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Home » p'''(1) = 24 imes 1 - 24 = 0 - Crosslake

p'''(1) = 24 imes 1 - 24 = 0 - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Planet Z is 4× farther than Y → R_Z = 4 × R_Y → R_Z³ = 64 × R_Y³ = 64 × 27 = <<64*27=1728>>1,728.

📌 T_Z² = (R_Z³) × (T_Y²)/(R_Y³) = 1,728 × (972 / 27) = 1,728 × 36 = <<1728*36=62208>>62,208.

📌 T_Z = √62,208 = <<sqrt(62208)=249.4>>≈ 249.4 Earth years.

📌 #### 249.4

📌 A geographer is mapping a region using GIS and finds that the population density increases exponentially from 50 people/km² at 10 km from a city center to 800 people/km² at 20 km. Assuming exponential growth, what is the density at 15 km?

📌 Model: D(r) = D₀ × e^(kr), where r is distance from center.

📌 D(10) = 50, D(20) = 800.

📌 800 = 50 × e^(10k) → e^(10k) = 16 → 10k = ln(16) = 2.7726 → k ≈ 0.27726.

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But we need an exact fraction. Compute step-by-step:
First, $ \frac{1}{10} + \frac{1}{17} = \frac{17 + 10}{170} = \frac{27}...
Next, $ \frac{27}{170} + \frac{1}{26} = \frac{27 \cdot 26 + 170}{170 \...
Then, $ \frac{218}{1105} + \frac{1}{37} = \frac{218 \cdot 37 + 1105}{1...
Check if reducible: GCD(9171,
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