You are asked to design a circuit that indicates if a 12 V battery has been connected the correct way around to two terminals, A and B. To do this, you have chosen to use a special kind of diode known as a LED.
A green LED will light to indicate the battery is connected using the correct orientation (positive to terminal A, negative to terminal B) and a red LED will light to show an incorrect connection (negative to terminal A and positive to terminal B).
A LED acts in a similar way to a rectifier diode, it lets current flow from the anode to the cathode, but blocks current from flowing from the cathode to the anode. When current is allowed to flow, the LED will also light up.
However, unlike a rectifier diode, which has a 0.6 - 0.8 volt drop across it when conducting, a LED typically has 2.5 volts drop. Too much current flowing through a LED can damage it; your circuit needs to limit the amount of current flowing through any LED used. After reading the specification sheet from your LED's supplier, you have found that to achieve a sufficient brightness without damaging the LED, a green LED should have 30mA flowing through it while a red LED should have 45mA.
Design the circuit and include all calculations necessary to show why you selected specific components. Draw your circuit twice, once for the correct connection of the battery and once for the incorrect connection. For each case, clearly show the voltage across each component and the current in each branch.