Question

The angles of depression of the top and the bottom of a 10 m tall building from the top of a multi-storeyed building are ${30}^{\circ }$ and ${45}^{\circ }$, respectively. Find the distance between the two buildings.

• A. 5($\sqrt{3}$ + 3) m
• B. 10 m
• C. 15 m
• D. 5 m

Answer 1

The correct option is A

5($\sqrt{3}$ + 3) m

The above figure represents the situation aptly
$\angle CBE=\angle BEFand\angle DAE=\angle AEF\left(alternateangles\right)tan\left(\angle EAD\right)=tan{45}^{\circ }=\frac{ED}{AD}=\frac{EC+CD}{AD}\Rightarrow AD\times tan{45}^{\circ }=EC+CD\dots \dots \left(1\right)andtan\left(\angle EBC\right)=tan{30}^{\circ }=\frac{EC}{CB}\Rightarrow CB\times tan{30}^{\circ }=EC\dots \dots \left(2\right)Subtractingeq\left(2\right)from\left(1\right)\Rightarrow AD\times tan{45}^{\circ }-CB\times tan{30}^{\circ }=CD\Rightarrow AD(tan{45}^{\circ }-tan{30}^{\circ })=CD(\because AD=CB)\Rightarrow AD(1-\frac{1}{\sqrt{3}})=10\Rightarrow AD=\frac{10\sqrt{3}}{\sqrt{3}-1}Rationlisingthedenominatorweget\Rightarrow AD=5(3+\sqrt{3})m$ Which is the distance between the two buildings.