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Two cylinders of equal volume have their heights in the ratio 2:12:1. What is the ratio of their radii ____

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Detailed Solution

Given that, Two cylinders of equal volume have their heights in the ratio 2:12:1.
Let us consider the figure as below and take the terms as
Question Image

Radius of the first cylinder is

r_{1}

,
Radius of the second cylinder is

r_{2}

,
Height of the first cylinder is

h_{1}

,
Height of the second cylinder is

h_{2}

.
∴∴ Volume of first cylinder is V1=π

{r_{1}}^{2} h_{1}

.
⇒⇒ Volume of second cylinder is V2=π

{r_{2}}^{2} h_{2}

.
But we have the volumes of both the cylinders the same.
∴V1= V2
⇒ π

{r_{1}}^{2} h_{1} =

{r_{2}}^{2} h_{2}

Cancelling the term ππ which is on both sides and rearranging the above equation as radii of both cylinders at one place, then we will have

\frac{{r_{1}}^{2}}{{r_{2}}^{2}} = \frac{h_{2}}{h_{1}}

⇒

\frac{r_{1}}{r_{2}} = \sqrt \frac{h_{2}}{h_{1}}

.. (i)
Given that, the heights of the cylinders are in the ratio 2:1.
∴

h_{1}

h_{2}

=2:1
⇒

\frac{h_{1}}{h_{2}} = \frac{2}{1}

⇒

h_{1}

h_{2}

Using the above value in equation (i), then we will have

\frac{r_{1}}{r_{2}} = \sqrt \frac{h_{2}}{2 h_{2}}

⇒

\frac{r_{1}}{r_{2}} = \sqrt \frac{1}{2}

⇒

\frac{r_{1}}{r_{2}} = \frac{1}{\sqrt 2}

∴ Ratio of radii of the two cylinders is 1:

\sqrt 2

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