In one circle, an arc subtends a central angle of 30°. The length of this arc is double the length of an arc in a second circle whose radius is three times the radius of the first circle. What is the measure of the central angle subtended by the arc in the second circle (in degrees)?

Introduction / Context:
This question tests the relationship between arc length, radius, and central angle in circles. It uses proportional reasoning across two different circles, which is a common theme in aptitude and geometry problems involving circles and arcs.

Given Data / Assumptions:

• Circle 1 has radius r₁ and an arc that subtends a central angle of 30°.
• Circle 2 has radius r₂ = 3r₁.
• The length of the arc in circle 1 is twice the length of an arc in circle 2.
• The arcs are compared by length, not by angle.
• We must find the central angle θ₂ of the arc in circle 2.

Concept / Approach:
Arc length is directly proportional to both the radius and the central angle (in the same units, typically radians). For a given unit system, we can write arc length s = k * r * θ, where k is a constant depending on unit choice. Since we compare ratios, the constant cancels out. We set up a proportion comparing the two arc lengths, using r₂ = 3r₁ and the fact that the arc in circle 1 is twice the arc in circle 2, then solve for θ₂ (in degrees).

Step-by-Step Solution:
Step 1: Let s₁ be the arc length in circle 1 and s₂ be the arc length in circle 2. Step 2: For circle 1: s₁ ∝ r₁ * θ₁, with θ₁ = 30°. Step 3: For circle 2: s₂ ∝ r₂ * θ₂ = 3r₁ * θ₂. Step 4: We are told s₁ = 2 * s₂. Step 5: Replace s₁ and s₂ with their proportional forms: r₁ * 30° = 2 * (3r₁ * θ₂). Step 6: Cancel r₁ from both sides: 30° = 2 * 3 * θ₂ = 6θ₂. Step 7: Solve for θ₂: θ₂ = 30° / 6 = 5°.

Verification / Alternative Check:
Think in terms of ratios. If radius is tripled (from r₁ to 3r₁) but the arc in the larger circle is half the length of the arc in the smaller circle, then the product radius × angle for the second circle must be one sixth of the product radius × angle for the first circle. Because 30° divided by 6 is 5°, this is consistent with the proportional relationship and supports the result θ₂ = 5°.

Why Other Options Are Wrong:
Angles 3°, 4°, and 6° would not satisfy the required ratio. For example, with θ₂ = 6°, the product 3r₁ * 6° = 18r₁ would make s₂ too large, so it would not be half of s₁. Only θ₂ = 5° makes the proportional equality hold true.

Common Pitfalls:
Some students mistakenly think that if the radius is tripled and the arc is half as long, the angle must be divided by 3 or 2 separately, leading to incorrect values. Others forget that arc length is proportional to the product of radius and angle, not to each separately. Setting up a clear proportional equation helps avoid these errors.

Final Answer:
The central angle in the second circle is 5°.