The difference between the circumferences of two circles, P and Q, is $132 ext{ cm}$. If the radius of circle Q is 4 times the radius of circle P, what is the area of circle Q (in square cm)?

Question

The difference between the circumferences of two circles, P and Q, is $132 	ext{ cm}$. If the radius of circle Q is 4 times the radius of circle P, what is the area of circle Q (in square cm)?

Accepted Answer

**Solution:**

1. Let the radius of circle P be $r$ cm and the radius of circle Q be $R$ cm.

2. Given that $R = 4r$.

3. The difference between their circumferences is given by:
   $$2 \pi R - 2 \pi r = 132$$

4. Substitute $R = 4r$:
   $$2 \pi (4r) - 2 \pi r = 132$$
   $$8 \pi r - 2 \pi r = 132$$
   $$6 \pi r = 132$$

5. Using $\pi = \frac{22}{7}$, solve for $r$:
   $$6 	imes \frac{22}{7} 	imes r = 132$$
   $$\Rightarrow r = \frac{132 	imes 7}{6 	imes 22} = 7 	ext{ cm}$$

6. Therefore, radius of circle Q is:
   $$R = 4r = 4 	imes 7 = 28 	ext{ cm}$$

7. Calculate the area of circle Q:
   $$	ext{Area} = \pi R^2 = \frac{22}{7} 	imes 28 	imes 28 = 2464 	ext{ cm}^2$$

**Answer:** $2464$

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