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Let's solve this step-by-step:

### Given:
1. The perimeter of the rectangular field \( P = 500 \) yards.
2. The area of the rectangular field \( A = 14400 \) square yards.

### Formulas:
For a rectangle:
1. **Perimeter** \( P = 2(l + b) \), where \( l \) is the length and \( b \) is the breadth.
2. **Area** \( A = l \times b \).

### Steps to Solve:
1. From the perimeter formula:
   \[
   2(l + b) = 500
   \]
   Dividing both sides by 2:
   \[
   l + b = 250
   \]
   So, we have:
   \[
   b = 250 - l
   \]

2. Substitute \( b = 250 - l \) into the area formula:
   \[
   l \times (250 - l) = 14400
   \]
   Expanding this equation:
   \[
   250l - l^2 = 14400
   \]
   Rearranging terms to form a standard quadratic equation:
   \[
   l^2 - 250l + 14400 = 0
   \]

3. Solve this quadratic equation using the quadratic formula:
   \[
   l = \frac{-(-250) \pm \sqrt{(-250)^2 - 4 \cdot 1 \cdot 14400}}{2 \cdot 1}
   \]
   Simplifying:
   \[
   l = \frac{250 \pm \sqrt{62500 - 57600}}{2}
   \]
   \[
   l = \frac{250 \pm \sqrt{4900}}{2}
   \]
   \[
   l = \frac{250 \pm 70}{2}
   \]
   So, we get two possible values for \( l \):
   \[
   l = \frac{250 + 70}{2} = 160 \quad \text{and} \quad l = \frac{250 - 70}{2} = 90
   \]

4. Substitute these values back to find \( b \):
   - If \( l = 160 \), then \( b = 250 - 160 = 90 \).
   - If \( l = 90 \), then \( b = 250 - 90 = 160 \).

### Answer:
The length and breadth of the field are \( 160 \) yards and \( 90 \) yards.

Would you like a detailed breakdown of any specific step, or do you have any questions?

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Here are 5 related questions you might find interesting:
1. How can we verify the solution by recalculating the perimeter and area?
2. What are some other ways to solve quadratic equations besides the quadratic formula?
3. How would the solution change if only the perimeter or only the area were given?
4. What if the perimeter and area values were different? How would that affect the possible dimensions?
5. Can we apply this method to solve for dimensions of non-rectangular shapes?

**Tip:** Always double-check your solutions by substituting them back into the original equations, especially in geometry problems like this one!

Perimeter of rectangular field is 500 yards and its area is 14400 sq yards. What is its length and breadth?

Find the length and breadth of a rectangular field with a perimeter of 500 yards and an area of 14400 sq yards. This problem involves algebraic and geometric concepts, including perimeter, area, and quadratic equations. Ideal for students in Grades 8-10.

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