Question

A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60∘, when he moves 40m from the bank he finds the angle to be 30∘. Find the height of the tree and the breadth of the river.

Answer 1

The student wants to calculate the height of a tree and the breadth of a river using angles of elevation measured at two different points. This is a trigonometry problem that utilizes the properties of right-angled triangles and the application of tangents for the angles provided.

Step-by-step explanation:

The student is asking a trigonometric question related to finding the height of the tree and the breadth of the river using angles of elevation and distances. This involves understanding the properties of right-angled triangles and applying trigonometric ratios.

Let the height of the tree be h and the breadth of the river be b. When the person observes the tree at a 60° angle of elevation, the tree, the person, and the point directly opposite the tree across the river form a right-angled triangle, with h as the opposite side and b as the adjacent side to the 60° angle. The tangent of 60° is equal to h/b.

When the person moves 40 meters away from the bank and observes the tree at a 30° angle, the point where the person now stands, the tree, and the point directly opposite the tree form another right-angled triangle. Now, the distance between the person and the opposite point of the tree is b+40 meters. The tangent of 30° is equal to h/(b+40). By setting up equations with these tangents and solving them simultaneously, we can determine h and b.