Question

The length of the shadow of a tower standing on a level plane is found to be 2x metres longer when the sun's altitude is ${30}^{\circ }$ then when it is ${45}^{\circ }$. The height of the tower = ___ m.

• A. $x(\sqrt{3}+1)$
• B. $x(\sqrt{3}-1)$
• C. $x\sqrt{3}$
• D. $\frac{\sqrt{3}+1}{x}$

Answer 1

The correct option is A

$x(\sqrt{3}+1)$

Let the weight of the tower = h

In $\Delta ABD$

$tan{45}^{\circ }=\frac{h}{y}$

$1=\frac{h}{y}$

$y=h$ ........ (i)

In $\Delta ABC$

$tan{30}^{\circ }=\frac{h}{y+2x}$

$\frac{1}{\sqrt{3}}=\frac{h}{h+2x}$

$2x+h=h\sqrt{3}$

$h(\sqrt{3}-1)=2x$

$h=\frac{2x}{\sqrt{3}-1}$

$h=\frac{2x(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}$

$h=\frac{2x(\sqrt{3}+1)}{3-1}$

$=\frac{2x(\sqrt{3}+1)}{2}$

$=x(\sqrt{3}+1)m$