125 identical cubes are cut from a big cube and all the smaller cubes are arranged in a row to form a long cuboid . What is the percentage increase in the total surface area of the cuboid over the total surface area of the cube ?

Options

A. `234 (2)/(3)%`
B. `235(1)/(3)%`
C. `134(2)/(3)%`
D. none of these

Answer

To solve the problem of finding the percentage increase in the total surface area of a cuboid formed by arranging 125 identical cubes cut from a larger cube, we can follow these steps:

### Step 1: Find the side length of the smaller cubes
Assuming each smaller cube has a side length of 1 cm, the volume of one smaller cube is:
\[ \text{Volume of one cube} = 1^3 = 1 \text{ cm}^3 \]
Thus, the total volume of 125 smaller cubes is:
\[ \text{Total volume} = 125 \times 1 = 125 \text{ cm}^3 \]

### Step 2: Find the side length of the larger cube
The volume of the larger cube is equal to the total volume of the smaller cubes, which is 125 cm³. The volume of a cube is given by \( V = a^3 \), where \( a \) is the side length. Therefore, we can find the side length of the larger cube:
\[ a^3 = 125 \implies a = \sqrt[3]{125} = 5 \text{ cm} \]

### Step 3: Calculate the total surface area of the larger cube
The total surface area \( A \) of a cube is given by:
\[ A = 6a^2 \]
Substituting the side length:
\[ A = 6 \times (5)^2 = 6 \times 25 = 150 \text{ cm}^2 \]

### Step 4: Determine the dimensions of the cuboid
When the 125 smaller cubes are arranged in a row, the dimensions of the cuboid will be:
- Length \( L = 125 \text{ cm} \) (since we have 125 cubes each of 1 cm side)
- Width \( B = 1 \text{ cm} \)
- Height \( H = 1 \text{ cm} \)

### Step 5: Calculate the total surface area of the cuboid
The total surface area \( A_c \) of a cuboid is given by:
\[ A_c = 2(LB + BH + HL) \]
Substituting the dimensions:
\[ A_c = 2(125 \times 1 + 1 \times 1 + 1 \times 125) = 2(125 + 1 + 125) = 2(251) = 502 \text{ cm}^2 \]

### Step 6: Calculate the increase in surface area
The increase in surface area is:
\[ \text{Increase} = A_c - A = 502 - 150 = 352 \text{ cm}^2 \]

### Step 7: Calculate the percentage increase
The percentage increase in surface area is calculated as:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{A} \right) \times 100 = \left( \frac{352}{150} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage Increase} = \frac{352 \times 100}{150} = \frac{35200}{150} \approx 234.67\% \]

### Final Answer
The percentage increase in the total surface area of the cuboid over the total surface area of the cube is approximately **234.67%**.

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