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 volume of a right circular cone in terms of the measured base diameter $$d$$ and height $$h$$ is

$$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\left(\frac{d}{2}\right)^{2}h=\frac{\pi}{12}\,d^{2}h$$

If a quantity $$Q$$ depends on the variables $$x,\,y,\,z$$ as $$Q=x^{m}y^{n}z^{p}$$, the maximum fractional (relative) error in $$Q$$ is

$$\frac{\Delta Q}{Q}=m\frac{\Delta x}{x}+n\frac{\Delta y}{y}+p\frac{\Delta z}{z}$$

Comparing $$V=\dfrac{\pi}{12}d^{2}h$$ with the general form, we have $$m=2$$ for $$d$$ and $$n=1$$ for $$h$$ (the constant $$\pi/12$$ does not affect errors).

Therefore, the maximum fractional error in volume is

$$\frac{\Delta V}{V}=2\frac{\Delta d}{d}+1\frac{\Delta h}{h}$$ $$-(1)$$

The measurements are:

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

The least count of the scale is $$2\text{ mm}=0.2\text{ cm}$$. In error calculations we take the full least count as the maximum absolute error, so

$$\Delta d = 0.2\text{ cm}, \qquad \Delta h = 0.2\text{ cm}$$

Compute the fractional errors:

$$\frac{\Delta d}{d}=\frac{0.2}{20.0}=0.01$$   (that is, $$1\%$$)

$$\frac{\Delta h}{h}=\frac{0.2}{20.0}=0.01$$   (that is, $$1\%$$)

Substitute into $$(1)$$:

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

Converting to percentage:

Maximum percentage error $$=0.03\times100\% = 3\%$$

Hence, the maximum percentage error in determining the volume of the cone is 3 %.