From the top of a tower 50m high the angle of depression of the top of a pole is 45degree - Maths - Some Applications of Trigonometry

Question

From the top of a tower 50m high the angle of depression of the
top of a pole is 45degree and from the foot of the pole,the angle
of elevation of the top of the tower is 60degree find the height of
the pole if the pole and tower stand on the same plane?

Lalit Mehra · Accepted Answer

Let AB be the pole and CD be the tower Suppose the height of pole is h m Given, âˆ ADF = 45Â°, âˆ CBD = 60Â° and CD = 50 m âˆ ADF = âˆ DAE = = 45Â° (Alternate angles) AB = CE = h m DE = CD â€“ CE = (50 â€“ h) m In Î”ADE, $\tan 45 ° = \frac{DE}{AE} ∴ 1 = \frac{50 - h}{AE} \Rightarrow AE = 50 - h (1)$ In Î”CBD, $\tan 60 ° = \frac{CD}{BC} ∴ \sqrt{3} = \frac{50 m}{BC} \Rightarrow BC = \frac{50}{\sqrt{3}} m = \frac{50}{3} \sqrt{3} m (2)$ Now, AE = BC $∴ 50 - h = \frac{50}{3} \sqrt{3} [Using (1) and (2)] \Rightarrow h = 50 - \frac{50 \sqrt{3}}{3} \Rightarrow h = \frac{150 - 50 \sqrt{3}}{3} m \Rightarrow h = \frac{50 \sqrt{3} (\sqrt{3} - 1)}{3} m$ Thus, the height of pole is $\frac{50 \sqrt{3} (\sqrt{3} - 1)}{3} m$

From the top of a tower 50m high the angle of depression of the top of a pole is 45degree and from the foot of the pole,the angle of elevation of the top of the tower is 60degree.find the height of the pole if the pole and tower stand on the same plane?