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To solve this problem, let's break it down step by step.

### Problem Overview
You have six circles arranged within a rectangle, and you're asked to find the radius of the largest circle given that the perimeter of the rectangle is 288.

### Step 1: Analyze the Circle Arrangement
Based on the image:
- There are six circles: three smaller circles on the bottom row, two medium-sized circles in the middle, and one large circle in the top left.
- The circles are tangent to each other and to the sides of the rectangle.

### Step 2: Set Up the Dimensions of the Rectangle
Let:
- \( r \) be the radius of the largest circle (top left).
- \( r_2 \) be the radius of the medium circles.
- \( r_3 \) be the radius of the small circles.

The width of the rectangle is the sum of the diameters of the circles along the bottom row, and the height of the rectangle is the sum of the diameters along the left side.

### Step 3: Express the Perimeter of the Rectangle
The perimeter \( P \) of the rectangle is given by:
\[
P = 2(\text{Width} + \text{Height}) = 288
\]

The width is:
\[
\text{Width} = 2r_3 + 2r_2 + 2r_3 = 2(r_2 + 2r_3)
\]

The height is:
\[
\text{Height} = 2r + 2r_2 = 2(r + r_2)
\]

Substituting these into the perimeter equation:
\[
2\left(2(r_2 + 2r_3) + 2(r + r_2)\right) = 288
\]

Simplify:
\[
4(r_2 + 2r_3 + r + r_2) = 288
\]
\[
r + 2r_2 + 2r_3 = 72
\]

### Step 4: Analyze the Circle Relationships
The smaller circles have the smallest radius \( r_3 \). Let’s consider the geometric relationship between the circles:
- The largest circle \( r \) and one medium circle \( r_2 \) are tangent.
- \( r_2 \) is the same for both medium circles.
- The medium and small circles are tangent as well.

The radii follow the proportional relationship:
\[
r = 2r_2 = 4r_3
\]

### Step 5: Substitute and Solve
Given \( r = 2r_2 \) and \( r_2 = 2r_3 \), we substitute:
\[
r + 2r_2 + 2r_3 = 72
\]
\[
r + r + \frac{r}{2} = 72
\]
\[
\frac{5r}{2} = 72
\]
\[
r = \frac{72 \times 2}{5} = \frac{144}{5} = 28.8
\]

### Final Answer
The radius of the largest circle is **28.8 units**.

Would you like further details or have any questions? Here are five related questions you might consider:

1. How can the radii of the medium and small circles be calculated from this result?
2. What would be the width and height of the rectangle based on these circle radii?
3. How does the relationship between circle radii help in other geometric problems?
4. Could this problem be solved using a different method, like using algebra or calculus?
5. How would this solution change if the perimeter of the rectangle were different?

**Tip:** Always check the consistency of your geometric relationships by verifying with the given constraints (like the perimeter). This helps avoid mistakes in setting up the equations.

Discover how to find the radius of the largest circle among six circles arranged within a rectangle, where the perimeter of the rectangle is 288 units. Follow a step-by-step explanation involving circle geometry and proportional relationships. Suitable for advanced high school students and anyone interested in circle geometry problems.

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