Question

From the top of a cliff $50$m high, the angles of depression of the top and bottom of a tower are observed to be $30°$ and $45°$.

The height of tower is

• A. $50$m
• B. $50\sqrt{3}$m
• C. $50(\sqrt{3}–1)$m
• D. $50(1–\frac{\sqrt{3}}{3})$m

Answer 1

The correct option is D

$50(1–\frac{\sqrt{3}}{3})$m

Explanation for the correct option:

Step 1. Draw the figure according to the question:

From the figure, we get

$BD=50m$

$AE=$Height of the tower ($h$)

$\angle DEC=30°$

$\angle DAB=45°$

$BA=CE$

$=x$ m

Step 2. From the right-angled $△DEC$,

$\tan 30°=\frac{DC}{CE}$

$=\frac{[50–h]}{x}$

$\Rightarrow$$\frac{1}{\sqrt{3}}=\frac{[50–h]}{x}$

$\Rightarrow$ $x=\sqrt{3}(50–h)$ — (i)

Step 3. From the right-angled $△BAD$,

$\tan 45°=\frac{DB}{BA}$

$\Rightarrow$ $1=\frac{50}{x}$

$\Rightarrow$ $x=50$— (ii)

Step 4. From (i) and (ii), we get

$50=(50–h)\sqrt{3}$

$50–50\sqrt{3}=–h\sqrt{3}$

$h=\frac{[50(\sqrt{3}–1\left)\right]}{\sqrt{3}}$

$h=\mathbf{50}\mathbf{}\mathbf{(}\mathbf{1}\mathbf{}\mathbf{–}\mathbf{}\frac{\sqrt{\mathbf{3}}\mathbf{}}{\mathbf{3}}\mathbf{)}\mathbf{}$m

Hence, Option ‘D’ is Correct.