Step 1: Understand the question.
We have $75\,\text{g}$ of ethane and must find the volume it takes up at S.T.P. in $\text{dm}^3$.
Step 2: Remember the S.T.P. fact.
At S.T.P. one mole of any gas fills $22.4\,\text{dm}^3$. So if we know the moles, we just multiply by $22.4$.
Step 3: Find the molar mass of ethane.
Ethane is $\text{C}_2\text{H}_6$. Its mass is $2 \times 12 + 6 \times 1 = 24 + 6 = 30\,\text{g mol}^{-1}$.
Step 4: Find the moles.
\[ n = \frac{75}{30} = 2.5\ \text{mol} \]
Step 5: Convert moles to volume.
\[ V = 2.5 \times 22.4 = 56.0\ \text{dm}^3 \]
Step 6: Pick the answer.
The volume is $56.0\,\text{dm}^3$, which is option (B).
\[ \boxed{56.0\ \text{dm}^3} \]