Question : The ratio of the volumes of two right circular cylinders A and B is $\frac{x}{y}$ and the ratio of their heights is a : b. What is the ratio of the radii of A and B?
Option 1: $\frac{y b}{x a}$
Option 2: $\sqrt{\frac{x a}{y b}}$
Option 3: $\frac{x b}{y a}$
Option 4: $\sqrt{\frac{x b}{y a}}$
Correct Answer: $\sqrt{\frac{x b}{y a}}$
Solution :
As we know,
The volume of a cylinder = $\pi r^2h$, where $r$ is the radius and $h$ is the height of the cylinder.
Let the ratio of the radius of two cylinders be $r_1:r_2$
Volume ratio of two cylinder $V_1:V_2$ = $x:y$
Height ratio of two cylinder $h_1:h_2$ = $a:b$
According to the question,
$\frac{V_1}{V_2}= \frac{\pi (r_1)^2 h_1}{\pi (r_2)^2 h_2}$
⇒ $\frac{x}{y} = \frac{r_1^2a}{r_2^2b}$
⇒ $\frac{r_1^2}{r_2^2}=\frac{xb}{ya}$
⇒ $\frac{r_1}{r_2}=\sqrt{\frac{xb}{ya}}$
Hence, the correct answer is $\sqrt{\frac{xb}{ya}}$.
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