Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is:

We have a hollow (thick-walled) cylinder whose length is the same everywhere, so its mass is distributed uniformly in the annular region between the inner radius and the outer radius. The data are

$$\text{Inner radius } r_1 = 10\ \text{cm},\qquad \text{Outer radius } r_2 = 20\ \text{cm},\qquad \text{Length } L = 30\ \text{cm}. $$

Let the mass of this hollow cylinder be $$M.$$ For any hollow cylinder rotating about its own axis (the line that passes through the centre and is perpendicular to the circular faces) the standard formula for the moment of inertia is first stated:

$$I_{\text{hollow}}=\frac12\,M\left(r_1^2+r_2^2\right).$$

Substituting the given numerical values, we obtain

$$I_{\text{hollow}}=\frac12\,M\left[(10\ \text{cm})^2+(20\ \text{cm})^2\right]$$

$$=\frac12\,M\left[100+400\right]$$

$$=\frac12\,M\left[500\right]$$

$$=250\,M.$$

This moment of inertia is denoted by the symbol $$I$$ in the question, so we may simply write

$$I = 250\,M.$$

Now we wish to replace the thick-walled cylinder by a thin cylinder that has the same mass $$M$$ and the same moment of inertia $$I$$ about its own axis. For a thin cylindrical shell (all the mass concentrated practically at one radius $$R$$) the standard result is:

$$I_{\text{thin}} = M\,R^2.$$

Because the problem asks that both moments of inertia be equal, we set

$$I_{\text{thin}} = I_{\text{hollow}}.$$

So

$$M\,R^2 = 250\,M.$$

Since the masses are equal and non-zero, they cancel out directly, leaving

$$R^2 = 250.$$

Taking the (positive) square root, we get

$$R=\sqrt{250}\ \text{cm}.$$

Now, $$250 = 25 \times 10,$$ so

$$R=\sqrt{25\times10}\ \text{cm}=5\sqrt{10}\ \text{cm}.$$

Numerically, $$\sqrt{10}\approx3.162$$, and therefore

$$R \approx 5 \times 3.162\ \text{cm}\approx15.81\ \text{cm}.$$

Rounding to the nearest whole centimetre that appears among the given options, we obtain

$$R\approx16\ \text{cm}.$$

Hence, the correct answer is Option A.