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?
The length is
  
enter your response here.
The width is
  
enter your response here.
(a) How many possible shapes can the garden have?(b) What are the areas of the possible gardens?The length is.The width isThe width is.

Answer 1

To answer the questions, let's start with the conditions given. The farmer has 12 feet of fencing and is constructing a rectangular garden with whole number side lengths. The perimeter \(P\) of a rectangle is given by the formula:

\[
P = 2 \cdot (L + W)
\]

where \(L\) is the length and \(W\) is the width of the rectangle. Given that \(P = 12\), we have:

\[
2 \cdot (L + W) = 12 \implies L + W = 6
\]

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

We can rearrange this equation to find the pairs of \(L\) and \(W\):

1. \(L = 1 \Rightarrow W = 5 \quad \Rightarrow (1, 5)\)
2. \(L = 2 \Rightarrow W = 4 \quad \Rightarrow (2, 4)\)
3. \(L = 3 \Rightarrow W = 3 \quad \Rightarrow (3, 3)\)
4. \(L = 4 \Rightarrow W = 2 \quad \Rightarrow (4, 2)\)
5. \(L = 5 \Rightarrow W = 1 \quad \Rightarrow (5, 1)\)

Thus, the possible pairs \((L, W)\) are:
- (1, 5)
- (2, 4)
- (3, 3)
- (4, 2)
- (5, 1)

That's a total of **5 unique shapes** (counting pairs that are the same).

**Answer: 5 unique shapes (C: 1×5, 2×4, 3×3).**

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

The area \(A\) of a rectangle is given by:

\[
A = L \times W
\]

Calculating for each pair:

1. \(1 \times 5 = 5\) sq ft
2. \(2 \times 4 = 8\) sq ft
3. \(3 \times 3 = 9\) sq ft
4. \(4 \times 2 = 8\) sq ft
5. \(5 \times 1 = 5\) sq ft

The areas of the possible gardens are:
- 5 sq ft
- 8 sq ft
- 9 sq ft

**Answer: Areas are 5 sq ft, 8 sq ft, and 9 sq ft.**

### (c) What are length and width of the shape that has the largest area?

From our calculations, the largest area is \(9\) sq ft, which occurs for:

\[
L = 3 \quad \text{and} \quad W = 3
\]

Thus the shape with the largest area is a square with:

**Length: 3 ft, Width: 3 ft.**

### Summary of Answers:

- (a) 5 shapes: 1×5, 2×4, 3×3, 4×2, 5×1
- (b) Areas: 5 sq ft, 8 sq ft, 9 sq ft
- (c) Length is 3 ft; Width is 3 ft.