The volume of a cone is 25 over three pi centimeters cubed what is the volume of a sphere if it’s a radius is the same as the cones in the height of the cone is equal to the severe diameter?
Given that the volume of the cone is $\frac{25}{3\pi}$ cm$^3$ and the radius of the sphere is the same as the cone's height, we can represent the height of the cone as $r$, and the radius of the cone as $h$.
The volume of a cone is given by the formula $V_{\text{cone}}=\frac{1}{3}\pi r^2h$, so we have $\frac{1}{3}\pi h^3 = \frac{25}{3\pi}$.
To find the volume of the sphere, we need to find the radius first. Since the height of the cone is equal to the diameter of the sphere, we can relate the radius ($r_{\text{sphere}}$) and height ($h_{\text{cone}}$) as $r_{\text{sphere}} = \frac{1}{2}h_{\text{cone}} = \frac{1}{2}r_{\text{cone}}$.
Substituting this relation into the equation for the cone's volume, we have:
$$\frac{1}{3}\pi (\frac{1}{2}r_{\text{cone}})^3 = \frac{25}{3\pi}.$$
Simplifying, we have:
$$\frac{1}{3}\pi \cdot \frac{1}{8}r_{\text{cone}}^3 = \frac{25}{3\pi}.$$
Multiply both sides by $\frac{3}{\pi}$:
$$\frac{1}{8}r_{\text{cone}}^3 = \frac{25}{\pi}.$$
Multiply both sides by $8$:
$$r_{\text{cone}}^3 = \frac{200}{\pi}.$$
Take the cube root of both sides to find the radius of the cone:
$$r_{\text{cone}} = \sqrt[3]{\frac{200}{\pi}}.$$
Since the radius of the sphere is half the radius of the cone, the radius of the sphere is:
$$r_{\text{sphere}} = \frac{1}{2}\sqrt[3]{\frac{200}{\pi}}.$$
The volume of a sphere is given by the formula $V_{\text{sphere}}=\frac{4}{3}\pi r_{\text{sphere}}^3$. Substituting in the value of $r_{\text{sphere}}$, we have:
$$V_{\text{sphere}} = \frac{4}{3}\pi \left(\frac{1}{2}\sqrt[3]{\frac{200}{\pi}}\right)^3 = \frac{4}{3}\pi \frac{1}{8}\sqrt[3]{\frac{200}{\pi}}^3 = \frac{1}{6}\pi\sqrt[3]{\frac{200}{\pi}}^3 = \frac{1}{6}\pi \cdot\frac{200}{\pi} = \frac{200}{6} = \boxed{\frac{100}{3}} \text{ cm}^3.$$