Question

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Answer 1

Question:-

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of with the ground. The distance between the foot of the tree to the point where the top touches the ground is m. Find the height of the tree.

Hint:-

In this question, the broken part of the tree forms a right-angled triangle with the ground and remaining part of the tree. So, we should have knowledge of trigonometric ratios to find the height of a tree.

$\huge\sf{\underline {\underline{Solution:-}}}$

Let us assume that the tree is positioned at A and after the tree breaks down from point C it touches the ground at point B, which forms a right-angled triangle.

Now, to find the height of the tree, we need to find the length of the side’s AC and BC, so that if we add them up, we will get the height of the tree.

Now, we are given that the top of the tree touches the ground and makes an angle of , with the ground.

⇒∠$360°$

So here inΔABC , we will apply trigonometric ratios to find the length of side AC and side BC.

We know that ,

$Tan\theta$  is always equal to the ratio of perpendicular to the base, mathematically it can be written as $tan\theta=\frac{perpendicular}{base}$

⇒$tan\theta$

We know that,∠ABC=$\theta$=30°and

tan30°=$\frac{1}{\sqrt{3} }$ therefore, tan$\theta$=$\frac{AC}{AB}$

⇒$tan30°=\frac{AC}{AB}$

We have been given that the top of the tree is at a distance of 8 m from the foot of tree, so, we get,

⇒$\frac{1}{\sqrt{3}}=\frac{AC}{8}$

⇒$\frac{8}{\sqrt{3}}$

In the same manner, we will find the length of BC by using cosine function, we will get,

$cos\theta= \frac{base}{hypotenuse}$

⇒$cos\theta=\frac{AB}{BC}$

⇒$cos30°=\frac{8}{BC}$

⇒$\frac{\sqrt{3}}{2}=\frac{8}{BC}$

⇒$BC=\frac{16}{\sqrt{3}}$

Now, we know that the height of the tree is the sum of lengths AC and BC, which we can write as

Height of tree=AC+BC

Now, we will put the values of AC and BC from equation (i) and (ii) to equation (iii), so, we get,

$\sf height\:of\:tree= \frac{8}{\sqrt{3}}+\frac{16}{\sqrt{3}}$

⇒height of tree=$\frac{24}{\sqrt{3}}$

⇒height of tree = $\frac{8}{\sqrt{3}}$

Therefore the height of tree is $\frac{8}{\sqrt{3}}$m.

Answer 2

Answer:

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