If each edge of a cube is increased by 25%, then the percentage increase in its surface area is: (a) 25% (b) 50% (c) 56.25%

Answer

To find the percentage increase in the surface area of a cube when each edge is increased by 25%, we can follow these steps:

### Step-by-Step Solution:

1. **Let the original edge length of the cube be \( A \)**.
- **Hint**: Define a variable for the original edge length to simplify calculations.

2. **Calculate the original surface area of the cube**.
\[ \text{Surface Area} = 6A^2 \]
- **Hint**: Remember that the surface area of a cube is calculated using the formula \( 6 \times \text{(edge length)}^2 \).

3. **Determine the new edge length after a 25% increase**.
- A 25% increase means the new edge length is:
\[ \text{New Edge Length} = A + \frac{25}{100}A = A + \frac{1}{4}A = \frac{5A}{4} \]
- **Hint**: To find the new edge length, calculate 25% of the original length and add it to the original length.

4. **Calculate the new surface area of the cube with the new edge length**.
\[ \text{New Surface Area} = 6 \left(\frac{5A}{4}\right)^2 = 6 \times \frac{25A^2}{16} = \frac{150A^2}{16} = \frac{75A^2}{8} \]
- **Hint**: Substitute the new edge length into the surface area formula and simplify.

5. **Find the increase in surface area**.
\[ \text{Increase in Surface Area} = \text{New Surface Area} - \text{Original Surface Area} = \frac{75A^2}{8} - 6A^2 \]
- Convert \( 6A^2 \) to have a common denominator:
\[ 6A^2 = \frac{48A^2}{8} \]
- Now compute the increase:
\[ \text{Increase} = \frac{75A^2}{8} - \frac{48A^2}{8} = \frac{27A^2}{8} \]
- **Hint**: To find the increase, subtract the original surface area from the new surface area.

6. **Calculate the percentage increase in surface area**.
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{\text{Original Surface Area}}\right) \times 100 = \left(\frac{\frac{27A^2}{8}}{6A^2}\right) \times 100 \]
- Simplifying this gives:
\[ = \left(\frac{27}{48}\right) \times 100 = \frac{27 \times 100}{48} = \frac{2700}{48} = 56.25\% \]
- **Hint**: To find the percentage, divide the increase by the original area and multiply by 100.

### Final Answer:
The percentage increase in the surface area of the cube is **56.25%**.