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Home » Grandpa’s Hilarious Struggles? Watch His Face When He Tells These Stories! - Carbonext

Grandpa’s Hilarious Struggles? Watch His Face When He Tells These Stories! - Carbonext

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 The radius $r$ of the incircle of a right triangle is given by:

📌 r = \frac{a + b - t}{2}

📌 \frac{\pi r^2}{\frac{1}{2}ab} = \frac{2\pi r^2}{ab}

📌 Substituting $r = \frac{a + b - t}{2}$:

📌 \frac{2\pi \left(\frac{a + b - t}{2}\right)^2}{ab} = \frac{2\pi \cdot \frac{(a + b - t)^2}{4}}{ab} = \frac{\pi (a + b - t)^2}{2ab}

📌 \boxed{\frac{\pi (a + b - t)^2}{2ab}}

📌 Question: An astronomer observes a planetary orbit modeled as an ellipse with semi-major axis $a = 10$ AU and semi-minor axis $b = 6$ AU. What is the approximate circumference of the elliptical orbit? (Use Ramanujan’s approximation: $C \approx \pi \left[3(a + b) - \sqrt{(3a + b)(a + 3b)}\right]$.)

📌 Solution: Using Ramanujan’s approximation for the circumference of an ellipse:

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