To find the height of the cylinder, we start with the formula for the volume of a cylinder:
\[ V = \text{Area of base} \cdot \text{Height} \]
Given that the volume \( V \) is \( 250\pi \) and the radius \( r \) is 5, we can compute the area of the base (which is a circle) using the formula:
\[ \text{Area of base} = \pi r^2 \]
Substituting the value of \( r \):
\[ \text{Area of base} = \pi (5)^2 = \pi \cdot 25 = 25\pi \]
Now, we can substitute this into the volume formula:
\[ 250\pi = 25\pi \cdot h \]
Next, we can divide both sides of the equation by \( 25\pi \):
\[ h = \frac{250\pi}{25\pi} \]
The \( \pi \) cancels out:
\[ h = \frac{250}{25} = 10 \]
Thus, the height of the cylinder is:
\[ \boxed{10} \]