Question

The volume of two cylinders are in the ratio $a:b$ and their heights are in ratio $c:d$. Find the ratio of their diameters.

• A. $\frac{ab}{bc}$
• B. $\frac{a{d}^2}{a{c}^2}$
• C. $\sqrt{\frac{ad}{bc}}$
• D. $\sqrt{\frac{a}{b}}\times \frac{c}{d}$

Answer 1

The correct option is C $\sqrt{\frac{ad}{bc}}$

Volume of a cylinder $=\pi r^{2}h$

Let $r_1$ and $h_1$ be the radius and height of the first cylinder and $r_2$ and $h_2$ be the radius and height of the second cylinder.

Now, ratio of their volumes is calculated as:

$\Rightarrow \frac{V_1}{V_2}=\frac{a}{b}$

$\Rightarrow \frac{\pi {r_1}^2{h}_1}{\pi {r_2}^2{h}_2}=\frac{a}{b}$

$\Rightarrow \frac{{r_1}^{2}c}{{r_2}^{2}d}=\frac{a}{b}$

$\Rightarrow \frac{{r_1}^2}{{r_2}^2}=\frac{ad}{bc}$

$\Rightarrow \frac{r_1}{r_2}=\sqrt{\frac{ad}{bc}}$

Ratio of diameters $=\frac{d_1}{d_2}=\frac{2r_1}{2r_2}=\frac{r_1}{r_2}=\sqrt{\frac{ad}{bc}}$