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 \(W\):
- \(L = 1 \Rightarrow W = 5 \quad \Rightarrow (1, 5)\)
- \(L = 2 \Rightarrow W = 4 \quad \Rightarrow (2, 4)\)
- \(L = 3 \Rightarrow W = 3 \quad \Rightarrow (3, 3)\)
- \(L = 4 \Rightarrow W = 2 \quad \Rightarrow (4, 2)\)
- \(L = 5 \Rightarrow W = 1 \quad \Rightarrow (5, 1)\)
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:
- \(1 \times 5 = 5\) sq ft
- \(2 \times 4 = 8\) sq ft
- \(3 \times 3 = 9\) sq ft
- \(4 \times 2 = 8\) sq ft
- \(5 \times 1 = 5\) 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.
1 answer