The volume of a cone is 25/3 pi cm³. What is the volume of a sphere if its radius is the same as the cones and the height of the cone is equal to the spheres diameter?
First, let's write the formula for the volume of a cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \( V_{\text{cone}} \) is the volume of the cone, \( r \) is the radius of the base of the cone, and \( h \) is the height of the cone.
Given that the volume of the cone is \( 25/3 \) \( \pi \) cm³, we can equate:
\[ \frac{25}{3} \pi = \frac{1}{3} \pi r^2 h \]
Now, we are told that the height of the cone, \( h \), is equal to the diameter of the sphere. Since the diameter is twice the radius (\( d = 2r \)), we have \( h = 2r \). Let's substitute \( h \) with \( 2r \) into the equation:
\[ \frac{25}{3} \pi = \frac{1}{3} \pi r^2 (2r) \]
\[ \frac{25}{3} \pi = \frac{2}{3} \pi r^3 \]
Now we can solve for \( r^3 \):
\[ r^3 = \frac{25}{3} \pi \times \frac{3}{2 \pi} \]
\[ r^3 = \frac{25}{2} \]
Once we have \( r^3 \), we can find the volume of the sphere using the formula for the volume of a sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Now, let's plug in \( r^3 = \frac{25}{2} \):
\[ V_{\text{sphere}} = \frac{4}{3} \pi \times \frac{25}{2} \]
\[ V_{\text{sphere}} = \frac{4}{3} \times \frac{25}{2} \pi \]
\[ V_{\text{sphere}} = \frac{4 \times 25 \pi}{3 \times 2} \]
\[ V_{\text{sphere}} = \frac{100 \pi}{6} \]
\[ V_{\text{sphere}} = \frac{50 \pi}{3} \]
Therefore, the volume of the sphere with the same radius as the cone is \( \frac{50}{3} \pi \) cm³.