question_answer

A man is standing on the deck of a ship, which is 10 m above water level. He observes the angle of elevation of the top of a hill as \[{{45}^{o}}\] and the angle of depression of the base of the hill as \[{{30}^{o}}\]. Calculate the distance of the hill from the ship and the height of the hill.

A)  \[17.32m,\text{ }27.3m\]

B)  \[18.32\text{ }m,\text{ }28.3\text{ }m\]

C)  \[17.89\text{ }m,\text{ }28.3\text{ }m\]

D)  \[8.32\text{ }m,\text{ }29.2\text{ }m\]

Correct Answer: A

Solution :

Let x be the distance of hill from man and \[h+10\]be height of hill which is required. In \[\Delta ABC,\] \[\tan {{45}^{o}}=\frac{AC}{BC}=\frac{h}{x}\Rightarrow 1=\frac{h}{x}\] In  \[\Delta BCD,\]                         \[\tan {{30}^{o}}=\frac{CD}{BC}=\frac{10}{x}\] \[\Rightarrow \]\[\frac{1}{\sqrt{3}}=\frac{10}{x}\Rightarrow x=10\sqrt{3}\] \[\therefore \]   Height of hill                         \[=10\sqrt{3}+10\] \[=10\times 1.73+10\] \[=27.3\,m\] Distance of ship from              hill \[=x=10\sqrt{3}\,m\] \[=17.32\text{ }m~\]