To find the root mean square (RMS) value of the given alternating current, follow these steps:

The current is represented as:

$ i = 5\sqrt{2} + 10 \cos \left(650\pi t + \frac{\pi}{6}\right) \, \text{Amp} $

Here, the time-independent DC component is $5\sqrt{2}$ and the AC component is $10 \cos \left(650\pi t + \frac{\pi}{6}\right)$.

Calculate the square of the current, $i^2$:

$ i^2 = \left(5\sqrt{2}\right)^2 + \left(10 \cos \left(650\pi t + \frac{\pi}{6}\right)\right)^2 + 2 \times 5\sqrt{2} \times 10 \cos \left(650\pi t + \frac{\pi}{6}\right) $

Simplifying, we have:

$ i^2 = 50 + 100 \cos^2 \left(650\pi t + \frac{\pi}{6}\right) + 100\sqrt{2} \cos \left(650\pi t + \frac{\pi}{6}\right) $

Find the average value $\langle i^2 \rangle$:

The average value of $\cos$ terms over a period is zero, simplifying our equation to:

$ \langle i^2 \rangle = 50 + \frac{100}{2} + 0 $

This simplifies to:

$ \langle i^2 \rangle = 50 + 50 = 100 $

Calculate the RMS current:

The RMS value is the square root of the mean of the squares of the current:

$ \langle i \rangle = \sqrt{100} = 10 \, \text{Amp} $

Thus, the RMS value of the current is 10 Amps.