Question

Sakshi drew a rangoli design for a competition as shown below:

Circles C₁, C₂, C₃ and Ca have a common centre P.

The radius of circle C₁ = 6 cm.

The table given below shows the radii of the circles in terms of the radius of circle C₁.

Radius of circle	Radius in terms of circle C1
C2	2×
C3	2.5×
C4	3.5×

1. The circumference of circle C₂ is:
   1. 12π cm
   2. 24π cm
   3. 15π cm
   4. 20π cm
2. The area enclosed between C₁ and C₂ and also between C₃ and C₄ has been painted. What area of the rangoli has been painted ?
   1. 85π cm²
   2. 108π cm²
   3. 324π cm²
   4. 846π cm²

Answer 1

Let r₁ be the radius of circle C₁. According to the problem statement:

r₁ = 6 cm

From the table, the radii of the other circles in terms of r₁ (letting x = r₁ = 6 cm) are:

• Radius of circle C₂: r₂ = 2x = 2(6) = 12 cm
• Radius of circle C₃: r₃ = 2.5x = 2.5(6) = 15 cm
• Radius of circle C₄: r₄ = 3.5x = 3.5(6) = 21 cm

(i)

The formula for the circumference of a circle is 2πr. Substituting the radius of C₂:

\[\text{Circumference} = 2\pi r_2 = 2\pi(12) = 24\pi\text{ cm}\]

Therefore, the correct option is (b).

(ii)

The painted regions are the area enclosed between C₁ and C₂, and the area enclosed between C₃ and C₄.

1. Area between C₁ and C₂:
   \[\text{Area}_1 = \pi r_2^2 - \pi r_1^2 = \pi (12^2 - 6^2) = \pi (144 - 36) = 108\pi\text{ cm}^2\]
2. Area between C₃ and C₄:
   \[\text{Area}_2 = \pi r_4^2 - \pi r_3^2 = \pi (21^2 - 15^2) = \pi (441 - 225) = 216\pi\text{ cm}^2\]
3. Total painted area:
   \[\text{Total Area} = 108\pi + 216\pi = 324\pi\text{ cm}^2\]

Therefore, the correct option is (c).