the formula:
\[ \text{Volume of cube} = \text{side}^3 \]
\[ \text{Volume of cube} = (1.7)^3 \]

Calculating this gives:
\[ (1.7)^3 = 4.913 \, \text{cm}^3 \]

### Step 5: Calculate the number of cubes that can be cut
To find the number of cubes (n) that can be cut from the cuboid, we use the formula:
\[ n = \frac{\text{Volume of cuboid}}{\text{Volume of cube}} \]
\[ n = \frac{73695}{4.913} \]

Calculating this gives:
\[ n \approx 15000 \, \text{cubes} \]

### Step 6: Calculate the total surface area of the cubes
The total surface area (TSA) of one cube is given by:
\[ \text{TSA of one cube} = 6 \times \text{side}^2 \]
\[ \text{TSA of one cube} = 6 \times (1.7)^2 \]

Calculating this gives:
\[ (1.7)^2 = 2.89 \]
\[ \text{TSA of one cube} = 6 \times 2.89 = 17.34 \, \text{cm}^2 \]

Now, the total surface area of all cubes is:
\[ \text{Total TSA} = n \times \text{TSA of one cube} \]
\[ \text{Total TSA} = 15000 \times 17.34 \]

Calculating this gives:
\[ \text{Total TSA} = 260100 \, \text{cm}^2 \]

### Final Answer
The total surface area of the identical cubes of the largest possible size that can be cut from the cuboid is **260100 cm²**.