Length of a rectangular land is twice the radius of a circle whose circumference is $132 ext{ cm}$. The area of the rectangular land increases by $144 ext{ sq.cm}$ when a square land is attached along its breadth. Find the area of the rectangle in $ ext{sq.cm}$.

Question

Length of a rectangular land is twice the radius of a circle whose circumference is $132 	ext{ cm}$. The area of the rectangular land increases by $144 	ext{ sq.cm}$ when a square land is attached along its breadth. Find the area of the rectangle in $	ext{sq.cm}$.

Accepted Answer

**Solution:**

1. Given the circumference of the circle is $132 	ext{ cm}$. Using the formula for circumference:
   $$2 \pi r = 132$$

2. Calculate the radius $r$:
   $$r = \frac{132}{2 \pi} = \frac{132}{2 	imes 3.14} = \frac{132}{6.28} \approx 21 	ext{ cm}$$

3. Length of the rectangle is twice the radius:
   $$	ext{Length} = 2r = 2 	imes 21 = 42 	ext{ cm}$$

4. Let the breadth of the rectangle be $b$ cm.

5. When a square land is attached along the breadth, the area increases by $144 	ext{ sq.cm}$. The square's side is equal to the breadth $b$.

6. The increase in area due to the square is:
   $$b^2 = 144 \implies b = \sqrt{144} = 12 	ext{ cm}$$

7. Now, area of the rectangle:
   $$	ext{Area} = 	ext{Length} 	imes 	ext{Breadth} = 42 	imes 12 = 504 	ext{ sq.cm}$$

**Answer:** $504$

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