Consider all rectangles such that the rectangle's length is greater than the rectangle's width and the length and width are whole numbers of inches. Which of the following perimeters, in inches, is NOT possible for such a rectangle with an area of 144 square inches?

Options

A. 48
B. 60
C. 80
D. 102

Answer

To solve the problem, we need to find the possible dimensions of rectangles with an area of 144 square inches, where the length is greater than the width, and both dimensions are whole numbers. We will then calculate the perimeters of these rectangles and identify which of the given options is not a possible perimeter.

### Step-by-Step Solution:

1. **Define the Area Condition**:
The area \( A \) of a rectangle is given by the formula:
\[ A = L \times W \]
where \( L \) is the length and \( W \) is the width. For this problem, we have:
\[ L \times W = 144 \]

2. **List Factor Pairs of 144**:
We need to find pairs of whole numbers \( (L, W) \) such that \( L \times W = 144 \) and \( L > W \). The factor pairs of 144 are:
- \( (1, 144) \)
- \( (2, 72) \)
- \( (3, 48) \)
- \( (4, 36) \)
- \( (6, 24) \)
- \( (8, 18) \)
- \( (9, 16) \)
- \( (12, 12) \) (not valid since \( L \) must be greater than \( W \))

3. **Valid Length and Width Combinations**:
From the factor pairs, we select only those where \( L > W \):
- \( (144, 1) \)
- \( (72, 2) \)
- \( (48, 3) \)
- \( (36, 4) \)
- \( (24, 6) \)
- \( (18, 8) \)
- \( (16, 9) \)

4. **Calculate the Perimeters**:
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(L + W) \]
Now we calculate the perimeter for each valid pair:
- For \( (144, 1) \):
\[ P = 2(144 + 1) = 290 \]
- For \( (72, 2) \):
\[ P = 2(72 + 2) = 148 \]
- For \( (48, 3) \):
\[ P = 2(48 + 3) = 102 \]
- For \( (36, 4) \):
\[ P = 2(36 + 4) = 80 \]
- For \( (24, 6) \):
\[ P = 2(24 + 6) = 60 \]
- For \( (18, 8) \):
\[ P = 2(18 + 8) = 52 \]
- For \( (16, 9) \):
\[ P = 2(16 + 9) = 50 \]

5. **List of Possible Perimeters**:
The possible perimeters we have calculated are:
- 290
- 148
- 102
- 80
- 60
- 52
- 50

6. **Identify the Impossible Perimeter**:
Now we compare the given options:
- 148 (possible)
- 260 (not possible)
- 380 (not possible)
- 102 (possible)

Out of these, the only perimeter that is not in our list of calculated perimeters is **260** and **380**.

### Conclusion:
The perimeters that are not possible for such a rectangle with an area of 144 square inches are **260** and **380**.