If the rms value of current `i=3+4 sin (omega t+pi / 3)` is `x` ampere, then find `x^(2)`.

Answer

To find the value of \( x^2 \) where \( x \) is the RMS value of the current given by the equation \( i = 3 + 4 \sin(\omega t + \frac{\pi}{3}) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components of the Current**:
The current \( i \) can be separated into a DC component and an AC component:
\[ i = 3 + 4 \sin(\omega t + \frac{\pi}{3}) \]
Here, the DC component \( a = 3 \) and the amplitude of the AC component \( b = 4 \).

2. **Calculate the RMS Value**:
The RMS value of a current that consists of a DC component and an AC component can be calculated using the formula:
\[ I_{\text{rms}} = \sqrt{a^2 + \left(\frac{b}{\sqrt{2}}\right)^2} \]
However, since we are dealing with the full amplitude of the sine wave, we can use:
\[ I_{\text{rms}} = \sqrt{a^2 + \frac{b^2}{2}} \]

3. **Substituting the Values**:
Substitute \( a = 3 \) and \( b = 4 \) into the formula:
\[ I_{\text{rms}} = \sqrt{3^2 + \frac{4^2}{2}} \]
Calculate \( 3^2 \) and \( 4^2 \):
\[ 3^2 = 9 \quad \text{and} \quad 4^2 = 16 \]
Therefore:
\[ I_{\text{rms}} = \sqrt{9 + \frac{16}{2}} = \sqrt{9 + 8} = \sqrt{17} \]

4. **Finding \( x^2 \)**:
Since \( I_{\text{rms}} = x \), we have:
\[ x = \sqrt{17} \]
To find \( x^2 \):
\[ x^2 = (\sqrt{17})^2 = 17 \]

### Final Answer:
Thus, the value of \( x^2 \) is:
\[ \boxed{17} \]