A man standing on a railway bridge 5 3 meters high, observes the engine of a train at an angle of depression 30 ° . But after 2 seconds , he observes the engine at an angle of depression 45 ° on the other side of the bridge. Find the speed of the train.

Given, height ( h ) of the bridge = A B = 5 3 m , the angle of depression from one side = ∠ A C B = 45 ° , the angle of depression from another side = ∠ A D B = 30 ° . Required time ( t ) = 2 seconds . We need to find the speed of the train. Now from the ∆ A C B , tan 45 ° = A B B C ⇒ 1 = 5 3 B C ⇒ B C = 5 3 m Now from the ∆ A D B , tan 30 ° = A B B D ⇒ 1 3 = 5 3 B D ⇒ B D = 15 m B C = B D - C D B C = 15 - 5 3 m Therefore, the distance covered by the train in 2 seconds is 15 - 5 3 m . Speed of the train = 15 - 5 3 2 = 3 . 17 m / s Hence, the speed of the train is 3 . 17 m / s .

The length of the shadow of a post becomes 3 meters smaller when the angle of elevation of the Sun increases from 45 ° to 60 ° . Find the height of the post.

Given, the length of the shadow of a post becomes 3 meters smaller when the angle of elevation of the Sun increases from 45 ° to 60 ° . We need to find the height of the post. Here, ∠ B C A = 45 ° , ∠ B D A = 60 ° and C D = 3 m . To find B D In △ A B D , A B B D = tan 60 ° ⇒ A B B D = 3 ⇒ A B = 3 × B D In △ A B C , A B B C = tan 45 ° ⇒ 3 × B D B D + 3 = 1 ⇒ 3 × B D = B D + 3 ⇒ 3 × B D - B D = 3 ⇒ B D 3 - 1 = 3 ⇒ B D = 3 3 - 1 Now by rationalising the denominator, we get 3 3 - 1 × 3 + 1 3 + 1 = 3 3 + 1 3 - 1 = 3 3 + 1 2 Now, A B = 3 3 3 + 1 2 ⇒ A B = 1 . 732 3 1 . 732 + 1 2 = 7 . 097736 Hence, the height of the post is 7 . 097736 .

If the ratio of the length of the shadow of a tower and height of the tower is 3 : 1 , find the angle of elevation of the sun.

Given that the ratio of the length of the shadow of a tower and height of the tower is 3 : 1 Let the height of the tower is AB , The length of the shadow is BC , angle of elevation is θ From the given question, AB BC = 3 1 = 3 From the ∆ ABC tan θ = AB BC tan θ = 3 tan θ = tan 60 ° ∵ tan 60 ° = 3 θ = 60 ° Hence, the angle of elevation of the sun is 60 ° .