Cylinder and Cone Volume Ratio Calculation

This problem asks for the ratio between the volume of a cylinder and a cone, given that they share the same base radius and height.

Volume Formulas

The standard formulas for the volume of a cylinder and a cone are:

• Volume of Cylinder ($V_{cyl}$): $\pi r^2 h$
• Volume of Cone ($V_{cone}$): $\frac{1}{3} \pi r^2 h$

Where '$r$' is the radius of the base and '$h$' is the height.

Ratio Calculation

We are given that the cylinder and the cone have the same base (meaning the same radius, '$r$') and the same height ('$h$'). To find the ratio of their volumes, we divide the volume of the cylinder by the volume of the cone:

$ \text{Ratio} = \frac{V_{cyl}}{V_{cone}} $

Substitute the formulas:

$ \text{Ratio} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h} $

Cancel out the common terms ($\pi r^2 h$):

$ \text{Ratio} = \frac{1}{\frac{1}{3}} $

Simplify the fraction:

$ \text{Ratio} = 1 \times \frac{3}{1} = 3 $

The ratio is 3, which can be expressed as $3:1$.

Conclusion

Therefore, the ratio of the volume of the cylinder to the volume of the cone is $3:1$.