Question

A farmer has 12 feet of fencing. He constructs a rectangular garden whose sides are whole numbers. He uses all the fence to enclose the garden.
​(a) How many possible shapes can the garden​ have?
​(b) What is the area of each possible​ garden?
​(c) Which shape has the largest​ area?

​(a) How many possible shapes can the garden​ have?
A.1 times 5 comma 2 times 4 comma 3 times 3
1 times 5 comma 2 times 4 comma 3 times 3
B.2 times 10 comma 4 times 8 comma 6 times 6
2 times 10 comma 4 times 8 comma 6 times 6
C.1 times 12 comma 2 times 6 comma 3 times 4
1 times 12 comma 2 times 6 comma 3 times 4

Answer 1

To solve the problem, we need to understand the relationship between the perimeter of a rectangle and its dimensions. The perimeter \( P \) of a rectangle is given by the formula:

\[ P = 2 \times (\text{length} + \text{width}) \]

In this case, we know the perimeter is 12 feet, so we can set up the equation:

\[ 2 \times (l + w) = 12 \]
\[ l + w = 6 \]

Where \( l \) is the length and \( w \) is the width of the rectangle. Since the sides must be whole numbers, we can find all possible integer pairs \( (l, w) \) that satisfy this equation.

Now let's break it down step by step.

### (a) How many possible shapes can the garden have?

We can find the possible pairs \( (l, w) \) with \( l + w = 6 \), where both \( l \) and \( w \) are positive integers:

1. \( l = 1, w = 5 \) → Shape: 1 x 5
2. \( l = 2, w = 4 \) → Shape: 2 x 4
3. \( l = 3, w = 3 \) → Shape: 3 x 3
4. \( l = 4, w = 2 \) → Shape: 4 x 2 (equivalent to 2 x 4)
5. \( l = 5, w = 1 \) → Shape: 5 x 1 (equivalent to 1 x 5)

However, the pairs \( (l, w) \) are the same as \( (w, l) \). Hence, the unique combinations are:

- 1 x 5
- 2 x 4
- 3 x 3

So there are **3 unique possible shapes.**

### (b) What is the area of each possible garden?

The area \( A \) of a rectangle is calculated as:

\[ A = l \times w \]

Calculating the areas for each shape:

1. **For 1 x 5:**
\[ A = 1 \times 5 = 5 \, \text{square feet} \]

2. **For 2 x 4:**
\[ A = 2 \times 4 = 8 \, \text{square feet} \]

3. **For 3 x 3:**
\[ A = 3 \times 3 = 9 \, \text{square feet} \]

### (c) Which shape has the largest area?

From our calculations:

- 1 x 5: Area = 5 square feet
- 2 x 4: Area = 8 square feet
- 3 x 3: Area = 9 square feet

Thus, the shape with the largest area is **3 x 3** with an area of **9 square feet**.

### Summary of the answers:
(a) There are **3 possible shapes**: 1 x 5, 2 x 4, 3 x 3.
(b) The areas of each possible garden are: 5, 8, and 9 square feet.
(c) The shape with the largest area is 3 x 3.