To solve the problem, we start by establishing that the perimeter of a rectangle is given by the formula:
\[ P = 2(l + w) \]
where \(l\) is the length and \(w\) is the width of the rectangle. Given that the farmer has 12 feet of fencing, we set the perimeter to 12:
\[ 2(l + w) = 12 \]
Dividing both sides by 2, we get:
\[ l + w = 6 \]
To find whole number pairs \((l, w)\), we can express \(w\) in terms of \(l\):
\[ w = 6 - l \]
Now we need to find all combinations of \(l\) and \(w\) where both values are whole numbers and non-negative. The pairs \((l, w)\) where both are positive integers that sum to 6 are:
- \(l = 1, w = 5\) → \((1, 5)\)
- \(l = 2, w = 4\) → \((2, 4)\)
- \(l = 3, w = 3\) → \((3, 3)\)
- \(l = 4, w = 2\) → \((4, 2)\)
- \(l = 5, w = 1\) → \((5, 1)\)
So the possible shapes (rectangles) are:
- \(1 \times 5\)
- \(2 \times 4\)
- \(3 \times 3\)
- \(4 \times 2\)
- \(5 \times 1\)
Now we can answer the parts of the question:
Part (a): How many possible shapes can the garden have?
The possible shapes of the garden based on the pairs derived above are:
- \(1 \times 5\)
- \(2 \times 4\)
- \(3 \times 3\)
- \(4 \times 2\)
- \(5 \times 1\)
So, there are 5 possible shapes.
Answer: 1 times 5, 2 times 4, 3 times 3, and their reverses.
Part (b): What is the area of each possible garden?
The area \(A\) of a rectangle can be calculated by:
\[ A = l \times w \]
Calculating the area for each rectangle:
- For \(1 \times 5\): \(A = 1 \times 5 = 5\)
- For \(2 \times 4\): \(A = 2 \times 4 = 8\)
- For \(3 \times 3\): \(A = 3 \times 3 = 9\)
- For \(4 \times 2\): \(A = 4 \times 2 = 8\)
- For \(5 \times 1\): \(A = 5 \times 1 = 5\)
So the areas of each possible garden shape are:
- \(1 \times 5\): Area = 5
- \(2 \times 4\): Area = 8
- \(3 \times 3\): Area = 9
- \(4 \times 2\): Area = 8
- \(5 \times 1\): Area = 5
Part (c): Which shape has the largest area?
The largest area calculated is for the shape \(3 \times 3\), which has an area of 9.
Answer: The shape with the largest area is \(3 \times 3\) with an area of 9.
2 answers