Heights And Distances

To solve the problem of finding the breadth of the river, we apply trigonometric concepts involving right-angled triangles. The man first sees the tree at a 60° angle and later at a 30° angle after moving 24m away from the riverbank.
Let the breadth of the river be meters and the height of the tree be meters.
The first observation forms a right triangle with the riverbank, where the angle is 60°. Thus, using the tangent function:

Since , we have:
(Equation 1)
After moving 24m away, the angle is now 30°. The distance from the man to the tree is now . Again using tangent:

Since , we have:
(Equation 2)
From Equation 1 and Equation 2, we equate the expressions for and solve for . From Equation 1:

Substitute in Equation 2:

Cross-multiply:

Simplify:

Thus:

Hence, the breadth of the river is meters.