To determine the possible shapes of the garden that can be constructed with 12 feet of fencing, we can denote the length of the garden as \( l \) and the width as \( w \). The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2l + 2w \]
Since the farmer uses all the fence, we have:
\[ 2l + 2w = 12 \]
Dividing both sides by 2 gives:
\[ l + w = 6 \]
Now, since \( l \) and \( w \) must be whole numbers, we can find the pairs of \( (l, w) \) that satisfy this equation.
The possible pairs of values 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 distinct shapes are:
- (1, 5)
- (2, 4)
- (3, 3)
- (4, 2)
- (5, 1)
(a) The answer is 5 possible shapes.
(b) Now, let's calculate the area of each possible garden. The area \( A \) of a rectangle is given by:
\[ A = l \times w \]
Calculating the area for each pair:
- For \( (1, 5) \): \( A = 1 \times 5 = 5 \)
- For \( (2, 4) \): \( A = 2 \times 4 = 8 \)
- For \( (3, 3) \): \( A = 3 \times 3 = 9 \)
- For \( (4, 2) \): \( A = 4 \times 2 = 8 \)
- For \( (5, 1) \): \( A = 5 \times 1 = 5 \)
Areas for each shape:
- (1, 5) → Area = 5
- (2, 4) → Area = 8
- (3, 3) → Area = 9
- (4, 2) → Area = 8
- (5, 1) → Area = 5
(c) To determine which shape has the largest area, we compare the areas calculated above. The largest area is:
The largest area is 9, which corresponds to the dimension 3 times 3 (a square).
In summary:
- (a) There are 5 possible shapes.
- (b) The areas are 5, 8, 9, 8, 5.
- (c) The shape with the largest area is the square 3 x 3 with an area of 9.
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