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The radii of two cylinders are in the ratio $2 : 3$ and their heights are in the ratio $5:3$. Calculate the ratio of their volumes and the ratio of their curved surfaces.
Given:
The radii of two cylinders are in the ratio $2 : 3$ and their heights are in the ratio $5:3$.
To do:
We have to find the ratio of their volumes and the ratio of their curved surfaces.
Solution:
Ratio in radii of two cylinders $= 2:3$
Ratio in their heights $= 5:3$
Let the radius of the first cylinder $(r_1) = 2x$ and the radius of the second cylinder $(r_2) = 3x$
Height of the first cylinder $(h_1) = 5y$ and the height of the second cylinder $(h_2) = 3y$
Volume of the first cylinder $= \pi r^2h$
$= \pi (2x)^2 \times 5y$
$= 20\pi x^2y$
Volume of the second cylinder $= \pi (3x)^2 \times 3y$
$= 27\pi x^2y$
Ratio in their volumes $= 20\pi x^2y : 27\pi x^2y$
$= 20 : 27$
Curved surface area of the first cylinder $= 2\pi rh$
$= 2\pi \times 2x \times 5y$
$=20\pi xy$
Curved surface area of the second cylinder = 2\pi \times 3x \times 3y$
$= 18\pi xy$
Therefore,
Ratio in their curved surface areas $= 20\pi xy : 18\pi xy$
$= 10 : 9$