Question
Question content area top
Part 1
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 1
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
Answers
GPT-4o mini
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:
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 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. \(1 \times 5\)
2. \(2 \times 4\)
3. \(3 \times 3\)
4. \(4 \times 2\)
5. \(5 \times 1\)
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:
1. For \(1 \times 5\): \(A = 1 \times 5 = 5\)
2. For \(2 \times 4\): \(A = 2 \times 4 = 8\)
3. For \(3 \times 3\): \(A = 3 \times 3 = 9\)
4. For \(4 \times 2\): \(A = 4 \times 2 = 8\)
5. 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**.
\[
P = 2(l + w)
\]
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:
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 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. \(1 \times 5\)
2. \(2 \times 4\)
3. \(3 \times 3\)
4. \(4 \times 2\)
5. \(5 \times 1\)
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:
1. For \(1 \times 5\): \(A = 1 \times 5 = 5\)
2. For \(2 \times 4\): \(A = 2 \times 4 = 8\)
3. For \(3 \times 3\): \(A = 3 \times 3 = 9\)
4. For \(4 \times 2\): \(A = 4 \times 2 = 8\)
5. 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**.