Suppose a sphere and a cone share the same radius and the height of the cone is twice the radius. Which ratio describes the relationship between the formulas for the volume of a cone to the volume of a sphere?(1 point)

Answer 1

To find the ratio between the volume of a cone and the volume of a sphere, we first need to use the formulas for the volumes of both shapes.

1. Volume of a sphere (V_s):
   The formula for the volume of a sphere is given by:
   \[ V_s = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere.

2. Volume of a cone (V_c):
   The formula for the volume of a cone is given by:
   \[ V_c = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

In this case, we are told that the height of the cone is twice the radius, so we have:
\[ h = 2r \]

Now we can substitute this into the formula for the volume of the cone:
\[ V_c = \frac{1}{3} \pi r^2 (2r) \]
\[ V_c = \frac{2}{3} \pi r^3 \]

Now we can set up the ratio of the volume of the cone to the volume of the sphere:
\[ \text{Ratio} = \frac{V_c}{V_s} = \frac{\frac{2}{3} \pi r^3}{\frac{4}{3} \pi r^3} \]

The \(\pi\) and \(r^3\) terms cancel out:
\[ \text{Ratio} = \frac{2/3}{4/3} = \frac{2}{4} = \frac{1}{2} \]

Therefore, the ratio of the volume of the cone to the volume of the sphere is:
\[\frac{1}{2}\]