Question

Two right circular solid cylinders have radii in the ratio $3:5$ and heights in the ratio $2:3$ . Find the ratio between their:
i) curved surface areas.
ii) volumes.

Answer 1

Suppose that two right circular solid cylinders A with radius $r$ and height $h$ and B with radius $R$ and height $Hthen$

Ratio of radii $r:R=3:5$
Let $r=3xandR=5x$
Ratio of heights $=h:H=2:3$
Let $h=2yandH=3y$
Curved Surface Area of cylinder $A=2\pi rh=2\pi \left(3x\right)\left(2y\right)=12\pi xy$
Curved Surface Area of cylinder $B=2\pi RH=2\pi \left(5x\right)\left(3x\right)=30\pi xy$
Ratio of their Curved Surface Area of $A$ and $B=\frac{12\pi xy}{30\pi xy}=\frac{2}{5}$
Hence, the ratio of the curved surface areas of cylinders is $2:5.$

Volume of cylinder $A=\pi r^{2}h=\pi \left(3x{)}^2\right(2y)=18\pi x^{2}y$
Volume of cylinder $B=\pi R^{2}H=\pi \left(5x{)}^2\right(3y)=75\pi x^{2}y$

Ratio of the Volume of $AandB=\frac{18\pi x^{2}y}{75\pi x^{2}y}=\frac{6}{25}$

Hence, the ratio of the volumes of cylinders is $6:25.$