The volume of a cone is 25/3 π cm^3 . What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?(1 point)
Responses 50/3 π cm^3
25/6 π cm^3
25 π cm^3
25/2 π cm^3
To find the volume of the sphere, we first need to find the radius and height of the cone.
Given that the volume of the cone is \( \frac{25}{3} \pi \, \text{cm}^3 \), we can use the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi r^2 h \]
Where \( r \) is the radius of the cone and \( h \) is the height of the cone.
Since the radius of the sphere is the same as the cone, let's denote it as \( r \). Also, since the height of the cone is equal to the diameter of the sphere, let's denote it as \( 2r \).
Now, we're given that the volume of the cone is \( \frac{25}{3} \pi \, \text{cm}^3 \):
\[ \frac{25}{3} \pi = \frac{1}{3} \pi r^2 (2r) \]
\[ \frac{25}{3} = 2r^3 \]
\[ r^3 = \frac{25}{6} \]
\[ r = \left( \frac{25}{6} \right)^{\frac{1}{3}} = \frac{\sqrt[3]{25}}{\sqrt[3]{6}} \]
Therefore, the volume of the sphere with radius \( r \) would be:
\[ V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left( \frac{\sqrt[3]{25}}{\sqrt[3]{6}} \right)^3 \]
\[ V = \frac{4}{3} \pi \frac{25}{6} \]
\[ V = \frac{100}{18} \pi \]
\[ V = \frac{50}{9} \pi \]
So, the volume of the sphere is \( \frac{50}{9} \pi \, \text{cm}^3 \).
The closest option to this value is \( \frac{25}{6} \pi \, \text{cm}^3 \).