Question

The angle of elevation of the tip of a flag staff from a point 10 m due South of its base is \[60^\circ \]. What is the height of the flag staff correct to the nearest meter?
A) 15 m
B) 16 m
C) 17 m
D) 18 m

Answer 1

Hint: Here, we will use the trigonometric ratio to find the height of the flag staff. We will first substitute the given values in the formula of the tangent of an angle \[\theta\]. Then we will simplify the equation to find the required answer. A trigonometric ratio is defined as the ratio of one side to the other side in a right-angled triangle.

Formula Used:
We will use the following formula:
1) Trigonometric ratio: \[\tan \theta = \dfrac{{{\text{Opposite side}}}}{{{\text{Adjacent side}}}}\]
2) Trigonometric angle: \[\tan 60^\circ = \sqrt 3 \]

Complete step by step solution:

Let \[A,B,C\] be the tip of the flag staff, ground and base of the flag staff respectively.
We are given that the angle of elevation of the tip of a flag staff is \[60^\circ \] and the distance between the base of the flag staff and the ground is 10 m.
We know that the side opposite to the right angle is called the hypotenuse, the side opposite to the given angle is called the opposite side, the side which is next to the given angle and is not hypotenuse is called the adjacent side.
From the diagram, we know that \[\theta = 60^\circ ,BC = 10{\rm{m}}\] and \[AC = h\].
So, the trigonometric ratio defining a relation between opposite side and adjacent side is the tangent.
In a right angled triangle, \[\Delta ABC\]
By using trigonometric ratio, we get
\[\tan \theta = \dfrac{{{\text{Opposite side}}}}{{{\text{Adjacent side}}}}\]
\[ \Rightarrow \tan \theta = \dfrac{{AC}}{{BC}}\]
Substituting \[\theta = 60^\circ ,BC = 10{\rm{m}}\] and \[AC = h\] in the above equation, we get
\[ \Rightarrow \tan 60^\circ = \dfrac{h}{{10}}\]
We know that the trigonometric value of \[\tan 60^\circ = \sqrt 3 \]. So, Substituting \[\tan 60^\circ = \sqrt 3 \] in the above equation, we get
\[ \Rightarrow \sqrt 3 = \dfrac{h}{{10}}\]
On cross multiplication, we get
\[ \Rightarrow h = 10 \times \sqrt 3 \]
\[ \Rightarrow h = 10 \times 1.732\]
By multiplying the terms, we get
\[ \Rightarrow h = 17.32{\rm{m}}\]
\[ \Rightarrow h \approx 17{\rm{m}}\]

Therefore, the height of the flag staff is 17m and thus Option(C) is the correct answer.

Note:
We know that the tangent is defined as the ratio of the opposite side to the adjacent side. Angle of elevation is the angle from the horizontal to the object at the top. We should convert it to the nearest meter by rounding off the digits. To round off the digits we need to consider following points:
If the number to be left is less than 5, then it can be left as it is.
If the number to be left is greater than 5, then 1 should be added to the preceding number.