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📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 3Question: In a right triangle, the inradius is $ r $ and one leg is $ a $. Express the hypotenuse $ c $ in terms of $ r $ and $ a $.

📌 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} $.

📌 Question: A circle is inscribed in a square with side length $ s $. What is the ratio of the circle’s area to the square’s area, expressed in terms of $ \pi $?

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