To compare the volume of a cone to the volume of a cylinder when they both have the same radius and height, we'll use the formulas for the volumes of these shapes and then compare. We have a radius (r) of 4 cm and a height (h) of 12 cm.
For the cylinder, the volume (V_cylinder) is given by the formula:
\[ V_{cylinder} = \pi r^2 h \]
So we plug in our values for the radius and height:
\[ V_{cylinder} = \pi (4\text{ cm})^2 (12\text{ cm}) \]
\[ V_{cylinder} = \pi (16\text{ cm}^2) (12\text{ cm}) \]
\[ V_{cylinder} = 192\pi\text{ cm}^3 \]
\[ V_{cylinder} \approx 603.19\text{ cm}^3 \] (rounded to two decimal places)
For the cone, the volume (V_cone) is calculated using the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Using the same values for r and h:
\[ V_{cone} = \frac{1}{3} \pi (4\text{ cm})^2 (12\text{ cm}) \]
\[ V_{cone} = \frac{1}{3} \pi (16\text{ cm}^2) (12\text{ cm}) \]
\[ V_{cone} = \frac{192}{3}\pi\text{ cm}^3 \]
\[ V_{cone} \approx 64\pi\text{ cm}^3 \]
\[ V_{cone} \approx 201.06\text{ cm}^3 \] (rounded to two decimal places)
Now, to find out the volume of the cone in terms of the cylinder's volume, we divide the volume of the cone by the volume of the cylinder:
\[ \text{Ratio} = \frac{V_{cone}}{V_{cylinder}} \]
Substituting the values, we get:
\[ \text{Ratio} \approx \frac{201.06\text{ cm}^3}{603.19\text{ cm}^3} \]
\[ \text{Ratio} \approx 0.333 \] (rounded to three decimal places)
Therefore, the volume of the cone is one-third (or approximately 0.333 times) the volume of the cylinder when they share the same radius and height.
