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

Question

Accepted Answer

To determine the percentage increase in the surface area after cutting a 4 cm sided cube into 1 cm sided cubes, let's follow these steps:

1. **Calculate the Surface Area of the Original Cube:**
   - The original cube has a side length of 4 cm.
   - Surface area of a cube = 6 * (side length)^2
   - Surface area of the original cube = 6 * (4 cm)^2 = 6 * 16 cm² = 96 cm²

2. **Calculate the Surface Area of the Smaller Cubes:**
   - When the 4 cm cube is cut into 1 cm cubes, each side of the original cube is divided into 4 equal parts, resulting in 4 x 4 x 4 = 64 smaller cubes.
   - Each smaller cube has a side length of 1 cm.
   - Surface area of one smaller cube = 6 * (1 cm)^2 = 6 cm²
   - Total surface area of all 64 smaller cubes = 64 * 6 cm² = 384 cm²

3. **Calculate the Percentage Increase in Surface Area:**
   - Increase in surface area = Total surface area of smaller cubes - Surface area of original cube
   - Increase in surface area = 384 cm² - 96 cm² = 288 cm²
   - Percentage increase = (Increase in surface area / Surface area of original cube) * 100%
   - Percentage increase = (288 cm² / 96 cm²) * 100% = 300%

Therefore, the percentage increase in the surface area after cutting the cube is 300%, which corresponds to Option 2.

**Explanation for Why Other Options are Incorrect:**

- **Option 1 (4%):** This option is incorrect because the calculated percentage increase is significantly higher than 4%. The increase in surface area is substantial due to the large number of smaller cubes created.

- **Option 3 (75%):** This option is incorrect because the actual percentage increase is much greater than 75%. The calculation shows a 300% increase, which is far beyond 75%.

- **Option 4 (400%):** This option is incorrect because the calculated percentage increase is 300%, not 400%. The increase is substantial but not as high as 400%.

In conclusion, the correct answer is Option 2, which accurately reflects the 300% increase in surface area after cutting the original cube into smaller cubes.