Question

3. A right circular cylinder is inscribed in a sphere of radius 10. Express its volume and surface area as functions of its
height h.​

Answer 1

Given :-

• A right circular cylinder is inscribed in a sphere of radius 10.

To Find :-

• Volume and Surface area in terms of height.

Formula used :-

${{ \boxed{\large{\bold\red{Volume_{(Cylinder)}\: = \:\pi r^2 h }}}}}$

${{ \boxed{\small{\bold\pink{Total \: Surface Area_{(Cylinder)}\: = \:2\pi r(r + h )}}}}}$

where,

• r = radius of cylinder
• h = height of cylinder

$\begin{gathered}\Large{\underline{\bf{\color{red}CaLcUlAtIoN}}} \end{gathered}$

$\begin{gathered}\bf\red{Let,} \end{gathered}$

$\begin{gathered}\longmapsto\:\:\bf{Radius\:is \:r\:units}. \end{gathered}$

$\begin{gathered}\longmapsto\:\:\bf{Height\:is \:2h\:units}. \end{gathered}$

Using Pythagoras Theorem,

$\begin{gathered}{\boxed{\bf{\pink{(Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2}}}}\end{gathered}$

$\bf \:⟶ \: {10}^{2} = {h}^{2} + {r}^{2}$

$\bf\implies \:100 = {r}^{2} + {h}^{2}$

$\bf\implies \: {r}^{2} = 100 - {h}^{2}$

$\bf\implies \:r = \sqrt{100 - {h}^{2} }$

$\begin{gathered}\bf\pink{So,}\end{gathered}$

${{ {\large{\bold\red{Volume_{(Cylinder)}\: = \:\pi r^2 (2h) }}}}}$

$\bf\implies \:Volume = \pi \: (100 - {h}^{2} )(2h)$

$\bf\implies \:Volume =2 \pi \: (100h - {h}^{3} ) \: cu. \: units$

$\begin{gathered}\bf\pink{Also,} \end{gathered}$

${{ \boxed{\small{\bold\pink{Total \: Surface Area_{(Cylinder)}\: = \:2\pi r(r + 2h )}}}}}$

$\bf\:Area = 2\pi \: \sqrt{100 - {h}^{2} } (2h + \sqrt{100 - {h}^{2} } )sq.un$

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