To solve the problem of finding the length and breadth of the rectangular park given the cost of fencing and their ratio, follow these steps:

1. Determine the total perimeter of the park:

   • The cost of fencing is given as ₹3648.
   • The cost per meter of fencing is ₹28.50.
   • The total length of the fencing (perimeter) can be found by dividing the total cost by the cost per meter.

   $\text{Total Perimeter} = \frac{3648}{28.50}$

2. Calculate the total perimeter:

   $\text{Total Perimeter} = \frac{3648}{28.50} = 128 \text{ meters}$

3. Set up the relationship between the length and breadth:

   • Let the length be $5x$ and the breadth be $3x$, based on the given ratio $5:3$.

4. Express the perimeter in terms of $x$:

   • The formula for the perimeter of a rectangle is $2 \times (\text{length} + \text{breadth})$.

   $2 \times (5x + 3x) = 128$

   Simplifying this:

   $2 \times 8x = 128$

   $16x = 128$

   $x = \frac{128}{16} = 8$

5. Find the length and breadth:

   • Length $= 5x = 5 \times 8 = 40$ meters
   • Breadth $= 3x = 3 \times 8 = 24$ meters

Thus, the length of the park is 40 meters and the breadth is 24 meters.

Would you like details or have any questions? Here are some related questions that could further your understanding:

1. How is the cost per meter of fencing used to find the total perimeter?
2. Can you derive the perimeter formula for different shapes?
3. How would the problem change if the cost per meter of fencing was different?
4. What if the ratio of length to breadth was different, say 4:3?
5. How would you calculate the area of the park once you know the length and breadth?
6. What are some real-life applications of understanding perimeter and area?
7. How would the solution differ if only the perimeter and ratio were given, but not the cost?
8. How can you verify the solution to ensure the length and breadth calculations are correct?

Tip: Always verify your solutions by plugging values back into the original problem to ensure consistency.