A 4 cm sided cube is cut into 1 cm sided cubes. What is the percentage increase in the surface area after such cutting?

Options

A. `4%`
B. `300%`
C. `75%`
D. `400%`

Answer

To find the percentage increase in the surface area after cutting a 4 cm sided cube into 1 cm sided cubes, we can follow these steps:

### Step 1: Calculate the surface area of the larger cube.
The formula for the surface area \( A \) of a cube with side length \( a \) is given by:
\[ A = 6a^2 \]
For the larger cube, where \( a = 4 \) cm:
\[ A = 6 \times (4)^2 = 6 \times 16 = 96 \text{ cm}^2 \]

### Step 2: Calculate the surface area of one smaller cube.
Using the same formula for the smaller cube with side length \( a = 1 \) cm:
\[ A = 6 \times (1)^2 = 6 \times 1 = 6 \text{ cm}^2 \]

### Step 3: Determine the number of smaller cubes formed.
The volume of the larger cube is:
\[ V = a^3 = (4)^3 = 64 \text{ cm}^3 \]
The volume of one smaller cube is:
\[ V = (1)^3 = 1 \text{ cm}^3 \]
Thus, the number of smaller cubes formed is:
\[ \text{Number of cubes} = \frac{\text{Volume of larger cube}}{\text{Volume of smaller cube}} = \frac{64}{1} = 64 \]

### Step 4: Calculate the total surface area of all smaller cubes.
The total surface area of all 64 smaller cubes is:
\[ \text{Total Surface Area} = 64 \times \text{Surface Area of one smaller cube} = 64 \times 6 = 384 \text{ cm}^2 \]

### Step 5: Calculate the increase in surface area.
The increase in surface area is:
\[ \text{Increase} = \text{Total Surface Area of smaller cubes} - \text{Surface Area of larger cube} = 384 - 96 = 288 \text{ cm}^2 \]

### Step 6: Calculate the percentage increase in surface area.
The percentage increase is given by:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Original Surface Area}} \right) \times 100 = \left( \frac{288}{96} \right) \times 100 = 300\% \]

### Final Answer:
The percentage increase in the surface area after cutting the cube is **300%**.

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