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6. Application of derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths6. Application of derivatives1 Mark
MCQ
A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is
  • ✓
    $1: 1$
  • B
    $1: 2$
  • C
    $2: 1$
  • D
    $\sqrt{2}: 1$
✓

Answer

Correct option: A.
$1: 1$
a
(a)

Surface area of solid i.e. $\quad S=2 \pi r^2+2 \pi r h+\pi r^2$

$S=3 \pi r^2+2 \pi r h$

$h=\frac{S-3 \pi r^2}{2 \pi r}$

Volume of solid

$\text { i.e. } \quad V=\pi r^2 h+\frac{2}{3} \pi r^3$

$\Rightarrow \quad V=\pi r^2\left(\frac{S-3 \pi r^2}{2 \pi r}\right)+\frac{2}{3} \pi r^3$

$\Rightarrow \quad \quad V=\frac{1}{2}\left(S r-3 \pi r^3\right)+\frac{2}{3} \pi r^3$

$\Rightarrow \quad \frac{d V}{d r}=\frac{S}{2}-\frac{9 \pi r^2}{2}+2 \pi r^2$

$\text { For maximum or minimum } \frac{d V}{d r}=0$

$\therefore \quad S=5 \pi r^2$

$\Rightarrow \quad h=\frac{S-3 \pi r^2-5 \pi r^2-3 \pi r^2}{2 \pi r}=r$

$\therefore \quad h=r \Rightarrow h: r=1: 1$

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