Question

The shadow of a vertical tower on the level ground increases by $10$ m when the altitude of the sun changes from 45${}^o$ to 30${}^o$. The height of the tower is _____ .

• A. $5($$\sqrt{3}$ $+1)$
• B. $10($$\sqrt{3}$ $-1)$
• C. $9$ m
• D. $13$ m

Answer 1

The correct option is A $5($$\sqrt{3}$ $+1)$

From the given figure,
AB = tower = h,
AC and AD are shadow at 45${}^o$ and 30${}^o$.
$\therefore$ $\angle$C = 45${}^o$ and $\angle$D = 30${}^o$.
Also, $AC=x$ m and $AD=(x+10)$ m

$\therefore$ cot 30${}^o$ = $\frac{x+10}{h}\Rightarrow$ x + 10 = h$\sqrt{3}$

Also, cot 45${}^o$ = $\frac{x}{h}\Rightarrow 1=\frac{x}{h}\Rightarrow x=h$

$\therefore$ x + 10 = h$\sqrt{3}$ & $x=h$

$\Rightarrow$ h + 10 = h$\sqrt{3}$

$\Rightarrow$ h($\sqrt{3}$ - 1) = 10 $\Rightarrow$ h = $\frac{10}{\sqrt{-}2}=\frac{10}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}=\frac{10(\sqrt{3}+1)}{3-1}$ = 5($\sqrt{3}$+ 1) m.

So, option A is correct.