Question

If a right circular cylinders volume =V
total surface area =S , height =h , ground radius is r then proof S=2V(1/h+1/r)


pls step by step answer and spam =ban​

Answer 1

Question :-

If a right circular cylinder has volume = V, total surface area = S, height =h, ground radius is r, then prove that

$\rm \: S = 2V\bigg(\dfrac{1}{h} + \dfrac{1}{r}\bigg)$

$\large\underline{\sf{Solution-}}$

We know that,

Volume of cylinder (V) of radius r and height h is

$\rm \: Volume_{(cylinder)} \: = \: V = \: \pi \: {r}^{2}h - - - - (1)$

Now, Total Surface area (S) of cylinder of radius r and height h is

$\rm \: S \: = \: 2\pi \: r \: (h \: + \: r) - - - - (2)$

Now, Consider

$\rm \: 2V\bigg(\dfrac{1}{h} + \dfrac{1}{r}\bigg)$

On substituting the value of V, from equation (1), we get

$\rm \: = \: 2\pi \: {r}^{2}h \bigg(\dfrac{r + h}{hr} \bigg)$

$\rm \: = \: 2 \: \pi \: r \: (h \: + \: r)$

$\rm \: = \: S$

Hence,

$\rm\implies \:\boxed{\rm{  \:\rm \: S = 2V\bigg(\dfrac{1}{h} + \dfrac{1}{r}\bigg) \: }}$

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Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{CSA_{(cylinder)} = 2\pi \: rh}\\ \\ \bigstar \: \bf{Volume_{(cylinder)} = \pi {r}^{2} h}\\ \\ \bigstar \: \bf{TSA_{(cylinder)} = 2\pi \: r(r + h)}\\ \\ \bigstar \: \bf{CSA_{(cone)} = \pi \: r \: l}\\ \\ \bigstar \: \bf{TSA_{(cone)} = \pi \: r \: (l + r)}\\ \\ \bigstar \: \bf{Volume_{(sphere)} = \dfrac{4}{3}\pi {r}^{3} }\\ \\ \bigstar \: \bf{Volume_{(cube)} = {(side)}^{3} }\\ \\ \bigstar \: \bf{CSA_{(cube)} = 4 {(side)}^{2} }\\ \\ \bigstar \: \bf{TSA_{(cube)} = 6 {(side)}^{2} }\\ \\ \bigstar \: \bf{Volume_{(cuboid)} = lbh}\\ \\ \bigstar \: \bf{CSA_{(cuboid)} = 2(l + b)h}\\ \\ \bigstar \: \bf{TSA_{(cuboid)} = 2(lb +bh+hl )}\\ \: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$