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📰 Solution: Original radius $ r = \sqrt{\frac{100\pi}{\pi}} = 10 \, \text{km} $. New radius $ r' = 8 \, \text{km} $. Original area $ 100\pi $, new area $ \pi (8)^2 = 64\pi $. The decrease is $ 100\pi - 64\pi = \boxed{36\pi} \, \text{km}^2 $. 📰 Question: A zoologist tracks an animal’s path forming a right triangle with hypotenuse $ 25 \, \text{m} $ and inradius $ 6 \, \text{m} $. What is the ratio of the inscribed circle’s area to the triangle’s area? 📰 Solution: For a right triangle, the inradius $ r = \frac{a + b - c}{2} $, where $ c $ is the hypotenuse. Let $ a $ and $ b $ be the legs. Then $ r = \frac{a + b - 25}{2} = 6 \Rightarrow a + b = 37 $. The area of the triangle is $ \frac{ab}{2} $, and the inradius formula $ r = \frac{a + b - c}{2} $ gives consistency. Using $ a^2 + b^2 = 625 $ and $ a + b = 37 $, solve for $ ab $: $ (a + b)^2 = 1369 = a^2 + b^2 + 2ab = 625 + 2ab \Rightarrow ab = 372 $. Area $ = 186 \, \text{m}^2 $. Circle area $ \pi r^2 = 36\pi $. Ratio $ \frac{36\pi}{186} = \boxed{\frac{6\pi}{31}} $. 📰 What Is A Backdoor Roth Ira 9991385 📰 How The Chf To Usd Exchange Rate Shocked Markets Todaywatch The History Unfold 3173690 📰 You Wont Let This Gifted Squilliam Fancyson Slip Awaydiscover His Hidden Talents 4523772 📰 Stop Strugglingyour Body Was Made To Stay Fit And Live Strong 2402075 📰 Youll Wake Up In Hyrule Warriors Like Never Beforewatch The Unbelievable Turnaround 2112389 📰 Hsa Bank App Review The Ultimate Tool For Smart Healthcare Spendingtry It Today 7293704 📰 Why Every Modern Business Needs Cutting Edge Supply Chain Software Click To Discover The Must Have Tool 9669597 📰 Youll Never Guess What Hidden Treasures Are Hidden In Your Microsoft Folder 5190147 📰 Is The Us Stock Market Open Today 9060417 📰 Big Movers Stocks Are Surprising Investorsheres How Theyre Transforming The Market 8362065 📰 Ntpc Limited Share Price 5398161 📰 Actually The Standard Method For Circular Arrangements Fixing One Person Eliminates Rotation Then Linear Permutation Of Remaining 7 6517430 📰 5Fix Stick Drift On Ps5 Gamers Are Raving About This Game Changing Hack 233840 📰 You Wont Believe How Much These Reading Quotes Boost Your Reading Game 8455464 📰 Good Programs For Mac 7929852