When a voltage `V_(s)=200 sqrt(2) sin (100 t) V` is applied to an ac circuit the current in the circuit is found to be `i=2 sin [omega t + (pi//4)]A`. Find the average power consumed in the circuit.

Explanation

To find the average power consumed in the AC circuit, we can follow these steps:

### Step 1: Identify the given quantities
We have the voltage:
\[ V_s = 200 \sqrt{2} \sin(100t) \, \text{V} \]
And the current:
\[ i = 2 \sin(\omega t + \frac{\pi}{4}) \, \text{A} \]

### Step 2: Compare with standard forms
The standard forms for voltage and current in an AC circuit are:
\[ V = V_0 \sin(\omega t + \phi) \]
\[ I = I_0 \sin(\omega t + \phi) \]

From the given equations, we can identify:
- Amplitude of voltage \( V_0 = 200 \sqrt{2} \, \text{V} \)
- Amplitude of current \( I_0 = 2 \, \text{A} \)
- Phase difference \( \phi = \frac{\pi}{4} \)

### Step 3: Calculate RMS values
The RMS (Root Mean Square) values for voltage and current are given by:
\[ V_{rms} = \frac{V_0}{\sqrt{2}} \]
\[ I_{rms} = \frac{I_0}{\sqrt{2}} \]

Calculating these:
\[ V_{rms} = \frac{200 \sqrt{2}}{\sqrt{2}} = 200 \, \text{V} \]
\[ I_{rms} = \frac{2}{\sqrt{2}} = \sqrt{2} \, \text{A} \]

### Step 4: Calculate average power
The average power consumed in the circuit is given by:
\[ P = V_{rms} \cdot I_{rms} \cdot \cos(\phi) \]

Substituting the values we found:
\[ P = 200 \cdot \sqrt{2} \cdot \cos\left(\frac{\pi}{4}\right) \]

Since \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):
\[ P = 200 \cdot \sqrt{2} \cdot \frac{1}{\sqrt{2}} \]

The \( \sqrt{2} \) terms cancel out:
\[ P = 200 \, \text{W} \]

### Final Answer
The average power consumed in the circuit is:
\[ P = 200 \, \text{W} \]

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