A sensor circuit needs a steady 5.1 V from a supply whose voltage is not steady…

Step 1. The greatest Zener current. $I_{z,max} = \dfrac{P}{V_z} = \dfrac{0.50\ \text{W}}{5.1\ \text{V}} = 98.0$ mA.

Step 2. The largest resistor. At the lowest supply and the heaviest load, the series current must be $I_{z,min} + I_{L,max} = 2.0\ \text{mA} + 30\ \text{mA} = 32$ mA. So $R_s \le \dfrac{9.0\ \text{V} - 5.1\ \text{V}}{0.032\ \text{A}} = 122\ \Omega$.

Step 3. The smallest resistor. At the highest supply and the lightest load, the series current must be at most $I_{z,max} + I_{L,min} = 98.0\ \text{mA} + 5.0\ \text{mA} = 103$ mA. So $R_s \ge \dfrac{12\ \text{V} - 5.1\ \text{V}}{0.103\ \text{A}} = 67.0\ \Omega$.

Step 4. The choice. $R_s$ must lie between $67.0\ \Omega$ and $122\ \Omega$. A resistor of $100\ \Omega$ works. A resistor of $150\ \Omega$ is too large, and the Zener diode would switch off when the supply is low and the load is heavy.