The perimeter of a square with side length $(A + 10) ext{ cm}$ is equal to the perimeter of a rectangle with length $A ext{ cm}$ and breadth $3A ext{ cm}$. Find the difference between the area of the rectangle and the square in square centimeters.

Question

The perimeter of a square with side length $(A + 10) 	ext{ cm}$ is equal to the perimeter of a rectangle with length $A 	ext{ cm}$ and breadth $3A 	ext{ cm}$. Find the difference between the area of the rectangle and the square in square centimeters.

Accepted Answer

**Solution:**

1. Given the perimeter of the square with side $(A + 10)$ cm is equal to the perimeter of the rectangle with length $A$ cm and breadth $3A$ cm.

2. Calculate the perimeter of the square:
   $$P_{square} = 4 	imes (A + 10) = 4A + 40$$

3. Calculate the perimeter of the rectangle:
   $$P_{rectangle} = 2 	imes (A + 3A) = 2 	imes 4A = 8A$$

4. Since the perimeters are equal:
   $$4A + 40 = 8A$$

5. Solve for $A$:
   $$40 = 8A - 4A = 4A \implies A = 10$$

6. Calculate the area of the rectangle:
   $$Area_{rectangle} = length 	imes breadth = A 	imes 3A = 3A^2 = 3 	imes 10^2 = 300$$

7. Calculate the area of the square:
   $$Area_{square} = (A + 10)^2 = (10 + 10)^2 = 20^2 = 400$$

8. Find the difference between the areas:
   $$|Area_{rectangle} - Area_{square}| = |300 - 400| = 100$$

**Answer:** $100$

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