Question

Find the ratio between the volumes of two cylindrical drums which have (i) same base but heights in the ratio 2 :3 (ii) same height but radii in the ratio 2 : 3.​

Answer 1

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

Let assume that radius of both the cylindrical drums be r units.

Given that, Height of cylindrical drums are in the ratio 2 : 3

Height of first cylindrical drum be 2h

Height of second cylindrical drum be 3h

We know, Volume of cylinder of radius r and height h is given by

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

Now,

$\rm \: Volume_{( {1}^{st} \: Cylindrical \: drum)} : Volume_{( {2}^{nd} \: Cylindrical \: drum)}$

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

$\rm \: = \: 2 : 3$

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

Let assume that height of both the cylindrical drums be h units.

Given that, radius of cylindrical drums are in the ratio 2 : 3

Radius of first cylindrical drum be 2r

Radius of second cylindrical drum be 3r

We know, Volume of cylinder of radius r and height h is given by

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

Now,

$\rm \: Volume_{( {1}^{st} \: Cylindrical \: drum)} : Volume_{( {2}^{nd} \: Cylindrical \: drum)}$

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

$\rm \: = \: 4 : 9$

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

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

Answer:

1:3

Step-by-step explanation:

we know that

$\Longrightarrow \boxed{\bold{Volume_{(cylinder)} = \pi r^2h}}$

Let 1st cylinder having radius x and 2nd having radius  y

also given that,

heights of cylinders are in the ratio of 1:9

1st cylinder having height = 1h and 2nd = 9h

now,

Both cylinders have equal volumes

⇒ πx²h = πy²×9h

⇒  x² = 9y² = (3y)²

⇒  x= 3y

Radii of cylinders are in ratio of 1:3