The area of a rectangle exceeds the area of a circle by $528 ext{ cm}^2$. The ratio of the length to the breadth of the rectangle is $26 : 11$. If the radius of the circle is $\frac{7}{11}$ of the breadth of the rectangle, find the radius of the circle (in cm).

Question

The area of a rectangle exceeds the area of a circle by $528 	ext{ cm}^2$. The ratio of the length to the breadth of the rectangle is $26 : 11$. If the radius of the circle is $\frac{7}{11}$ of the breadth of the rectangle, find the radius of the circle (in cm).

Accepted Answer

**Solution:**

1. Let the length and breadth of the rectangle be $26x$ and $11x$ respectively.

2. Given the radius of the circle is $\frac{7}{11}$ of the breadth, so radius $r = 11x 	imes \frac{7}{11} = 7x$.

3. Area of rectangle = $26x 	imes 11x = 286x^2$.

4. Area of circle = $\pi r^2 = \frac{22}{7} 	imes (7x)^2 = \frac{22}{7} 	imes 49x^2 = 154x^2$.

5. According to the question, the difference in areas is $528$:
   $$286x^2 - 154x^2 = 528$$
   $$132x^2 = 528$$
   $$x^2 = \frac{528}{132} = 4$$
   $$x = 2$$

6. Therefore, the radius of the circle is:
   $$r = 7x = 7 	imes 2 = 14 	ext{ cm}$$

**Note:** Rechecking the original solution, the value of $x$ was found as 4, so let's verify calculations carefully:

- Step 3: Area of rectangle = $26x 	imes 11x = 286x^2$ (correct)
- Step 4: Area of circle = $\frac{22}{7} 	imes (7x)^2 = \frac{22}{7} 	imes 49x^2 = 154x^2$ (correct)
- Step 5: Difference = $286x^2 - 154x^2 = 132x^2 = 528$
- So, $132x^2 = 528 \Rightarrow x^2 = 4 \Rightarrow x = 2$

7. Now, radius $r = 7x = 7 	imes 2 = 14$ cm.

**However, the original answer given is 28 cm, which suggests $x=4$. Let's check the original problem statement carefully:**

- The original solution states $132x = 528$, which is incorrect because the difference involves $x^2$, not $x$.

- Correct equation should be:
  $$26x 	imes 11x - \frac{22}{7} 	imes (7x)^2 = 528$$
  $$286x^2 - 154x^2 = 528$$
  $$132x^2 = 528$$
  $$x^2 = 4$$
  $$x = 2$$

- So radius = $7x = 7 	imes 2 = 14$ cm.

**Therefore, the correct radius is $14$ cm, option (a).**

**Answer:** $14$ cm

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