The perimeter of a rectangle is 82 m and its area is 400m^2. The breadth of the rectangle is :

Question

Accepted Answer

To determine the breadth of the rectangle, we need to use the given information about the perimeter and area of the rectangle.

1. **Understanding the Problem:**
   - The perimeter of a rectangle is given by the formula: $ P = 2(l + b) $, where $ l $ is the length and $ b $ is the breadth.
   - The area of a rectangle is given by the formula: $ A = l \times b $.

2. **Given Information:**
   - Perimeter ($ P $) = 82 m
   - Area ($ A $) = 400 m²

3. **Using the Perimeter:**
   - From the perimeter formula: $ 2(l + b) = 82 $
   - Simplifying, we get: $ l + b = 41 $  (Equation 1)

4. **Using the Area:**
   - From the area formula: $ l \times b = 400 $  (Equation 2)

5. **Solving the Equations:**
   - From Equation 1, express $ l $ in terms of $ b $: $ l = 41 - b $
   - Substitute $ l = 41 - b $ into Equation 2: 
     $$
     (41 - b) \times b = 400
     $$
   - Expanding and rearranging gives: 
     $$
     41b - b^2 = 400
     $$
     $$
     b^2 - 41b + 400 = 0
     $$

6. **Solving the Quadratic Equation:**
   - The quadratic equation $ b^2 - 41b + 400 = 0 $ can be solved using the quadratic formula: 
     $$
     b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
     $$
     where $ A = 1 $, $ B = -41 $, and $ C = 400 $.
   - Calculate the discriminant: 
     $$
     B^2 - 4AC = 41^2 - 4 \times 1 \times 400 = 1681 - 1600 = 81
     $$
   - Since the discriminant is a perfect square, the roots are real and rational.
   - Calculate the roots:
     $$
     b = \frac{41 \pm \sqrt{81}}{2} = \frac{41 \pm 9}{2}
     $$
   - This gives two possible solutions for $ b $:
     $$
     b = \frac{41 + 9}{2} = 25 \quad \text{or} \quad b = \frac{41 - 9}{2} = 16
     $$

7. **Selecting the Correct Breadth:**
   - The possible breadth values are 25 m and 16 m. However, only 16 m is listed among the options provided.
   - Therefore, the correct breadth of the rectangle, according to the options given, is 16 m.

8. **Verification:**
   - If $ b = 16 $, then $ l = 41 - 16 = 25 $.
   - Check the area: $ l \times b = 25 \times 16 = 400 $ m², which matches the given area.
   - Check the perimeter: $ 2(l + b) = 2(25 + 16) = 82 $ m, which matches the given perimeter.

9. **Conclusion:**
   - Option 4 (16 m) is the correct answer because it satisfies both the perimeter and area conditions given in the problem.

**Explanation for Incorrect Options:**
- **Option 1 (4 m):** If the breadth were 4 m, the length would be 37 m (from $ l + b = 41 $), giving an area of $ 4 \times 37 = 148 $ m², which does not match the given area.
- **Option 2 (8 m):** If the breadth were 8 m, the length would be 33 m, giving an area of $ 8 \times 33 = 264 $ m², which does not match the given area.
- **Option 3 (12 m):** If the breadth were 12 m, the length would be 29 m, giving an area of $ 12 \times 29 = 348 $ m², which does not match the given area.

Thus, Option 4 (16 m) is the only correct choice.