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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. 📰 Supercharge Puppy Training Crate Like A Seasoned Owner Today 5490789 📰 1 To Rub This Mind Blowing Exchange Rate Will Change How You Spend 5071969 📰 Tampa Airport To St Pete Beach 9450186 📰 Never Guess This Yuzu Sauce Will Take Your Dishes From Blah To Bold 1683478 📰 You Wont Believe What Happens When You Join This Hidden Rel Chat 6277227 📰 Gamegame Like Never Before Hackers Are Playing This Viral Game Everywhere 1376550 📰 My Teaching Strategies 7600956 📰 Lottery Powerball 3861464 📰 Grand Rapids First 3945478 📰 Shoprite Marlboro New Jersey 154779 📰 Master Every Move In This Immersive Football Game Play Today 409157 📰 A G T Judges 2521682 📰 Car Financing Interest Rate 2056558 📰 Read In The Dark Like A Pro Unlock The Secret With This Breakthrough Reading Light App 3809011 📰 Flies Wont Stand A Chancethis Method You Never Knew About 3818635