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This problem involves determining the dimensions of a rectangular animal enclosure, with the given conditions:

- A total of 132 feet of chain-link fencing is used to enclose the perimeter and subdivide it into two smaller rectangular regions.
- The total enclosed area is 576 square feet.
- The enclosure is divided in half along its length, as shown in the diagram.

Let:
- \( l \) = length of the enclosure
- \( w \) = width of the enclosure

### Step 1: Perimeter Equation
The total fencing includes the perimeter and the fence that divides the enclosure into two smaller regions. Since the enclosure is subdivided along its length, the fencing equation can be set up as:

\[
2l + 3w = 132
\]

This is because:
- There are two lengths (one on each side) contributing \( 2l \).
- There are three widths (one on each side and one in the middle dividing the region), contributing \( 3w \).

### Step 2: Area Equation
The area of the entire enclosure is:

\[
l \times w = 576
\]

### Step 3: Solving the System of Equations
We now have two equations:
1. \( 2l + 3w = 132 \)
2. \( lw = 576 \)

We can solve this system of equations by first expressing \( l \) in terms of \( w \) using the perimeter equation and substituting it into the area equation. Let's solve this system now.The solutions to the system of equations are two possible sets of dimensions for the enclosure:

1. \( l = 18 \) feet and \( w = 32 \) feet
2. \( l = 48 \) feet and \( w = 12 \) feet

Both of these dimensions satisfy the conditions of the problem. Since the problem doesn't specify which dimension corresponds to length or width, either pair is valid, depending on how the dimensions are interpreted.

Would you like further details on the steps involved, or do you have any questions?

Here are 5 related questions to explore further:
1. What would happen if the area were larger but the same amount of fencing was used?
2. How would the dimensions change if the enclosure were divided into three smaller regions instead of two?
3. What if the fencing constraint was increased to 150 feet? How would the dimensions change?
4. How does the relationship between area and perimeter change when we increase or decrease the number of subdivisions?
5. Can this problem be generalized to find dimensions for different rectangular areas and perimeters?

**Tip:** When solving problems with both area and perimeter constraints, it's often helpful to reduce the system to one variable before solving the system of equations.

A veterinarian wishes to use 132 feet of chain-link fencing to enclose a rectangular region and subdivide the region into two smaller rectangular regions. If the total enclosed area is 576 square feet, find the dimensions of the enclosed region.

This problem involves using 132 feet of fencing to enclose a rectangular area subdivided into two smaller regions, with a total enclosed area of 576 square feet. Learn how to solve the system of equations to determine the dimensions of the enclosure. Suitable for Grades 9-12.

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