To find the radius of a cylinder given its height and volume, we use the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
where \(V\) is the volume, \(r\) is the radius, and \(h\) is the height.
In this case, the volume \(V = 81\pi\) cubic units and the height \(h = 4\) units. Plugging these values into the volume formula, we get:
\[ 81\pi = \pi r^2 (4) \]
We can simplify this equation by dividing both sides by \(\pi\):
\[ 81 = r^2 (4) \]
Now, divide both sides by 4:
\[ r^2 = \frac{81}{4} \]
Taking the square root of both sides to find \(r\):
\[ r = \sqrt{\frac{81}{4}} = \frac{\sqrt{81}}{\sqrt{4}} = \frac{9}{2} = 4.5 \]
The radius of the cylinder is \(4.5\) units.
Therefore, the correct response is:
4.5 units.
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