Write the formulas for surface area and volume of a sphere

$S = 4\pi r^2$ $V = \frac{4}{3}\pi r^3$

Use error propagation for powers

If a quantity depends on $r^n$, then the relative error is:

$\frac{\Delta Q}{Q} = n\frac{\Delta r}{r}$

Given relative error in surface area

Since $S \propto r^2$ we have $\frac{\Delta S}{S} = 2\frac{\Delta r}{r}$

Given that relative error in surface area is $\alpha$,

$\alpha = 2\frac{\Delta r}{r}$

Therefore,

$\frac{\Delta r}{r} = \frac{\alpha}{2}$

Find relative error in volume

Since $V \propto r^3$ so $\frac{\Delta V}{V} = 3\frac{\Delta r}{r}$

Substitute $\frac{\Delta r}{r} = \frac{\alpha}{2}$:

$\frac{\Delta V}{V} = 3\cdot \frac{\alpha}{2} = \frac{3}{2}\alpha$

Match with the options

$\boxed{\frac{3}{2}\alpha}$

So the correct option is A.