Question

suppose a cylinder and a cone share the same radius and the same height, which is twice the radius. which ratio describes the relationship between the formulas for the volume of a cone to the volume of a cylinder? (1 point) check answer remaining attempts : 3

Explanation:

Step1: Define variables and formulas

Let the radius of the cylinder and cone be \( r \), and the height \( h = 2r \) (since height is twice the radius).
The volume of a cylinder \( V_{cylinder} = \pi r^2 h \).
The volume of a cone \( V_{cone} = \frac{1}{3}\pi r^2 h \).

Step2: Substitute \( h = 2r \) into both formulas

For the cylinder:
\( V_{cylinder} = \pi r^2 (2r) = 2\pi r^3 \).

For the cone:
\( V_{cone} = \frac{1}{3}\pi r^2 (2r) = \frac{2}{3}\pi r^3 \).

Step3: Find the ratio \( \frac{V_{cone}}{V_{cylinder}} \)

Substitute the volumes:
\( \frac{V_{cone}}{V_{cylinder}} = \frac{\frac{2}{3}\pi r^3}{2\pi r^3} \).

Simplify: The \( \pi r^3 \) terms cancel, and \( \frac{\frac{2}{3}}{2} = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} \).

Answer:

The ratio of the volume of the cone to the volume of the cylinder is \( \frac{1}{3} \), so the ratio \( V_{cone}:V_{cylinder} = 1:3 \).