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.

Question

​(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
Part 2
​(b) What are the areas of the possible​ gardens?

Answer 1

To solve the problem, we need to analyze the constraints provided by the 12 feet of fencing and determine the dimensions of the rectangular garden based on that.

Given the perimeter \( P = 12 \) feet of the rectangular garden, we know that:

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

Setting up our equation:

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

This simplifies to:

\[ l + w = 6 \]

where \( l \) is the length and \( w \) is the width. Since both \( l \) and \( w \) have to be whole numbers, we can list the pairs of \( (l, w) \):

Possible dimensions:

1. \( l = 1 \), \( w = 5 \)
2. \( l = 2 \), \( w = 4 \)
3. \( l = 3 \), \( w = 3 \)
4. \( l = 4 \), \( w = 2 \)
5. \( l = 5 \), \( w = 1 \)

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

Considering all the unique pairs, we find that there are 5 unique combinations:

1. \( 1 \times 5 \)
2. \( 2 \times 4 \)
3. \( 3 \times 3 \)
4. \( 4 \times 2 \)
5. \( 5 \times 1 \)

So, the answer is:

• Option C: \( 1 \times 12, 2 \times 6, 3 \times 4 \), but this is the only unique in the pairs given as \( 12 \) is not a combination considered here.

(b) What are the areas of the possible gardens?

Now, let's calculate the area for each possible shape:

1. For \( 1 \times 5 \): \[ \text{Area} = 1 \times 5 = 5 \]

2. For \( 2 \times 4 \): \[ \text{Area} = 2 \times 4 = 8 \]

3. For \( 3 \times 3 \): \[ \text{Area} = 3 \times 3 = 9 \]

4. For \( 4 \times 2 \): \[ \text{Area} = 4 \times 2 = 8 \]

5. For \( 5 \times 1 \): \[ \text{Area} = 5 \times 1 = 5 \]

List of Areas:

• \( 1 \times 5 \) : \( 5 \) square feet
• \( 2 \times 4 \) : \( 8 \) square feet
• \( 3 \times 3 \) : \( 9 \) square feet
• \( 4 \times 2 \) : \( 8 \) square feet
• \( 5 \times 1 \) : \( 5 \) square feet

(c) Which shape has the largest area?

From our calculations, the largest area is for the shape \( 3 \times 3 \) which gives:

• Area = \( 9 \) square feet.

Summary of Answers

(a) There are 5 possible shapes.
(b) Areas of the possible gardens: 5, 8, 9, 8, 5 square feet.
(c) The shape with the largest area is \( 3 \times 3 \) with an area of 9 square feet.