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Home » Justice League vs. Teen Titans – The Ultimate Showdown That Explosively Changed DC! - Aurero

Justice League vs. Teen Titans – The Ultimate Showdown That Explosively Changed DC! - Aurero

📅 March 15, 2026 👤 scraface

Mar 15, 2026

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📌 Question: A quantum dot (modeled as a sphere) has radius $ r $. If its surface area equals the area of a circle with radius $ \sqrt{2}r $, find $ r $ in terms of the circle’s radius.

📌 Solution: Sphere surface area: $ 4\pi r^2 $. Circle area: $ \pi (\sqrt{2}r)^2 = 2\pi r^2 $. Setting equal: $ 4\pi r^2 = 2\pi r^2 $. This implies $ 4 = 2 $, a contradiction. Thus, no solution exists unless the circle’s radius is adjusted. However, if the problem states equivalence, the only possibility is $ r = 0 $, which is trivial. Rechecking the question reveals a misstatement; assuming the circle’s radius is $ R $, then $ 4\pi r^2 = \pi R^2 \Rightarrow R = 2r $. The original question’s setup is inconsistent, but if forced, $ r = \frac{R}{2} $, so $ \boxed{r = \dfrac{R}{2}} $.

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