Question

Two circular cylinders of equal volume have their heights in the ratio \( 1 : 2 \). The ratio of their radii is:

• A. \( 1 : \sqrt{2} \)
• B. \( \sqrt{2} : 1 \)
• C. \( 1 : 2 \)
• D. \( 1 : 4 \)

Hint:

When the volumes of two cylinders are equal and their heights are in a known ratio, the ratio of their radii is the square root of the inverse ratio of their heights.

Answer 1

The Correct Option is A

The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h, \] where \( r \) is the radius and \( h \) is the height. Let the radii and heights of the two cylinders be \( r_1, r_2 \) and \( h_1, h_2 \), respectively. We are given that the volumes of the cylinders are equal, and the ratio of their heights is \( h_1 : h_2 = 1 : 2 \). Therefore, the ratio of their volumes is: \[ \frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} = 1. \] Substitute \( h_1 = 1 \) and \( h_2 = 2 \): \[ \frac{r_1^2}{r_2^2} = \frac{2}{1} \quad \Rightarrow \quad \left( \frac{r_1}{r_2} \right)^2 = 2 \quad \Rightarrow \quad \frac{r_1}{r_2} = \sqrt{2}. \] Thus, the ratio of the radii is \( \boxed{1 : \sqrt{2}} \).