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📰 Solution: For a right triangle with legs $ a $, $ b $, hypotenuse $ c $, the inradius is $ r = \frac{a + b - c}{2} $. Using the Pythagorean theorem $ c = \sqrt{a^2 + b^2} $, substitute $ b = \frac{2r + c - a}{1} $ from rearranging the inradius formula. Squaring and solving yields $ c = a + \frac{2r^2 + 2ar}{2r + 2a} $. Simplifying, $ c = \frac{a(2r + a)}{r} $. Thus, the hypotenuse is $ \boxed{\dfrac{a(a + 2r)}{r}} $. 📰 Question: A carbon nanotube is modeled as a cylinder with radius $ r $ and height $ h $. If its lateral surface area equals the area of a circle with radius $ 2r $, what is the ratio $ \frac{h}{r} $? 📰 Solution: The lateral surface area of a cylinder is $ 2\pi r h $. The area of a circle with radius $ 2r $ is $ \pi (2r)^2 = 4\pi r^2 $. Setting them equal: $ 2\pi r h = 4\pi r^2 $. Cancel $ 2\pi r $ to get $ h = 2r $. Thus, the ratio $ \frac{h}{r} = \boxed{2} $. 📰 Roll In Shower Hack Thats Changing How We Relaxsee If Youre Missing It 2385143 📰 No Way They Saw How Fast They Could Create What Theyd Dreamed Offind Out The Secret To Building Your Legacy Instantly 4187233 📰 Finding Dea Numbers By Npi Made Easy Watch The Fast Lookup Reveal Everything 7849109 📰 Spangram Today 8473877 📰 Free Crosshair 977185 📰 Finiah Line Near Me 1396359 📰 Harry Houdini 7873773 📰 How To Plan A Wedding Ceremony 9280354 📰 The Stream East App 704821 📰 A Circle Is Inscribed In A Square With Side Length 14 Cm What Is The Area Of The Region Between The Circle And The Square 4835224 📰 This Booboo The Fool Moment Will Make You Rantwhy Everyones Freaking Out 4823660 📰 My Breast Friend Sally 1000053 📰 7 Of Cups Reversed You Wont Believe What This Mind Reading Revelation Means 8073844 📰 3 Nsaqubt News Just Broke The News Everyones Been Waiting Fordont Miss It 6733130 📰 You Wont Believe What Happens When You Land At Fidelity Larkspur Landingthis Hidden Gem Is Cut Right From The Regions Best 3809124