In this question, we're told that a battery connected circuit is generating a certain amount of power across its resistor, and we're being asked to determine the value of this circuit's resistance.

To start with, recall that the power generated by a circuit is proportional to the applied voltage as well as to the current flowing through the circuit. Written in equation form, we have:

From the above expression, we have values for the power and voltage terms, but we do not have a value for the current. However, we can make use of Ohm's law in combination with the above expression to solve for resistance.

By plugging in this expression for current into the above expression for power, we obtain:

Next, we can solve for the answer by rearranging and plugging in values.

To solve this problem, we need an appropriate equation for power that relates current, power, and resistance. 

This is given by

, where  is power,  is current, and  is resistance. 

We see that current and power are proportional via:

Since power is changed by a factor of , current  changes by

This can be written alternatively as:

The power supply (in this case what is providing the emf) will have a power output depending on what is connected to it. A battery or lab power supply is generally designed to put out a constant voltage. The different circuit elements connected will alter the equivalent resistance of the entire circuit, and the power supply will provide a current and power needed to keep the potential (voltage) constant.

There are 3 equations for power dissipation for a resistor in a circuit. They first is: 

Where the power dissipated is equal to the product of the current going through the resistor and the voltage drop across it. The second is:

The third is: 

Notice that for each equation we need to know only 2 out of the 3 variables in Ohm's law. Let's chose the second equation with current and resistance. To find the current, notice that there is only a single loop in the circuit since both resistors are connected in series. This means that the total current coming from the power supply is equal to the current going through  and the current going through . To find the current coming from the power supply let's find the total equivalent resistance of the circuit.

There are only two resistors connected in series. The equivalent resistance is just the sum of the series resistors:

The current from the power supply is found using Ohm's law:

The power dissipated by the second resistor is then:

The power in a circuit is determined by the equation , where  is the power of the circuit,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

Assuming the resistance stays the same, if the current is doubled, the power will be four times larger.

Expressed mathematically,

If 

The power in a circuit is determined by the equation , where  is the power of the circuit,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

Assuming the current stays the same, if the resistance is doubled, the power will also double.

Expressed mathematically,

If 

The power in a circuit is determined by the equation , where  is the power of the circuit,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

In our example,

The power in a circuit is determined by the equation , where  is the power of the circuit,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

Since we are given current and voltage, we will also need Ohm's law, , where  is the voltage,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

Solving Ohm's law for resistance gives us .

Substituting this form of Ohm's law into the power equation gives us

The power equation is now in a form that we can solve with the information we are given.

The power in a circuit is determined by the equation , where  is the power of the circuit,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

Since we are given resistance and voltage, we will also need Ohm's law, , where  is the voltage,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

Solving Ohm's law for current gives us .

Substituting this form of Ohm's law into the power equation gives us

The power equation is now in a form that we can solve with the information we are given.

The power in a circuit is determined by the equation , where  is the power of the circuit,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

To relate power to voltage, we will also need Ohm's law, , where  is the voltage,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

Solving Ohm's law for current gives us .

Substituting this form of Ohm's law into the power equation gives us

Assuming the current stays the same, if the voltage is doubled, the power will be four times larger.

Expressed mathematically,

If 

The power in a circuit is determined by the equation , where  is the power of the circuit,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

To relate power to voltage, we will also need Ohm's law, , where  is the voltage,  is the current in the circuit, and  is the equivalent  resistance of the circuit.

Solving Ohm's law for current gives us .

Substituting this form of Ohm's law into the power equation gives us

Energy is related to power by the equation , where  is energy,  is the power of the circuit, and  is time.

Substituting theequation for power into the equation for energy gives

For our problem