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.