Answer

Verified

449.1k+ 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.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How many crores make 10 million class 7 maths CBSE

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

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