If each edge of a cube is increased by 5 cm , then the lateral surface area of the cube increases by _________.

Options

A. `100 cm^(2)`
B. `150 cm^(2)`
C. ` 50 cm^(2)`
D. Cannot be determined

Answer

To solve the problem of how much the lateral surface area of a cube increases when each edge is increased by 5 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Lateral Surface Area of a Cube**:
The lateral surface area (LSA) of a cube is given by the formula:
\[ \text{LSA} = 4a^2 \]
where \( a \) is the length of one edge of the cube.

2. **Calculate the Initial Lateral Surface Area**:
Let the original length of the edge of the cube be \( a \). Thus, the initial lateral surface area is:
\[ \text{LSA}_{\text{initial}} = 4a^2 \]

3. **Increasing the Edge Length**:
If each edge of the cube is increased by 5 cm, the new length of the edge becomes:
\[ a + 5 \]

4. **Calculate the New Lateral Surface Area**:
The new lateral surface area with the increased edge length is:
\[ \text{LSA}_{\text{new}} = 4(a + 5)^2 \]
Expanding this:
\[ \text{LSA}_{\text{new}} = 4(a^2 + 10a + 25) = 4a^2 + 40a + 100 \]

5. **Finding the Increase in Lateral Surface Area**:
To find the increase in lateral surface area, we subtract the initial lateral surface area from the new lateral surface area:
\[ \text{Increase} = \text{LSA}_{\text{new}} - \text{LSA}_{\text{initial}} = (4a^2 + 40a + 100) - 4a^2 \]
Simplifying this gives:
\[ \text{Increase} = 40a + 100 \]

6. **Conclusion**:
The increase in the lateral surface area of the cube when each edge is increased by 5 cm is:
\[ 40a + 100 \text{ cm}^2 \]
Since we do not have a specific value for \( a \), we cannot determine a numerical answer without knowing the original edge length.