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Home » Volume of the sphere is \( V = \frac43 \pi R^3 \). - Baxtercollege

Volume of the sphere is \( V = \frac43 \pi R^3 \). - Baxtercollege

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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Second term:
-6t^2(2t - t^4) = -12t^3 + 6t^6.
- 2t^3 - 4t^6 - 12t^3 + 6t^6 = 2 - 14t^3 + 2t^6.
I''(t) = rac{2t^6 - 14t^3 + 2}{(1 + t^3)^3}.
Set numerator of \( I''(t) = 0 \):
2t^6 - 14t^3 + 2 = 0 \quad \Rightarrow \quad t^6 - 7t^3 + 1 = 0.
Let \( u = t^3 \), then:
u^2 - 7u + 1 = 0 \quad \Rightarrow \quad u = rac{7 \pm \sqrt{49 - 4}}...
Since \( t > 0 \), we take the positive root:
t^3 = rac{7 + 3\sqrt{5}}{2} \quad \Rightarrow \quad t = \left( rac{7...

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