Question

The height of two cylinders are in the ratio 7 : 9 and their curved surface areas are in the ratio 7 : 6, then the ratio of their volumes is

Answer 1

I think the ratio of their volume is 7 : 3

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Answer 2

Given:

Ratio of height of the two cylinders,

$\[{h_1}:{h_2} = 7:9\]$

$\[ \Rightarrow \frac{{{h_1}}}{{{h_2}}} = \frac{7}{9}\]$

Ratio of the CSA of the two cylinders,

$\[\begin{array}{l}{A_1}:{A_2} = 7:6\\ \Rightarrow \frac{{{A_1}}}{{{A_2}}} = \frac{7}{6}\end{array}\]$

To find: The ratio of volume of two cyclinders.

Solution:

Find the ratio of the radius of the two cylinders.

$\[\begin{array}{l}\frac{{{A_1}}}{{{A_2}}} = \frac{{2\pi {r_1}{h_1}}}{{2\pi {r_2}{h_2}}}\\ \Rightarrow \frac{7}{6} = \frac{{2\pi \times {r_1} \times 7}}{{2\pi \times {r_2} \times 9}}\\ \Rightarrow \frac{{{r_1}}}{{{r_2}}} = \frac{{7 \times 9}}{{7 \times 6}}\\ \Rightarrow \frac{{{r_1}}}{{{r_2}}} = \frac{3}{2}\end{array}\]$

Find the ratio of volume of the two cylinders.

$\[\begin{array}{l}\frac{{{V_1}}}{{{V_2}}} = \frac{{\pi r_1^2{h_1}}}{{\pi r_2^2{h_2}}}\\ \Rightarrow \frac{{{V_1}}}{{{V_2}}} = \frac{{\pi {{\left( 3 \right)}^2} \times 7}}{{\pi {{\left( 2 \right)}^2} \times 9}}\\ \Rightarrow \frac{{{V_1}}}{{{V_2}}} = \frac{{9 \times 7}}{{4 \times 9}}\\ \Rightarrow \frac{{{V_1}}}{{{V_2}}} = \frac{7}{4}\end{array}\]$

Hence, the ratio of volume of the two cylinders is $\[{V_1}:{V_2} = 7:4\]$.