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Home » Area: \( 12 \times 16 = 192 \) square cm - Groen Casting

Area: \( 12 \times 16 = 192 \) square cm - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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]

📌 \sqrt{36 \cdot 28} = \sqrt{1008} \approx 31.75

📌 C \approx \pi (48 - 31.75) = \pi (16.25)

📌 \boxed{16.25\pi}

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