The angles of depression of the top and the bottom of an 8 m tall building from the top of another multistoried building are $30°$ and $45°$, respectively. Find the height of the multistoried building

Question

The angles of depression of the top and the bottom of an 8 m tall building from the top of another multistoried building are $30°$ and $45°$, respectively. Find the height of the multistoried building and the distance between the two buildings.

Accepted Answer

Step 1: Let CD represent the multistoried building. Correct figure.
Step 2: In right $	riangle DEA$:
$\frac{DE}{AE} = 	an 30° = \frac{1}{\sqrt{3}}$
$\Rightarrow AE = \sqrt{3} \cdot DE$ ... (1)
Step 3: In right $	riangle DCB$:
$\frac{DC}{BC} = 	an 45° = 1$
$\Rightarrow BC = DC$ ... (2)
From figure, $AE = BC$
Step 4: $	herefore \sqrt{3} \cdot DE = DC$
$\sqrt{3}(DC - 8) = DC$
$\Rightarrow DC = \frac{8\sqrt{3}}{\sqrt{3} - 1} 	imes \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{8\sqrt{3}(\sqrt{3} + 1)}{2} = (12 + 4\sqrt{3})$ m
Step 5: From (2), $BC = (12 + 4\sqrt{3})$ m

Question

Solution