1. Use the standard expressions for molecular speeds

For a gas at temperature $T$:

• Average speed, $v_{\text{avg}}=\sqrt{\frac{8kT}{\pi m}}$

• RMS speed, $v_{\text{rms}}=\sqrt{\frac{3kT}{m}}$

Here, the question denotes the average speed by $v$.

So, $v=\sqrt{\frac{8kT}{\pi m}}$

and $v_{\text{rms}}=\sqrt{\frac{3kT}{m}}$

2. Relate $v_{\text{rms}}$ to $v$

We write $\frac{v_{\text{rms}}}{v}=\sqrt{\frac{3kT/m}{8kT/(\pi m)}}=\sqrt{\frac{3\pi}{8}}$

Hence, $v_{\text{rms}}=\sqrt{\frac{3\pi}{8}}\,v$

3. Use the given form

According to the question, $v_{\text{rms}}=\left(1+\frac{5}{x}\right)^{1/2}v$

Therefore, $1+\frac{5}{x}=\frac{3\pi}{8}$

Using $\pi=\frac{22}{7}$, $\frac{3\pi}{8}=\frac{3}{8}\cdot\frac{22}{7}=\frac{66}{56}=\frac{33}{28}$

So, $1+\frac{5}{x}=\frac{33}{28}$

Subtract 1: $\frac{5}{x}=\frac{33}{28}-1=\frac{33-28}{28}=\frac{5}{28}$

Thus, $x=28$

4. Check with options

The correct option is: $\boxed{\text{C: }28}$