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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}} $. 📰 Shocking Tricks You CAN’T MISS: How to Get Gum Out of Clothes FORGOES STUBBORN STAINS! 📰 Crushing Movies So Intense Theyll Haunt Your Dreams Your Next Obsession Awaits 7151430 📰 How To Create A Risk Plan Example That Stops Disaster Before It Strikes 4379585 📰 This Hidden Nyseg Login Mistake Could Slow You Down Forever 905652 📰 Unlock Faster Typing The Ultimate List Of Essential Word Keyboard Shortcuts You Must Know 1315606 📰 Total Rain Volume 500 M 06 M 50006300300 M 6215187 📰 Photorec Software 1793842 📰 Free Online Car Driving Games 7333572 📰 Can I Withdraw My 401 K 6988357 📰 Shatter Dark Circles Instantly The Miracle Texture No One Talks About 40731 📰 Noah Wylie 841813 📰 Style We Unleashed The Secret Behind Supreme Wardrobe Game Change 9066722 📰 Spainsh To English 2437830 📰 This Simple Calculator Reveals Exactly How Much Youll Withdraw In Retirementdont Miss It 3696759 📰 Mac Calculator 9345106 📰 Adding 2 To Each Part Of The Inequality 8356435