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 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?
Question content area bottom
Part 1
​(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?
A.12 sq ft comma 12 sq ft comma 12 sq ft
12 sq ft comma 12 sq ft comma 12 sq ft
B.5 sq ft comma 8 sq ft comma 9 sq ft
5 sq ft comma 8 sq ft comma 9 sq ft
C.20 sq ft comma 32 sq ft comma 36 sq ft
20 sq ft comma 32 sq ft comma 36 sq ft
Part 3
​(c) What are the length and width of the shape that has the largest​ area?

GPT-4o mini · Answer

To solve the farmer's fencing problem, we'll start by establishing the relationship between the perimeter and the dimensions of the rectangular garden.

Given:

The total length of fencing is 12 feet.
The formula for the perimeter $P$ of a rectangle is $P = 2 (w + l)$ where $w$ is the width and $l$ is the length.

Setting up the equation: $2 (w + l) = 12$ Dividing both sides by 2: $w + l = 6$

Part 1: How many possible shapes can the garden have?

Since both $w$ and $l$ must be whole numbers, we can list the pairs $(w, l)$ that satisfy $w + l = 6$ :

$1 + 5 = 6$ (Garden dimensions: 1 ft x 5 ft)
$2 + 4 = 6$ (Garden dimensions: 2 ft x 4 ft)
$3 + 3 = 6$ (Garden dimensions: 3 ft x 3 ft)
$4 + 2 = 6$ (Garden dimensions: 4 ft x 2 ft)
$5 + 1 = 6$ (Garden dimensions: 5 ft x 1 ft)

This gives us the distinct pairs of dimensions:

(1, 5)
(2, 4)
(3, 3)

Thus, the number of possible shapes is 3, corresponding to:

$1 \times 5$
$2 \times 4$
$3 \times 3$

Answer (Part 1): C. 1 times 5, 2 times 4, 3 times 3

Part 2: What are the areas of the possible gardens?

Now we compute the area for each garden shape using the formula $A = w \times l$ :

For $1 \times 5$ : $A = 1 \cdot 5 = 5 sq ft$
For $2 \times 4$ : $A = 2 \cdot 4 = 8 sq ft$
For $3 \times 3$ : $A = 3 \cdot 3 = 9 sq ft$

Therefore, the areas for the possible gardens are:

5 sq ft for $1 \times 5$
8 sq ft for $2 \times 4$
9 sq ft for $3 \times 3$

Answer (Part 2): The areas are: 5 sq ft, 8 sq ft, 9 sq ft.

Part 3: What are the length and width of the shape that has the largest area?

From the computed areas, we see that the maximum area is $9, sq ft$ which corresponds to the dimensions of the shape:

Answer (Part 3): The shape with the largest area is $3$ ft by $3$ ft.

In conclusion, the farmer can create three distinct shapes, with areas being 5 sq ft, 8 sq ft, and 9 sq ft, and the largest area corresponds to a square garden measuring 3 ft by 3 ft.