Question

two cylinder have their radii in the ratio 3:1 but their heights are in the ratio 1:3 compare their volumes

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step by step explanation ​

Answer 1

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

Given that,

• The radii of the bases of a two cylinder are in the ratio 3:1

Let assume that

• Radius of first cylinder = 3r

• Radius of second cylinder = r

Further, given that

• Height of two cylinders are in the ratio 1 : 3

Let assume that

• Height of first cylinder = h

• Height of second cylinder = 3h

So, Volume of cylinder of radius 3r and height h is given by

$\rm \: Volume_{(Cylinder \: 1)} = \: \pi \: {(3r)}^{2} \: h \:$

$\rm\implies \:Volume_{(Cylinder \: 1)} = 9\: \pi \: {r}^{2} \: h \:$

Now, Volume of cylinder of radius r and height 3h is given by

$\rm \: Volume_{(Cylinder \: 2)} = \: \pi \: {r}^{2} \: (3h) \:$

$\rm\implies \:Volume_{(Cylinder \: 2)} = 3\: \pi \: {r}^{2} \: h \:$

Hence,

$\rm \: Volume_{(Cylinder \: 1)} : Volume_{(Cylinder \: 2)}$

$\rm \: = \: \:9 \: \pi \: {r}^{2} \: h \: : 3 \: \: \pi \: {r}^{2} \: h \:$

$\rm \: = \: 3 : 1$

Hence,

$\rm\implies \:\boxed{ \rm{ Volume_{(Cylinder \: 1)} : Volume_{(Cylinder \: 2)} = 3 : 1 \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

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

Answer 2

$\: \ \red {answer}$

Given that,

• The radii of the bases of a two cylinder are in the ratio 3:1

Let assume that

• Radius of first cylinder = 3r
• Radius of second cylinder = r

Further, given that

• Height of two cylinders are in the ratio 1 : 3

Let assume that

• Height of first cylinder = h
• Height of second cylinder = 3h

So, Volume of cylinder of radius 3r and height h is given by

$Volume(Cylinder2)=πr2(3h)</p><p>$

$⟹Volume(Cylinder2)=3πr2h</p><p></p><p></p><p>$

Hence,

$=9πr2h:3πr2h</p><p>$

=3:1

Hence,

$⟹Volume(Cylinder1):Volume(Cylinder2)=3:1</p><p></p><p>\rule{190pt}{2pt}</p><p></p><p>$