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📰 Solution: Let original side be $ s $. Original area: $ \frac{\sqrt{3}}{4} s^2 $. New side $ s - 3 $, new area: $ \frac{\sqrt{3}}{4} (s - 3)^2 $. The difference: $ \frac{\sqrt{3}}{4} [s^2 - (s - 3)^2] = 15\sqrt{3} $. Simplify: $ \frac{\sqrt{3}}{4} (6s - 9) = 15\sqrt{3} $. Cancel $ \sqrt{3} $ and solve $ \frac{6s - 9}{4} = 15 $, leading to $ 6s - 9 = 60 $, so $ s = \frac{69}{6} = 11.5 $. Original side length is $ \boxed{11.5} \, \text{cm} $.
📰 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}} $.
