`{:("Column A",a = 49"," b = 59,"Column B"),((a^2 - b^2)/(a - b),,(a^2 - b^2)/(a + b)):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to evaluate the expressions in Column A and Column B using the given values of \( a = 49 \) and \( b = 59 \).

### Step-by-Step Solution:

1. **Identify the expressions in Column A and Column B**:
- Column A: \( \frac{a^2 - b^2}{a - b} \)
- Column B: \( \frac{a^2 - b^2}{a + b} \)

2. **Use the difference of squares formula**:
We know that \( a^2 - b^2 \) can be factored as \( (a - b)(a + b) \). Thus:
\[ a^2 - b^2 = (a - b)(a + b) \]

3. **Simplify Column A**:
Substitute the factorization into Column A:
\[ \text{Column A} = \frac{(a - b)(a + b)}{a - b} \]
Since \( a - b \) is not zero, we can cancel \( a - b \):
\[ \text{Column A} = a + b \]

4. **Simplify Column B**:
Substitute the factorization into Column B:
\[ \text{Column B} = \frac{(a - b)(a + b)}{a + b} \]
Since \( a + b \) is not zero, we can cancel \( a + b \):
\[ \text{Column B} = a - b \]

5. **Substitute the values of \( a \) and \( b \)**:
Now substitute \( a = 49 \) and \( b = 59 \):
- For Column A:
\[ \text{Column A} = 49 + 59 = 108 \]
- For Column B:
\[ \text{Column B} = 49 - 59 = -10 \]

6. **Compare the values of Column A and Column B**:
- Column A = 108
- Column B = -10

Since \( 108 > -10 \), we conclude that Column A is larger than Column B.

### Final Answer:
Column A is larger than Column B.