Question

Volumes of two cylinders of radii $R,\dfrac{R}{2}$ and heights $H,h$ respectively are equal.
Then $H:h = $___
(A) $1:2$
(B) $1:4$
(C) $2:1$
(D) $4:1$

Answer 1

Hint: In this question, we have to equate the volumes of the two cylinders having different values of radius and height, then find the ratio between the heights of the cylinders. We know that for a cylinder having a radius of the base circle be $r$ and the height of the cylinder be $h$ , then the volume of the cylinder is given by-

Complete step-by-step answer:
Given:
The dimensions of two cylinders are given and the volume of both the cylinders are equal. So,
The radius of the first cylinder is $R$ and the height of the first cylinder is $H$.
So, the volume of the first cylinder is given by,
${V_1} = \pi {R^2}H$
And the radius of the second cylinder is $\dfrac{R}{2}$ and the height of the second cylinder is $h$.
So, the volume of the second cylinder is given by,
${V_2} = \pi {\left( {\dfrac{R}{2}} \right)^2}h$
Now, according to the question, the volumes of both cylinders are the same. So
The volume of the first cylinder = The volume of the second cylinder
${V_1} = {V_2}$
Substituting the values of ${V_1}{\rm{ and }}{V_2}$, we get,
$\pi {R^2}H = \pi {\left( {\dfrac{R}{2}} \right)^2}h$
Now, solving it step by step, we get,
$\begin{array}{c}
\pi {R^2}H = \pi \dfrac{R^2}{4}h\\
H = \dfrac{h}{4}
\end{array}$
We can also write this as-
$H:h = 1:4$
Therefore, the value of $H:h$ is $1:4$ and the correct option is (B) $1:4$.
So, the correct answer is “Option b”.

Note: If the volume of a cylinder is equal to the volume of another cylinder, then it does not mean that the surface areas of both the cylinders are equal because the cross-section of a cylinder could be different from one another. It means that if the diameter of one cylinder is smaller than the other cylinder, then its height would be bigger than the other cylinder.