`{:("Column A",1 < p < 3,"Column B),(p^2,,2p):}`

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 compare the expressions in Column A and Column B for values of \( p \) that lie in the interval \( 1 < p < 3 \).

### Step 1: Define the expressions
- Column A: \( p^2 \)
- Column B: \( 2p \)

### Step 2: Analyze the interval
We know that \( p \) can take any value between 1 and 3. We will check the expressions at specific points within this interval.

### Step 3: Test \( p = 2 \)
- For \( p = 2 \):
- Column A: \( p^2 = 2^2 = 4 \)
- Column B: \( 2p = 2 \times 2 = 4 \)

Here, Column A equals Column B.

### Step 4: Test \( p = 1.5 \)
- For \( p = 1.5 \):
- Column A: \( p^2 = (1.5)^2 = 2.25 \)
- Column B: \( 2p = 2 \times 1.5 = 3 \)

In this case, Column B is greater than Column A.

### Step 5: Test \( p = 2.5 \)
- For \( p = 2.5 \):
- Column A: \( p^2 = (2.5)^2 = 6.25 \)
- Column B: \( 2p = 2 \times 2.5 = 5 \)

Here, Column A is greater than Column B.

### Step 6: Conclusion
From the tests:
- At \( p = 2 \), both columns are equal.
- At \( p = 1.5 \), Column B is greater.
- At \( p = 2.5 \), Column A is greater.

Since the relationship between Column A and Column B changes depending on the value of \( p \) within the interval \( 1 < p < 3 \), we conclude that there is not enough information to definitively state which column is larger for all values of \( p \) in the given range.

### Final Answer
The correct option is 4: There is not enough information.

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