Question

If the angles of depression of the top and bottom of an 8 meter tall building from the top of a multistoried building are \(30^\circ\) and \(45^\circ\) respectively, then the height (in meters) of that multistoried building is

• A. \(12\)
• B. \(8\sqrt3\)
• C. \(4(1+\sqrt3)\)
• D. \(4(3+\sqrt3)\)

Hint:

For angle of depression, use the same angle as angle of elevation and form right triangles.

Answer 1

The Correct Option is D

Let the height of multistoried building be \(h\) m and horizontal distance be \(x\) m. From angle of depression to bottom: \[ \tan45^\circ=\frac{h}{x} \Rightarrow 1=\frac{h}{x} \Rightarrow x=h \] From angle of depression to top of 8 m building: \[ \tan30^\circ=\frac{h-8}{x} \] Substitute \(x=h\): \[ \frac1{\sqrt3}=\frac{h-8}{h} \] \[ h=\sqrt3(h-8) \] \[ h=\sqrt3 h-8\sqrt3 \] \[ h(\sqrt3-1)=8\sqrt3 \] \[ h=\frac{8\sqrt3}{\sqrt3-1} \] Rationalizing: \[ h=\frac{8\sqrt3(\sqrt3+1)}{3-1} =\frac{8(3+\sqrt3)}{2} =4(3+\sqrt3) \] Hence, \[ \boxed{4(3+\sqrt3)} \]