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.