1. The current is given by $i(t)=6+\sqrt{56}\sin\left(100\pi t+\frac{\pi}{3}\right)$

This is of the form $i(t)=I_0+I_m\sin(\omega t+\phi)$ where

• DC component: $I_0=6$
• AC amplitude: $I_m=\sqrt{56}$

2. For a current having both DC and AC components, the rms value is $I_{\text{rms}}=\sqrt{I_0^2+\frac{I_m^2}{2}}$

This follows because the average of $\sin(\omega t+\phi)$ over a full cycle is $0$, and the average of $\sin^2(\omega t+\phi)$ is $\frac12$.

3. Substitute the values: $I_{\text{rms}}=\sqrt{6^2+\frac{(\sqrt{56})^2}{2}}$

$=\sqrt{36+\frac{56}{2}}$

$=\sqrt{36+28}$

$=\sqrt{64}=8$

4. Therefore, the rms value of the current is $\boxed{8\ \text{A}}$