Step 1: A 5.1 V Zener diode regulates a supply. Input varies 9–12 V, load draws 20 mA. Choose the series resistor R_s for I_Z between 5 and 40 mA. Step 2: Worst case is minimum input (9 V), where total current is smallest. Design at V_in = 9 V with I_Z = 10 mA (safe margin above minimum). Step 3: I_total = I_Z + I_load = 10 + 20 = 30 mA. Step 4: R_s = (V_in − V_Z)/I_total = (9 − 5.1)/0.030 = 3.9/0.030 = 130 Ω. Step 5: Check at V_in = 12 V: I_total = (12 − 5.1)/130 = 53.1 mA; I_Z = 53.1 − 20 = 33.1 mA < 40 mA ✓. Step 6: Zener power: P_Z = 5.1 × 0.0331 = 0.169 W — use a 0.5 W diode for margin. Result: R_s = 130 Ω; output stays 5.1 V while input swings 9–12 V. The I–V curve's sharp reverse knee at V_Z is what makes this possible.