Question

The ratio of radii of two cylinders is $1:2$ and heights are in the ratio $2:3$. The ratio of there volumes is

• A. $1:6$
• B. $1:9$
• C. $1:3$
• D. $2:9$

Answer 1

The correct option is A

$1:6$

Step 1: Cylinder

1. Cylinder is the $3$d shapes, in geometry, which has $2$ parallel circular bases, which are joined by a curved surface, at a fixed distance.
2. The line segment joining the center of two circular bases is the axis of the cylinder.
3. The perpendicular distance between the bases is the height, "$h$“ and the distance from the axis to the outer surface is the radius ”$r$" of the cylinder.

Step 2: Explanation for correct option

Option (a): $1:6$

Assume that $r$ and $R$ be the radii of the two cylinders and $h$ and $H$ be the height of the two cylinders.

It is given that

$\frac{r}{R}=\frac{1}{2}$

$\frac{h}{H}=\frac{2}{3}$

We know that the volume of a cylinder $={\mathrm{\pi r}}^2\mathrm{nh}$

Now,

$\Rightarrow \frac{v}{V}=\frac{\mathrm{\pi r}2\mathrm{h}}{\mathrm{\pi R}2\mathrm{H}}$

$\Rightarrow \frac{v}{V}={\left(\frac{r}{R}\right)}^2\left(\frac{h}{H}\right)$

$\Rightarrow \frac{v}{V}={\left(\frac{1}{2}\right)}^2\left(\frac{2}{3}\right)$

$\Rightarrow \frac{v}{V}=\left(\frac{1}{4}\right)\left(\frac{2}{3}\right)$

$\Rightarrow \frac{v}{V}=\frac{1}{6}$

Thus, the ratio of there volume is $1:6$.

Therefore, option (a) is the correct answer.

Step 3: Explanation for incorrect option

Option (b): $1:9$

Option (c): $1:3$

Option (d): $2:9$

By calculating the ratio of two cylinders volume, we get $1:6$.

Therefore, option (b), (c) and (d) are incorrect answers.

Hence, option (a) is the correct answer.