Question

The radii of two cylinders are in the ratio $3:4$, and their height are in the ratio $4:3$. Find the ratio of their volumes.

Answer 1

Hint: In this question, we have been given the ratio of the radii and heights of two cylinders and we have been asked the ratio of their volumes. Since the values given to us are in ratio and we have been asked the answer in ratio as well, we are only supposed to put the required values in the ratio of their volumes, as they are given. On simplifying the ratio, we will get our required answer.

Formula used: Volume of cylinder $ = \pi {r^2}h$

Complete step-by-step solution:
We have been given the ratio of the radii and the heights of the two cylinders. Using these ratios, we have to find the ratio of their volumes.
Let cylinder 1 have R as radius and H as height. Let cylinder 2 have r as radius and h as height.
Therefore, the volume of cylinder 1 $ = \pi {R^2}H$, and volume of cylinder 2 $ = \pi {r^2}h$
Let the radii of two cylinders be $3x$ and $4x$, and their ratio is $R:r$; and their heights be $4y$ and $3y$, and their ratio is $H:h$.
We have to find the value of $\dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{\pi {R^2}H}}{{\pi {r^2}h}}$
Putting all the values, we will get,
$ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{\pi {{\left( {3x} \right)}^2}4y}}{{\pi {{\left( {4x} \right)}^2}3y}}$
On simplifying,
$ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{9{x^2} \times 4y}}{{16{x^2} \times 3y}}$
$ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{3}{4}$

$\therefore $ The ratio of their volumes is $3:4$.

Note: We have to remember that the volume of a cylinder is the measure of the amount of space occupied by a cylinder, or the measure of the capacity of a cylinder. In geometry, a cylinder is a three dimensional shape with two equal and parallel circles which are joined by a curved surface. The distance the circular faces of a cylinder is known as the height of a cylinder. The top and bottom of a cylinder are two congruent circles whose radius or diameter are denoted as $r$ and $d$ respectively.
There is a shortcut which will reduce these steps. You will not be required to assume the radii as $3x$ and $4x$.
If the values given to us in the question are in the form of ratio and the answer has also been asked in the form of ratio, then we can simply put the given in the formula, without assuming anything.