The angles of depression of the top and the bottom of a $\displaystyle 8$ m tall building from the top of a multi-storeyed building are $\displaystyle 30$° and $\displaystyle 45$° respectively. Find the height of the multi-storeyed building and the distance between the two buildings.

Let height of multi storeyed building AB be H m and CD is a tall building. Let the distance between the two buildings be X m. In \(\displaystyle \triangle \mathrm{ABD}\) \[\begin{aligned} & \tan 45^{\circ}=1=\frac{\mathrm{H}}{\mathrm{X}} \\ & \Rightarrow \mathrm{H}=\mathrm{X} \end{aligned} \] In \(\displaystyle \triangle \mathrm{AEC}\) \[\tan 30^{\circ}=\frac{1}{\sqrt{3}}=\frac{\mathrm{H}-8}{\mathrm{X}} \] \[\Longrightarrow \mathrm{X}=\sqrt{3}(\mathrm{H}-8) \] Solving equations (i) & (ii) \[X=4(3+\sqrt{3}) \] and \(\displaystyle \mathrm{H}=4(3+\sqrt{3})\) The height of the multi storeyed building is \(\displaystyle 4(3+\sqrt{3}) \mathrm{m}\) and the distance between the two buildings is \(\displaystyle 4(3+\sqrt{3}) \mathrm{m}\).