Understanding/analyzing a power supply circuit
Seems like your exercise is awkwardly worded...
The LED specification seems clear enough: your are required to choose Rled so that 15 mA. will flow through LED and Rled. With 15 mA. flowing through LED, you are told that the voltage drop across LED terminals is 2.0 volts.
The specification for RL makes no sense: "RL=12V" ?? Am suspecting that you must choose the ohmic value of RL so that it dissipates 1200mW when 12v appears across its terminals.
I would also assume that input voltage V1 is a 220v rms sinusoidal waveform. It is clear that its frequency is 60 Hz. and you should assume that the transformer is lossless, stepping down voltage by a factor of 15. The 7812 voltage regulator is simplified as well to a lossless component, consuming no current of its own - generating exactly 12v DC at its output terminal, provided input voltage is above 15v and below 27v. Whatever current flows into its input terminal emerges from its output terminal.
Be aware that since the regulator provides a constant output voltage, then current will be constant too, regardless of an input voltage that varies. This means that your exponential equation doesn't apply here. Think carefully about this before proceeding to solve for voltage across C.