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\( 700 - 105 = 595 \) hours - Nurtured Nest

📅 May 9, 2026 👤 scraface

May 09, 2026

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📌 Let \(r = \sqrt{x^2 + y^2} = c \sin^2\phi\)

📌 But from \(z = c \sin\phi \cos\phi = c \sin\phi \sqrt{1 - \sin^2\phi}\)

📌 Set \(u = \sin\phi\), then:

📌 r = c u^2, \quad z = c u \sqrt{1 - u^2}

📌 u = rac{r^{1/2}}{\sqrt{c}}, \quad z = c \cdot rac{r^{1/2}}{\sqrt{c}} \cdot \sqrt{1 - rac{r^2}{c^2}} = \sqrt{c} \cdot r^{1/2} \sqrt{1 - rac{r^2}{c^2}}

📌 z^2 = c \cdot r \left(1 - rac{r^2}{c^2}

📌 ight) = c r - rac{r^3}{c}

📌 But \(r = rac{\sqrt{x^2 + y^2}}{1}\), so:

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