[66.67 m] From the top of a hill 100 m high, the angles of depression of the top and bottom of a pole are 30°

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From the top of a hill 100 m high, the angles of depression of the top and bottom of a pole are 30° and 60° respectively. What is the height of the pole?

52 m

50 m

66.67 m

33.33 m

Explanation:

Consider the diagram shown above. AC represents the hill and DE represents the pole

Given that AC = 100 m
$\angle$XAD = $\angle$ADB = 30° (∵ AX || BD )
$\angle$XAE = $\angle$AEC = 60° (∵ AX || CE)

Let DE = h

Then, BC = DE = h,
AB = (100-h) (∵ AC=100 and BC = h),
BD = CE

tan60°=$\dfrac{AC}{CE}$

=>√3=$\dfrac{100}{CE}$

=>CE=$\dfrac{100}{\sqrt{3}} $ ⋯(1)

tan30°=$\dfrac{AB}{BD}$

=>$\dfrac{1}{\sqrt{3}} =\dfrac{100-h}{\left(\dfrac{100}{\sqrt{3}}\right)}$ (∵ BD = CE and substituted the value of CE from equation 1)

=>(100−h)=$\dfrac{1}{\sqrt{3}} \times \dfrac{100}{\sqrt{3}}$

=$\dfrac{100}{3}=33.33$

=>h=100−33.33=66.67 m
i.e., the height of the pole = 66.67 m

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