SAT Math: In a Circle with Central Angle θ, Arc Length is 8π and Sector Area is 48π

Exact Question Text

In a circle with center O, central angle theta intercepts an arc of length 8pi. The sector area is 48pi. What is the radius of the circle?

1. Step-by-Step Formal Solution

1. s = r * theta = 8pi.
2. Area = 1/2 r^2 theta = 1/2 r (r theta) = 1/2 r * (8pi) = 48pi.
3. 4pi * r = 48pi => r = 12.

2. Fast Desmos / Calculator Shortcut

Type 2 * (48pi) / (8pi) in Desmos => outputs 12.

3. Distractor Trap Analysis

• Trap A (Sign Flip): Forgetting to reverse inequality or dropping negative sign.
• Trap B (Partial Evaluation): Solving for intermediate variable and stopping early.
• Trap C (Extreme Assumption): Over-inferring beyond textual evidence.
• Option Correct: Accurately satisfies all algebraic and logical constraints.