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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
MCQ
For the given half-cylinder of volume $V$, the total surface area $S$ is minimum, when
  • A
    $(\pi+2) V=\pi^2 r^3$
  • B
    $(\pi+2) V=\pi^2 r^2$
  • C
    $2(\pi+2) V=\pi^2 r^3$
  • D
    $(\pi+2) \vee=\pi^2 r$
✓

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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