
The top of a broken tree has it’s top-end touching the ground at a distance 15m from the bottom, the angle made by the broken end with the ground is ${30^\circ }$. Then the length of the broken part is.
A) ${\text{10m}}$
B) $\sqrt {\text{3}} {\text{m}}$
C) ${\text{5}}\sqrt {\text{3}} {\text{m}}$
D) ${\text{10}}\sqrt {\text{3}} {\text{m}}$
Answer
485.1k+ views
Hint: We can draw a diagram with the given details. Then we can form a trigonometric ratio and solve the equation to get the required length. According to the question the, appropriate trigonometric ratio will be tan.
Complete step by step solution: We can draw a diagram with the given details
In the figure, AC is the broken part of the tree, B is the foot of the tree and c is the point where the top end touches the ground.
Consider right triangle ABC, by trigonometry,
${\text{cosC = }}\dfrac{{{\text{adjecent side}}}}{{{\text{hypotenuse}}}}{\text{ = }}\dfrac{{{\text{BC}}}}{{{\text{AC}}}}$
${\text{cos30 = }}\dfrac{{{\text{15}}}}{{{\text{AB}}}}$
We know ${\text{cos30 = }}\dfrac{{\sqrt {\text{3}} }}{{\text{2}}}$. Using this in the above equation, we get,
$\dfrac{{\sqrt {\text{3}} }}{{\text{2}}}{\text{ = }}\dfrac{{{\text{15}}}}{{{\text{AB}}}}$
\[
\Rightarrow {\text{AB = }}\dfrac{{{\text{15} \times 2}}}{{\sqrt {\text{3}} }}{\text{ = }}\dfrac{{{\text{30}}}}{{\sqrt {\text{3}} }} \\
\Rightarrow {\text{AB = 10}}\sqrt {\text{3}} {\text{ m}} \\
\]
So, length of the broken part is \[{\text{10}}\sqrt {\text{3}} {\text{ m}}\]
Therefore, the correct answer is option D.
Note: Drawing a diagram with the given details is very important. The concept of simple trigonometry is used to find the length of the broken piece. Trigonometric values of important angles must be known. We must understand which angle and sides of the right-angled triangle are given in the question.
Complete step by step solution: We can draw a diagram with the given details

In the figure, AC is the broken part of the tree, B is the foot of the tree and c is the point where the top end touches the ground.
Consider right triangle ABC, by trigonometry,
${\text{cosC = }}\dfrac{{{\text{adjecent side}}}}{{{\text{hypotenuse}}}}{\text{ = }}\dfrac{{{\text{BC}}}}{{{\text{AC}}}}$
${\text{cos30 = }}\dfrac{{{\text{15}}}}{{{\text{AB}}}}$
We know ${\text{cos30 = }}\dfrac{{\sqrt {\text{3}} }}{{\text{2}}}$. Using this in the above equation, we get,
$\dfrac{{\sqrt {\text{3}} }}{{\text{2}}}{\text{ = }}\dfrac{{{\text{15}}}}{{{\text{AB}}}}$
\[
\Rightarrow {\text{AB = }}\dfrac{{{\text{15} \times 2}}}{{\sqrt {\text{3}} }}{\text{ = }}\dfrac{{{\text{30}}}}{{\sqrt {\text{3}} }} \\
\Rightarrow {\text{AB = 10}}\sqrt {\text{3}} {\text{ m}} \\
\]
So, length of the broken part is \[{\text{10}}\sqrt {\text{3}} {\text{ m}}\]
Therefore, the correct answer is option D.
Note: Drawing a diagram with the given details is very important. The concept of simple trigonometry is used to find the length of the broken piece. Trigonometric values of important angles must be known. We must understand which angle and sides of the right-angled triangle are given in the question.
Recently Updated Pages
Master Class 12 Chemistry: Engaging Questions & Answers for Success

Class 12 Question and Answer - Your Ultimate Solutions Guide

Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

Master Class 12 Biology: Engaging Questions & Answers for Success

Master Class 12 Physics: Engaging Questions & Answers for Success

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What constitutes the central nervous system How are class 10 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Explain the Treaty of Vienna of 1815 class 10 social science CBSE
