# A tree 12 m high, is broken by the wind in such a way that its top touches the ground and makes an angle ${{60}^{\circ }}$ with the ground. At what height from the bottom the tree is broken by the wind?

(a) 5.569 m

(b) 1.732 m

(c) 5.916 m

(d) 2.456 m

Answer

Verified

361.8k+ views

Hint: Assume a variable x which represents the height from the bottom of the tree at which the tree is broken. Since the total height of the tree is 12 m, the length of the remaining part is 12 – x. We have a trigonometric function sin which is the ratio of length of perpendicular and the hypotenuse of a right triangle. Use this sin function to solve this question.

Before proceeding with the question, we must know all the formulas that will be required to solve this question.

In a right angle triangle ABC, for the angle x, we have,

$\sin x=\dfrac{AB}{AC}$ . . . . . . . . . . . . . . . . (1)

In the question, we are given a 12 m high tree that is broken by the wind in such a way that its top touches the ground and makes an angle ${{60}^{\circ }}$ with the ground. We are required to find the height from the bottom at which the tree is broken by the wind.

Let us assume B is the bottom of the tree and the tree is broken at the point A. Also, let us assume that C is the top of the tree that is touching the ground. Since the top is making an angle ${{60}^{\circ }}$ with the ground, we can say that $\angle ACB={{60}^{\circ }}$.

Let us assume that the tree is broken at the height x from the bottom. So, AB = x. Since the tree was 12 m height, we can say the remaining part of the tree i.e. AC = 12 – x. Using formula (1), we can say,

$\sin {{60}^{\circ }}=\dfrac{AB}{AC}$

From trigonometry, we know that $\sin {{60}^{\circ }}=\dfrac{\sqrt{3}}{2}$ . Also, AB = x and AC = 12 – x. Substituting these in the above equation, we get,

$\begin{align}

& \dfrac{\sqrt{3}}{2}=\dfrac{x}{12-x} \\

& \Rightarrow \sqrt{3}\left( 12-x \right)=2x \\

& \Rightarrow 12\sqrt{3}-\sqrt{3}x=2x \\

& \Rightarrow x\left( 2+\sqrt{3} \right)=12\sqrt{3} \\

& \Rightarrow x=\dfrac{12\sqrt{3}}{2+\sqrt{3}} \\

\end{align}$

Since $\sqrt{3}=1.732$, we get,

$\begin{align}

& x=\dfrac{12\left( 1.732 \right)}{2+1.732} \\

& \Rightarrow x=5.569 \\

\end{align}$

Hence, the answer is option (a).

Note: In the question, there are two options that are close to the answer obtained. There is a possibility that one may select the wrong one in a hurry to solve the question. So, one must be careful while marking the option.

__Complete step-by-step answer:__Before proceeding with the question, we must know all the formulas that will be required to solve this question.

In a right angle triangle ABC, for the angle x, we have,

$\sin x=\dfrac{AB}{AC}$ . . . . . . . . . . . . . . . . (1)

In the question, we are given a 12 m high tree that is broken by the wind in such a way that its top touches the ground and makes an angle ${{60}^{\circ }}$ with the ground. We are required to find the height from the bottom at which the tree is broken by the wind.

Let us assume B is the bottom of the tree and the tree is broken at the point A. Also, let us assume that C is the top of the tree that is touching the ground. Since the top is making an angle ${{60}^{\circ }}$ with the ground, we can say that $\angle ACB={{60}^{\circ }}$.

Let us assume that the tree is broken at the height x from the bottom. So, AB = x. Since the tree was 12 m height, we can say the remaining part of the tree i.e. AC = 12 – x. Using formula (1), we can say,

$\sin {{60}^{\circ }}=\dfrac{AB}{AC}$

From trigonometry, we know that $\sin {{60}^{\circ }}=\dfrac{\sqrt{3}}{2}$ . Also, AB = x and AC = 12 – x. Substituting these in the above equation, we get,

$\begin{align}

& \dfrac{\sqrt{3}}{2}=\dfrac{x}{12-x} \\

& \Rightarrow \sqrt{3}\left( 12-x \right)=2x \\

& \Rightarrow 12\sqrt{3}-\sqrt{3}x=2x \\

& \Rightarrow x\left( 2+\sqrt{3} \right)=12\sqrt{3} \\

& \Rightarrow x=\dfrac{12\sqrt{3}}{2+\sqrt{3}} \\

\end{align}$

Since $\sqrt{3}=1.732$, we get,

$\begin{align}

& x=\dfrac{12\left( 1.732 \right)}{2+1.732} \\

& \Rightarrow x=5.569 \\

\end{align}$

Hence, the answer is option (a).

Note: In the question, there are two options that are close to the answer obtained. There is a possibility that one may select the wrong one in a hurry to solve the question. So, one must be careful while marking the option.

Last updated date: 26th Sep 2023

•

Total views: 361.8k

•

Views today: 4.61k

Recently Updated Pages

What do you mean by public facilities

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

10 Slogans on Save the Tiger

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

What is the IUPAC name of CH3CH CH COOH A 2Butenoic class 11 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

The dimensions of potential gradient are A MLT 3A 1 class 11 physics CBSE

Define electric potential and write down its dimen class 9 physics CBSE

Why is the electric field perpendicular to the equipotential class 12 physics CBSE