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First, calculate the number of scenarios successfully resolved initially: - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 \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]

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

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

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\( l^2 + 25 = 169 \)
\( l^2 = 144 \)
\( l = 12 \).
تقطع سيارة 150 كم في ساعتين و30 دقيقة. ما هي سرعتها المتوسطة بالكيلومت...
تحويل الوقت: 2 ساعة 30 دقيقة = 2.5 ساعة.
السرعة المتوسطة = \( \frac{150}{2.5} = 60 \) كم / ساعة.
يحتوي خزان أسطواني على ارتفاع 10 أمتار وقطر 4 أمتار. ما حجم الخزان بال...
نصف القطر = 2 متر.
الحجم = \( \pi r^2 h = \pi (2^2)(10) = 40\pi \).
تقريب \( \pi \approx 3.14 \):

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