Question

The top of a broken tree has its top and touches the ground at a distance 15m from the bottom, the angle made by the broken end with the ground is 30. Then the length of the broken part is ___.
A. 10 m
B. $\sqrt 3 $m
C. $5\sqrt 3 $m
D. $10\sqrt 3 $ m

Answer 1

Hint:
First draw the corresponding figure. Then, use the ratio of cosine, $\cos \theta = \dfrac{{{\text{Base}}}}{{{\text{Hypotenuse}}}}$ to form an equation involving the length of broken tree. Then, solve the equation to find the required length.

Complete step by step solution:
First of all, we draw a corresponding figure for the given question.

Let $y$ be the length of the broken tree.
The angle made by the broken part of the tree is ${30^ \circ }$ from the ground.
Consider the triangle, ABC, we will find the value of $\cos {30^ \circ }$ from triangle $ABC$, when
$\cos \theta = \dfrac{{{\text{Base}}}}{{{\text{Hypotenuse}}}}$
Then, we have,
$\cos {30^ \circ } = \dfrac{{15}}{y}$
We know that the value of $\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$
On substituting the value, we will get,
$
   \Rightarrow \dfrac{{\sqrt 3 }}{2} = \dfrac{{15}}{y} \\
   \Rightarrow y = \dfrac{{30}}{{\sqrt 3 }} \\
$
Rationalise the above expression by multiplying and dividing by $\sqrt 3 $
$
   \Rightarrow y = \dfrac{{30\sqrt 3 }}{3} \\
   \Rightarrow y = 10\sqrt 3 \\
$
Hence, the value of the broken part of the tree is $10\sqrt 3 $ metres.

Thus, option D is correct.

Note:
One must draw the diagram correctly. Use the trigonometry ratio such that the required value and a known value can be used. Here, we have used the cosine ratio as the value of base was given and we had to calculate the value of hypotenuse of the given triangle and the ratio involving base and hypotenuse is cosine.