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📰 Question: An equilateral triangle’s side is reduced by 3 cm, decreasing its area by $ 15\sqrt{3} \, \text{cm}^2 $. What was the original side length? 📰 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. 📰 Hotels In Beloit Wi 4597323 📰 The Secret Power That Changes Your Life Overnight 2283178 📰 Bargain Hunt 2060597 📰 Sourcetree Git Client 7572444 📰 Labubu Mystery Box Price 3998429 📰 Allegiant Bank Of America Login 7664856 📰 License Win 10 2240508 📰 Food Sector Earnings Soarbut Margin Pressures Are Crushing Profits In 2024 2216867 📰 South Ireland News 236751 📰 The Crumbling Credit Adjust These Simple Habits In Huntington Beach 8279761 📰 This High Performance Scyther Just Sold Outwhat Makes It Unstoppable 8868779 📰 Jokes With Dad 3710209 📰 Brown Cardigan Code The Must Have Piece That Every Wardrobe Needs 4206877 📰 Love Far Apart These Long Distance Couple Apps Will Bring You Closer Instantly 1069077 📰 Hotel Riu Plaza Miami Beach Collins Avenue Miami Fl 4479884