Question

The radii of the two cylinders are in the ratio 2:3 and their heights are in the ratio 5:3. Find the ratio of their volumes.

Answer 1

Hint: We solve this question by first assuming the radius and height of one cylinder as $r$ and $h$ respectively and the radius and height of the second cylinder as $R$ and $H$ respectively. Then we get equations with a ratio of $r$ and $R$ and the ratio of $h$ and $H$. Then we consider the formula for volume of the cylinder, $Volume=\pi \times {{\left( Radius \right)}^{2}}\times \left( Height \right)$. Then using it we find the volume of the two cylinders. Then we take the ratio of their volumes and use the equations obtained before to find the value of the ratio of volumes.

Complete step by step answer:
We are given that the ratio of radii of two cylinders is 2:3.
We are also given that the ratio of their heights is 5:3.
Let us assume that the cylinders are one with radius and height $r$ and $h$ respectively and the second with radius and height $R$ and $H$ respectively.

So, as we are given that the ratio of the radii is 2:3, we can say that,
$\Rightarrow \dfrac{r}{R}=\dfrac{2}{3}.........\left( 1 \right)$
As we are given that the ratio of heights of cylinders is 5:3, we can say that,
$\Rightarrow \dfrac{h}{H}=\dfrac{5}{3}.........\left( 2 \right)$
Now let us consider the formula for volume of the cylinder.
$Volume=\pi \times {{\left( Radius \right)}^{2}}\times \left( Height \right)$
Using this formula, we get the volume of first cylinder as,
$\Rightarrow Volume=\pi {{r}^{2}}h$
Using the above formula, we get the volume of second cylinder as,
$\Rightarrow Volume=\pi {{R}^{2}}H$
Now let us consider the ratio of volumes of the cylinders.
\[\begin{align}
  & \Rightarrow \dfrac{\pi {{r}^{2}}h}{\pi {{R}^{2}}H} \\
 & \Rightarrow \left( \dfrac{{{r}^{2}}}{{{R}^{2}}} \right)\times \dfrac{h}{H} \\
 & \Rightarrow {{\left( \dfrac{r}{R} \right)}^{2}}\times \dfrac{h}{H} \\
\end{align}\]
Using equations (1) and (2) we can write the above ratio as,
\[\begin{align}
  & \Rightarrow {{\left( \dfrac{2}{3} \right)}^{2}}\times \dfrac{5}{3} \\
 & \Rightarrow \dfrac{4}{9}\times \dfrac{5}{3} \\
 & \Rightarrow \dfrac{20}{27} \\
\end{align}\]
So, we get the ratio of volumes of the two cylinders as \[\dfrac{20}{27}\].

So, the correct answer is “Option A”.

Note: The common mistake one makes while solving this problem is one might take the formula for the volume of the cylinder wrongly as, $Volume=\dfrac{1}{3}\times \pi \times {{\left( Radius \right)}^{2}}\times \left( Height \right)$. But it is the formula for the volume of cones not for the volume of the cylinder.