Two poles standing on a horizontal ground are of heights 5m and 10m...

Let, $DC$ be the pole of height $5m$ and $AB$ be the pole of height $10m .$

Also, let $DM ⊥ AB,$ then $ACDM$ is a rectangle and hence, $AM = DC = 5m$ and $BM = AB - AM = 10 - 5 = 5m$ and the distance between the poles is $AC = MD = K ($let$) .$

In $\delta BDM$
$\tan 15 ° = \dfrac{BM}{MD} = \dfrac{5}{K}$

$⇒ K = \dfrac{5}{\tan 15 °}$

$⇒ K = \dfrac{5\left(\sqrt{3} + 1\right)}{\sqrt{3} - 1} \times \dfrac{\sqrt{3} + 1}{\sqrt{3} + 1}$

$⇒ K = \dfrac{5\left(3 + 1 + 2\sqrt{3}\right)}{3 - 1}$

$⇒ K = 5\left(2 + \sqrt{3}\right)m .$