Given current

$i(t)=\left(\sqrt{42}\sin\left(\frac{2\pi}{T}t\right)+10\right)\,\text{A}$

This is a combination of:

• an alternating part: $\sqrt{42}\sin\left(\frac{2\pi}{T}t\right)$
• a constant (DC) part: $10$

Formula for RMS value

For a current of the form $i(t)=I_0+I_m\sin\omega t,$ the RMS value is $I_{\rm rms}=\sqrt{I_0^2+\frac{I_m^2}{2}}$

because over one complete cycle:

• $\langle \sin\omega t \rangle = 0$
• $\langle \sin^2\omega t \rangle = \frac{1}{2}$

Identify values

Here, $I_0=10, \qquad I_m=\sqrt{42}$

Compute RMS

$I_{\rm rms}=\sqrt{10^2+\frac{(\sqrt{42})^2}{2}}$

$=\sqrt{100+\frac{42}{2}}$

$=\sqrt{100+21}$

$=\sqrt{121}=11\,\text{A}$

Final answer

$\boxed{11}$

Comparison with stored answer

Stored correct answer = $11$

This matches the derived answer.