A surveyor must find the breadth of a straight river

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Trigonometry · Applications of Trigonometry

Problem

A surveyor must find the breadth of a straight river. Standing at point

A

on one bank, she sights a tree directly opposite on the other bank and measures the angle of elevation to the top of the tree as

38^\circ

. She then walks back

45\,\text{m}

along a line perpendicular to the bank to point

B

and now finds the angle of elevation to be

25^\circ

. Assume the ground is level and the tree stands vertically. (i) Draw a labelled diagram showing the two positions

A

and

B

, the tree

T

, the horizontal ground, and the line of sight for each observation. (ii) Denote the breadth of the river by

d

metres and the height of the tree by

h

metres. Using the two observations, set up two separate trigonometric equations in

d

and

h

. (iii) Solve the two equations simultaneously to obtain the values of

d

and

h

correct to one decimal place.