(This is the current that starts to charge the capacitor; as the voltage on the capacitor builds, the current through the 600 ohm resistor will grow accordingly.)
The current decays as i(t) = i0 exp(-t/RC). The initial current is 20 mA from part b. We seek the time that it takes for i(t)/i0 to reach 1.0 mA. Solving algebraically, that is the time RC ln(20) = 450 ms.