To determine the maximum percentage error in the volume of the cone, we first need to understand the dependence of the cone's volume on its dimensions: the diameter of the base and its height.

The volume $ V $ of a cone is given by the formula:

$$ V = \frac{1}{3} \pi r^2 h $$

where $ r $ is the radius of the base and $ h $ is the height.

Given that both the diameter ($ D $) of the base and the height ($ h $) are measured to be $ 20.0 \mathrm{~cm} $, we can find the radius $ r $ as follows:

$$ r = \frac{D}{2} = \frac{20.0 \mathrm{~cm}}{2} = 10.0 \mathrm{~cm} $$

Now, let's denote the errors in measuring the diameter and height as $ \Delta D $ and $ \Delta h $ respectively.

Since the measurements are taken with a scale having a least count of $ 2 \mathrm{~mm} $, we have:

$$ \Delta D = 2 \mathrm{~mm} = 0.2 \mathrm{~cm} $$

and

$$ \Delta h = 2 \mathrm{~mm} = 0.2 \mathrm{~cm} $$

To find the maximum percentage error in the volume, we need to use the formula for the propagation of relative errors. Considering the volume formula $ V = \frac{1}{3} \pi r^2 h $, the relative errors in $ r $ and $ h $ will propagate into the volume as follows:

The relative error in the radius $ \Delta r / r $ is:

$$ \left(\frac{\Delta r}{r}\right) = \left(\frac{\Delta D / 2}{D / 2}\right) = \left(\frac{\Delta D}{D}\right) = \frac{0.2 \mathrm{~cm}}{20.0 \mathrm{~cm}} = 0.01 $$

The relative error in the height $ \Delta h / h $ is:

$$ \left(\frac{\Delta h}{h}\right) = \frac{0.2 \mathrm{~cm}}{20.0 \mathrm{~cm}} = 0.01 $$

Since $ V $ is proportional to $ r^2 $ and $ h $, the total relative error in the volume is given by:

$$ \left(\frac{\Delta V}{V}\right) = 2 \left(\frac{\Delta r}{r}\right) + \left(\frac{\Delta h}{h}\right) $$

Substituting the relative errors, we get:

$$ \left(\frac{\Delta V}{V}\right) = 2 (0.01) + 0.01 = 0.02 + 0.01 = 0.03 $$

Hence, the maximum percentage error in the volume is:

$$ 0.03 \times 100\% = 3\% $$

Therefore, the maximum percentage error in the determination of the volume is 3%.