Digital SAT Math
SAT Math: In a Circle with Central Angle θ, Arc Length is 8π and Sector Area is 48π
• Direct Answer / Key Takeaway
Area = 1/2 * r * s => 48pi = 1/2 * r * (8pi) => 48pi = 4pi * r => r = 12.
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
- s = r * theta = 8pi.
- Area = 1/2 r^2 theta = 1/2 r (r theta) = 1/2 r * (8pi) = 48pi.
- 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.
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