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 ______.

The correct answer is 3.

Let's break down the step-by-step calculation to find the maximum percentage error in the determination of the volume of the cone.

The formula for the volume $V$ of a cone with base radius $r$ and height $h$ is:
$V=\frac{1}{3}\pi r^{2}h$
Since the base diameter $d$ is measured, we can express the radius in terms of diameter as $r=\frac{d}{2}$. Substituting this into the volume formula gives:
$V=\frac{1}{3}\pi {\left(\frac{d}{2}\right)}^{2}h=\frac{\pi }{12}{d}^{2}h$

To find the maximum fractional error in volume, we take the natural logarithm on both sides and differentiate:
$\ln \left(V\right)=\ln \left(\frac{\pi }{12}\right)+2\ln \left(d\right)+\ln \left(h\right)$
Differentiating both sides, we get the relative error relation:
$\frac{\Delta V}{V}=2\frac{\Delta d}{d}+\frac{\Delta h}{h}$

The maximum percentage error is obtained by multiplying the fractional errors by 100:
$\frac{\Delta V}{V}\times 100=\left(2\frac{\Delta d}{d}+\frac{\Delta h}{h}\right)\times 100$

Now, we look at the values given in the problem:
1. Least count of the scale (which represents the absolute error in measurement, $\Delta d$ and $\Delta h$) is 2 mm. Converting this to cm:
$\Delta d=\Delta h=2\text{mm}=0.2\text{cm}$
2. The measured diameter of the base $d$ is 20.0 cm.
3. The measured height $h$ is 20.0 cm.

Substitute these values into the percentage error equation:
$\frac{\Delta V}{V}\times 100=\left(2\times \frac{0.2}{20.0}+\frac{0.2}{20.0}\right)\times 100$
$\frac{\Delta V}{V}\times 100=\left(2\times 0.01+0.01\right)\times 100$
$\frac{\Delta V}{V}\times 100=\left(0.02+0.01\right)\times 100=0.03\times 100=3\%$

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