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

Answer 1

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:

1. \( l = 1, w = 5 \) → (1, 5)
2. \( l = 2, w = 4 \) → (2, 4)
3. \( l = 3, w = 3 \) → (3, 3)
4. \( l = 4, w = 2 \) → (4, 2)
5. \( l = 5, w = 1 \) → (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:

1. For \( (1, 5) \): \( A = 1 \times 5 = 5 \)
2. For \( (2, 4) \): \( A = 2 \times 4 = 8 \)
3. For \( (3, 3) \): \( A = 3 \times 3 = 9 \)
4. For \( (4, 2) \): \( A = 4 \times 2 = 8 \)
5. 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**.