Question

Two circular cylinders of equal volume have their heights in the ratio 1 : 2. Ratio of their radii is
A. 1 :
B. : 1
C. 1: 2
D. 1 : 4

Answer 1

$\text{Let the radii of the two cylinders be r_1\; and \;r_2 }$

$\text{and heights be h_1\; and \;h_2 }$

$\textbf{Given:}$

$\pi\,{r_1}^2\,h_1=\pi\,{r_2}^2\,h_2$

${r_1}^2\,h_1={r_2}^2\,h_2$

$\frac{h_1}{h_2}=\frac{{r_2}^2}{{r_1}^2}$

$\frac{1}{2}=\frac{{r_2}^2}{{r_1}^2}$

$\frac{{r_2}^2}{{r_1}^2}=\frac{1}{2}$

$\text{Taking reciprocals, we get}$

$\frac{{r_1}^2}{{r_2}^2}=\frac{2}{1}$

$\implies\frac{r_1}{r_2}=\frac{\sqrt{2}}{1}$

$\implies\bf\;r_1:r_2=\sqrt{2}:1$

$\therefore\textbf{Ratio of their radii is }\bf\;\sqrt2:1$

Answer 2

Answer:

\text{Let the radii of the two cylinders be $r_1\;$and$\;r_2$ }Let the radii of the two cylinders be r

1

andr

2

\text{and heights be $h_1\;$and$\;h_2$}and heights be h

1

andh

2

\textbf{Given:}Given:

\pi\,{r_1}^2\,h_1=\pi\,{r_2}^2\,h_2πr

1

2

h

1

=πr

2

2

h

2

{r_1}^2\,h_1={r_2}^2\,h_2r

1

2

h

1

=r

2

2

h

2

\frac{h_1}{h_2}=\frac{{r_2}^2}{{r_1}^2}

h

2

h

1

=

r

1

2

r

2

2

\frac{1}{2}=\frac{{r_2}^2}{{r_1}^2}

2

1

=

r

1

2

r

2

2

\frac{{r_2}^2}{{r_1}^2}=\frac{1}{2}

r

1

2

r

2

2

=

2

1

\text{Taking reciprocals, we get}Taking reciprocals, we get

\frac{{r_1}^2}{{r_2}^2}=\frac{2}{1}

r

2

2

r

1

2

=

1

2

\implies\frac{r_1}{r_2}=\frac{\sqrt{2}}{1}⟹

r

2

r

1

=

1

2

\implies\bf\;r_1:r_2=\sqrt{2}:1⟹r

1

:r

2

=

2

:1

\therefore\textbf{Ratio of their radii is }\bf\;\sqrt2:1∴Ratio of their radii is

2

:1