The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is $60°$. The angle of elevation of the top of second tower from the foot of the first tower is

Question

The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is $60°$. The angle of elevation of the top of second tower from the foot of the first tower is $30°$. Find the distance between two towers and the height of the other tower. Also, find the length of the wire attached to the tops of both the towers.

Accepted Answer

Step 1: Let $AB$ and $CD$ be the given towers.
$	an 30° = \frac{h}{x} \Rightarrow x = h\sqrt{3}$ ... (i)
Step 2: $	an 60° = \sqrt{3} = \frac{24}{x} \Rightarrow x = \frac{24}{\sqrt{3}}$ or $8\sqrt{3}$ ... (ii)
Step 3: Using (i) and (ii):
$x = 8\sqrt{3}$ and $h = 8$
Step 4: Length of wire $= \sqrt{BE^2 + x^2} = \sqrt{256 + 192} = \sqrt{448}$ m $= 8\sqrt{7}$ m

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