If ${\cos }^{3}110°+{\cos }^{3}10°+{\cos }^{3}130°$  is $\frac{3\sqrt{3}}{k}$ , then $k$  is

$={\cos }^{3}10+{\cos }^3\left[120-10\right]+{\cos }^3\left[120+10\right]$

We know that

$\cos 10+\cos \left[120-10\right]+\cos \left[120+10\right]=0$

$\left[\because \text{if}a+b+c=0\text{then}{a}^3+b^3+c^3=3abc\right]$

$=3\cos 10\times \cos \left[120-10\right].\cos \left[120+10\right]$

$=\frac{3}{4}\cos 3\times 10$

$=\frac{3}{4}\cos 30$

$=\frac{3\sqrt{3}}{8}$  

$\therefore \frac{3\sqrt{3}}{k}=\frac{3\sqrt{3}}{8}$