Two poles standing on a horizontal ground are of heights 5   m and 10   m respectively. The line…

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

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

In $\Delta BDM$
$\mathrm{t}\mathrm{a}\mathrm{n}15°=\frac{BM}{MD}=\frac{5}{K}$

$\Rightarrow K=\frac{5}{\mathrm{t}\mathrm{a}\mathrm{n}15°}$

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

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

$\Rightarrow K=5\left(2+\sqrt{3}\right) m.$