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Application of Derivatives — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSApplication of Derivatives1 Mark
Question
For the given half-cylinder of volume $V$, the total surface area $S$ is minimum, when
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Answer

$\because S=\pi r^2+\frac{2 V(\pi+2)}{\pi r} \Rightarrow \frac{d S}{d r}=2 \pi r-\frac{2 V(\pi+2)}{\pi} \times \frac{1}{r^2}$ For $S$ to be minimum, $\frac{d S}{d r}=0$ $\Rightarrow 2 \pi r=\frac{2 V(\pi+2)}{\pi r^2} \Rightarrow \pi^2 r^3=V(\pi+2)$

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