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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 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:

📌 C \approx \pi \left[3(a + b) - \sqrt{(3a + b)(a + 3b)}\right]

📌 Substitute $a = 10$ and $b = 6$:

📌 C \approx \pi \left[3(10 + 6) - \sqrt{(3 \cdot 10 + 6)(10 + 3 \cdot 6)}\right] = \pi \left[48 - \sqrt{(36)(28)}\right]

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