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Home » = 1/2 × 9 × height. - jntua results

= 1/2 × 9 × height. - jntua results

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Area of the Square:

📌 The side length of the square is \( 2r \), so its area is:

📌 A_{\text{square}} = (2r)^2 = 4r^2

📌 Ratio of Areas:

📌 The ratio of the area of the circle to the area of the square is:

📌 \text{Ratio} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi r^2}{4r^2} = \frac{\pi}{4}

📌 Thus, the ratio of the area of the circle to the area of the square is \(\boxed{\frac{\pi}{4}}\).

📌 A computational evolutionary ecology AI researcher is analyzing the growth patterns of circular plant colonies in Patagonia. One colony grows to form a circle with radius \( x \), while another expands to form an octagon with each side equal to \( x \). What is the ratio of the area of the circle to the area of the octagon?

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We need to solve: 0.05n log₂(n) < 0.01n².
Divide both sides by n (n > 0): 0.05 log₂(n) < 0.01n → log₂(n) < 0.2n.
We test values:
n = 10: log₂(10) ≈ 3.32, 0.2×10 = 2 → 3.32 > 2 → A faster
n = 20: log₂(20) ≈ 4.32, 0.2×20 = 4 → 4.32 > 4
n = 25: log₂(25) ≈ 4.64, 0.2×25 = 5 → 4.64 < 5 → B faster
n = 24: log₂(24) ≈ 4.58, 0.2×24 = 4.8 → 4.58 < 4.8 → B faster
n = 23: log₂(23) ≈ 4.52, 0.2×23 = 4.6 → 4.52 < 4.6 → B faster
n = 22: log₂(22) ≈ 4.46, 0.2×22 = 4.4 → 4.46 > 4.4 → A faster

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