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The root-mean-square (RMS) voltage is a way of expressing the effective value of an alternating current (AC) voltage. This comes handy because AC current and voltage fluctuate over time, unlike direct current (DC).

In the exercise, the voltage amplitude is given as 310 V. To find the RMS voltage, you use the following formula: \[ V_{rms} = \frac{V_m}{\sqrt{2}} \] where \( V_m \) is the amplitude of the voltage. For our problem, substituting the values gives us \( V_{rms} = \frac{310}{\sqrt{2}} \).

This operation gives us an RMS voltage of approximately 219.2 V, which represents the equivalent DC voltage that would deliver the same power to the motor as the fluctuating AC voltage does. The RMS value is lower than the peak amplitude because it accounts for the "average power effect" inherent in AC systems.

RMS current is similar to RMS voltage in that it represents the effective value of a fluctuating current in an AC circuit. Calculating the RMS current helps us understand how much current would be equivalent if it were a steady DC value.

In the provided exercise, the current amplitude is 10.0 A. Applying the formula \( I_{rms} = \frac{I_m}{\sqrt{2}} \), where \( I_m \) is the current amplitude, we find: \[ I_{rms} = \frac{10.0}{\sqrt{2}} \].

This calculation gives us an RMS current of approximately 7.07 A. This "average" value is what actually heats a resistor in the circuit. It provides a more useful measure for controlling the current in circuits that operate on AC power, ensuring devices run efficiently and safely.

Average power in an AC circuit refers to the actual power consumed over time, which can differ from instantaneous power due to its fluctuating nature. Calculating average power requires knowing both the effective RMS values of voltage and current, along with the power factor \( \cos\phi \).

In many practical problems like our exercise, it is often assumed \( \cos\phi = 1 \) unless a specific value is provided. This assumption implies that the voltage and current are perfectly in phase, and all the power is used effectively.

Using the formula \( P = V_{rms} \times I_{rms} \times \cos\phi \), with the values \( V_{rms} = 219.2 \) V and \( I_{rms} = 7.07 \) A, we calculate: \[ P = 219.2 \times 7.07 \times 1 \].

This yields an average power of approximately 1549.74 W. This number represents the actual power consumption that leads to the desired output/work. Understanding average power is critical for energy efficiency and managing electricity costs effectively in homes and industries.