A spherical body of radius r and density $$\sigma$$ falls freely through a viscous liquid having density $$\rho$$ and viscosity $$\eta$$ and attains a terminal velocity $$\upsilon_{0}$$. Estimated maximum error in the quantity $$\eta$$ is: (Ignore errors associated with $$\sigma,\rho$$ and g, gravitational acceleration)

Terminal velocity of a sphere: $$v_0 = \frac{2r^2(\sigma - \rho)g}{9\eta}$$, so $$\eta = \frac{2r^2(\sigma - \rho)g}{9v_0}$$.

Ignoring errors in $$\sigma, \rho, g$$:

$$\frac{\Delta\eta}{\eta} = 2\frac{\Delta r}{r} + \frac{\Delta v_0}{v_0}$$

(Since $$\eta \propto r^2/v_0$$, the maximum relative error is $$2\frac{\Delta r}{r} + \frac{\Delta v_0}{v_0}$$.)

The answer is Option 1: $$\frac{2\Delta r}{r} + \frac{\Delta v_0}{v_0}$$.