Question 2.SP.30: The Zener diode in the voltage-regulator circuit of Fig. 2-4......

By Ohm’s law
i_L = \frac{v_L}{R_L} = \frac{V_Z}{R_L} = \frac{8.2}{9} = 0.911  \text{A}

Now an application of KVL gives
R_S = \frac{V_b  –  V_Z}{i_Z  +  i_L}             (1)

and we use (1) to size R_S for maximum Zener current I_Z at the largest value of V_b:
R_S = \frac{(1.1)(12)  –  8.2}{1  +  0.911} = 2.62  \Omega

Now we check to see if i_Z ≥ 75  \text{mA} at the lowest value of V_b:
i_Z = \frac{v_b  –  v_Z }{R_S}  –  i_L = \frac{(0.9)(12)  –  8.2}{2.62}  –  0.911 = 81.3  \text{mA}

Since i_Z > 75  \text{mA},  v_Z = V_Z = 8.2  \text{V} and regulation is preserved.