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📅 May 14, 2026 👤 scraface

May 14, 2026

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To minimize, take derivative:
C'(r) = -\frac{40}{r^2} + 220\pi r
220\pi r = \frac{40}{r^2} \Rightarrow r^3 = \frac{40}{220\pi} = \frac{...
\Rightarrow r = \left( \frac{2}{11\pi} \right)^{1/3}
Compute numerically: \( \frac{2}{11\pi} \approx \frac{2}{34.5575} \app...
Then \( h = \frac{0.5}{\pi (0.387)^2} \approx \frac{0.5}{0.470} \appro...
Now compute areas:
Lateral area: \( 2\pi r h \approx 2\pi \cdot 0.387 \cdot 1.064 \approx...
Top and bottom: \( 2\pi r^2 \approx 2\pi \cdot 0.150 \approx 0.942 \)...
Cost: \( 40 \cdot 2.58 + 55 \cdot 0.942 \approx 103.2 + 51.81 = 155.01...

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