Question

A person standing on the bank of the river observes that the angle of elevation of the top of a tree standing on the opposite bank is ${60^ \circ }$. When he was $40m$ away from the bank he found the angle of elevation to be ${30^ \circ }$. Find the width of the river.
A.$20m$
B.$10m$
C.$5m$
D.$1m$

Answer 1

Hint : Firstly, construct a diagram using the information as given in the question. Use the basic trigonometric formulas related to sides of a triangle and the angle between them. Find the measurements of the required side. Initiate with solving the triangle with ${60^ \circ }$, find an equation and substitute it in the other equation.
Formula used
I.$\tan {60^ \circ } = \sqrt 3$
II.$\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}$

Complete step-by-step answer :
Constructing the diagram with the help of given information.
The angle of elevation is an angle that is formed between the horizontal line and the line of sight.

Let $CD = h$ be the height of the tree and $BC = x$ be the breadth of the river.
From the figure $\angle DAC = {30^ \circ }$ and $\angle DBC = {60^ \circ }$.
In right angled triangle $\vartriangle BCD$, we can write as: $\tan {60^ \circ } = \dfrac{{DC}}{{BC}}$
Since, we know that $\tan {60^ \circ } = \sqrt 3$.
Therefore, we get $\sqrt 3 = \dfrac{h}{x}$
Solving for h we, get:
$h = x\sqrt 3$ … ( $1$)
In right angled triangle $\vartriangle ACD$, $\tan {30^ \circ } = \dfrac{h}{{40 + x}}$
Since, we know that $\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}$.
Therefore, we get $\dfrac{1}{{\sqrt 3 }} = \dfrac{h}{{40 + x}}$
Hence, In terms of h, we can write as:
$\sqrt 3 h = 40 + x$ … ( $2$)
Writing the value of h from ( $1$) in ( $2$) we get
$\Rightarrow \sqrt 3 \left( {x\sqrt 3 } \right) = 40 + x$
$\Rightarrow 3x = 40 + x$
Subtracting both sides by x:
$\Rightarrow 3x - x = 40 + x - x$
$\Rightarrow 2x = 40$
Dividing both the sides by $2$:
$\Rightarrow \dfrac{{2x}}{2} = \dfrac{{40}}{2}$
$\Rightarrow x = 20$
Solving this equation, we get $x = 20$
Therefore, the width of the river is $20m$.
So, the correct answer is “Option A”.

Note : If a person stands and looks up at an object, the angle of elevation is the angle between the horizontal line of sight and the object. If a person stands and looks down at an object, the angle of depression is the angle between the horizontal line of sight and the object.