The perimeter of a rectangle is 72 cm. If the sides are positive integers, maximum how many distinct areas can it have?

Options

A. 18
B. 9
C. 36
D. 35

Answer

To solve the problem, we need to determine the maximum number of distinct areas that a rectangle can have given that its perimeter is 72 cm and its sides are positive integers.

### Step-by-Step Solution:

1. **Understanding the Perimeter Formula**:
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(l + w) \]
where \( l \) is the length and \( w \) is the width of the rectangle.

2. **Setting Up the Equation**:
Given that the perimeter is 72 cm, we can set up the equation:
\[ 2(l + w) = 72 \]
Dividing both sides by 2 gives:
\[ l + w = 36 \]

3. **Finding Possible Integer Pairs**:
Since \( l \) and \( w \) are positive integers, we can express \( w \) in terms of \( l \):
\[ w = 36 - l \]
To ensure both \( l \) and \( w \) are positive integers, \( l \) must satisfy:
\[ 1 \leq l < 36 \]

4. **Calculating Areas**:
The area \( A \) of the rectangle is given by:
\[ A = l \times w = l \times (36 - l) = 36l - l^2 \]
This is a quadratic function in terms of \( l \).

5. **Finding Distinct Areas**:
We need to find the distinct values of \( A \) for integer values of \( l \) from 1 to 35. The expression \( A = 36l - l^2 \) is a downward-opening parabola, which means it will have a maximum area at its vertex.

6. **Finding the Vertex**:
The vertex of the quadratic \( A = -l^2 + 36l \) occurs at:
\[ l = -\frac{b}{2a} = -\frac{36}{2 \times -1} = 18 \]
At \( l = 18 \):
\[ w = 36 - 18 = 18 \]
So, the maximum area is:
\[ A = 18 \times 18 = 324 \text{ cm}^2 \]

7. **Calculating Distinct Areas**:
As \( l \) varies from 1 to 35, the areas will be:
- For \( l = 1 \), \( A = 1 \times 35 = 35 \)
- For \( l = 2 \), \( A = 2 \times 34 = 68 \)
- For \( l = 3 \), \( A = 3 \times 33 = 99 \)
- ...
- For \( l = 17 \), \( A = 17 \times 19 = 323 \)
- For \( l = 18 \), \( A = 18 \times 18 = 324 \)
- For \( l = 19 \), \( A = 19 \times 17 = 323 \)
- ...
- For \( l = 35 \), \( A = 35 \times 1 = 35 \)

The areas will start repeating after \( l = 18 \).

8. **Counting Distinct Areas**:
The distinct areas can be calculated by evaluating the areas for \( l \) from 1 to 18, which gives us:
\[ A = 35, 68, 99, 128, 155, 180, 203, 224, 243, 260, 275, 288, 299, 308, 315, 320, 323, 324 \]
This results in 18 distinct areas.

### Conclusion:
The maximum number of distinct areas that the rectangle can have is **18**.