The dimensions of a cone are measured using a scale with a least count of 2 mm. The diameter of the base and the height are both measured to be 20.0 cm. The maximum percentage error in the determination of the volume is ______.

Step 1: Identify the given values and absolute errors.

The least count of the scale represents the absolute error in the direct measurements of diameter $d$ and height $h$.

$$\Delta d = \Delta h = 2\text{ mm} = 0.2\text{ cm}$$

The measured values are:

$$d = 20.0\text{ cm}$$ $$h = 20.0\text{ cm}$$

Step 2: Calculate the radius and its absolute error.

The radius $r$ is half of the diameter.

$$r = \frac{d}{2} = 10.0\text{ cm}$$

The absolute error in the radius is:

$$\Delta r = \frac{\Delta d}{2} = 0.1\text{ cm}$$

Step 3: Express the relative error in volume.

The volume of a cone is given by the formula:

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

The fractional error in volume is:

$$\frac{\Delta V}{V} = 2\frac{\Delta r}{r} + \frac{\Delta h}{h}$$

Step 4: Calculate the individual fractional errors.

The contribution from the radius is:

$$2\frac{\Delta r}{r} = 2 \times \frac{0.1}{10.0} = 0.02$$

The contribution from the height is:

$$\frac{\Delta h}{h} = \frac{0.2}{20.0} = 0.01$$

Step 5: Calculate the maximum percentage error.

Combine the fractional errors and convert to a percentage.

$$\frac{\Delta V}{V} = 0.02 + 0.01 = 0.03$$ $$\frac{\Delta V}{V} \times 100\% = 3\%$$

Answer: The maximum percentage error in the determination of the volume is $3\%$.