Cylinder Cone Equal Volume Radius Ratio

Determine the ratio of the radius of a cylinder's base to the radius of a cone's base when both shapes have the same height and equal volumes.

Volume Formulas Used

The volume ($V$) formulas for a cylinder and a right circular cone are:

• Cylinder: $V_{cylinder} = \pi r_{cylinder}^2 h$
• Cone: $V_{cone} = \frac{1}{3} \pi r_{cone}^2 h$

Where $r_{cylinder}$ and $r_{cone}$ are the radii of the bases, and $h$ is the common height.

Deriving the Radius Ratio

Given that $V_{cylinder} = V_{cone}$ and the height $h$ is the same for both:

$ \pi r_{cylinder}^2 h = \frac{1}{3} \pi r_{cone}^2 h $

Cancel $\pi$ and $h$ from both sides (since $h \neq 0$):

$ r_{cylinder}^2 = \frac{1}{3} r_{cone}^2 $

Rearrange to find the ratio of the squared radii:

$ \frac{r_{cylinder}^2}{r_{cone}^2} = \frac{1}{3} $

Take the square root of both sides to find the ratio of the radii:

$ \frac{r_{cylinder}}{r_{cone}} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} $

To express this ratio $\frac{1}{\sqrt{3}}$ in the form $a:b$ commonly seen in options, we can simplify it. Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$:

$ \frac{1}{\sqrt{3}} = \frac{1 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{3}}{3} $

This corresponds to the ratio $\sqrt{3} : 3$. Therefore, the ratio of the radius of the base of the cylinder to that of the cone is $\sqrt{3} : 3$.