Question

The radius and height of a cylinder are in the ratio 3:2 and its volume is $19404\,c{{m}^{2}}$. Find its radius and height.

Answer 1

Hint: Assume a proportionality constant (say x) for the given ratio of radius and height of cylinder. Calculate volume in terms of x using formula “volume of cylinder = $\pi {{r}^{2}}h$”, where r is the radius of the cylinder and ‘h’ is the height of the cylinder. Equate the obtained volume with the given volume to get an equation in x and solve for x.

Complete step-by-step solution -
We have to find the radius and height of the cylinder.
Given ratio: - radius: height=3:2.
Let us assume that the proportionality constant of this ratio to be x.
So, radius of the cylinder will be 3x and height of the cylinder will be 2x.

We know that: -
Volume of cylinder= $\pi {{r}^{2}}h$ where
r = radius of the cylinder and
h = height of cylinder.
r = 3x and
h = 2x
So, volume = $\pi {{\left( 3x \right)}^{2}}2x$
but according to the equation, volume is= $19404\,c{{m}^{2}}$
So, $\pi {{\left( 3x \right)}^{2}}2x$= $19404\,c{{m}^{2}}$
Taking \[\pi =\dfrac{22}{7}\], we will get-
$\Rightarrow \dfrac{22}{7}\times \left( 3x \right)\times \left( 3x \right)\times \left( 2x \right)=19404$
\[\] $\Rightarrow \left( \dfrac{22\times 3\times 3\times 2}{7} \right)\times {{x}^{3}}=19404$
Dividing both sides by \[\left( \dfrac{22\times 3\times 3\times 2}{7} \right)\], we will get-
 \[\begin{align}
  & \Rightarrow {{x}^{3}}=\dfrac{19404}{\left( \dfrac{22\times 3\times 3\times 2}{7} \right)} \\
 & \Rightarrow {{x}^{3}}=\dfrac{19404\times 7}{22\times 3\times 3\times 2} \\
\end{align}\]
\[\Rightarrow {{x}^{{}}}=343\]
On taking cube root of both sides of equation, we will get-
\[\begin{align}
  & \Rightarrow x=\sqrt{343} \\
 & \Rightarrow x=7\,cm \\
\end{align}\]
Hence,
Radius of the cylinder = \[3x=\left( 3\times 7 \right)cm=21cm\]
Height of the cylinder= \[2x=\left( 2\times 7 \right)cm=14cm\]

Note: We can also solve this question without assuming a proportionality constant.
We can write the height of the cylinder in terms of radius using the ratio.
Given ratio- $\dfrac{radius}{height}=\dfrac{3}{2}$
In cross multiplying, 2radius=3height
\[\begin{align}
  & \Rightarrow height=\dfrac{2\times radius}{3}..............(i) \\
 & \\
\end{align}\]
Volume of cylinder = \[2{{\left( radius \right)}^{2}}\times height.\]
\[\Rightarrow 19404c{{m}^{3}}=\pi {{\left( radius \right)}^{2}}\times height.....................(ii)\]
These are two equations and two variables (height and radius). Solve the two equations to get the value of radius and height.