Question

solve the following question

Answer 1

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Answer 2

Question:

A tree broke due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Answer:

The height of the tree is 13.84 m.

Step-by-step-explanation:

In figure,

Seg AB represents the tree before the storm came.

After the storm, the tree gets broken at point C.

Seg AC is the broken part of tree which reaches at the ground at point D.

As the broken part AC meets the ground becoming CD,

AC = CD - - - ( 1 )

We have given that,

$\sf\:BD\:=\:8\:m\\\\\sf\:\angle\:CDB\:=\:30^{\circ}$

In right angled $\sf\:\triangle\:CBD\:\:\:-\:-\:[\:Assuming\:tree\:is\:perpendicular\:to\:ground\:]$

$\sf\:\tan\:30^{\circ}\:=\:\frac{CB}{BD}\:\:\:-\:-\:[\:By\:de fi nition\:]\\\\\implies\sf\:\frac{1}{\sqrt{3}}\:=\:\frac{CB}{8}\:\:\:-\:-\:[\:\tan\:30^{\circ}\:=\:\frac{1}{\sqrt{3}}\:]\\\\\implies\boxed{\red{\sf\:CB\:=\:\frac{8}{\sqrt{3}}}}\\\\\sf\:Also,\\\\\sf\:\sin\:30^{\circ}\:=\:\frac{CB}{CD}\:\:\:-\:-\:[\:By\ de fi nition\:]\\\\\implies\sf\:\frac{1}{2}\:=\:\dfrac{\frac{8}{\sqrt{3}}}{CD}\:\:\:-\:-\:-\:[\:\because\:\sin\:30^{\circ}\:=\:\frac{1}{2}\:]\\\\\implies\sf\:CD\:=\:\frac{8}{\sqrt{3}}\:\times\:2\\\\\implies\boxed{\red{\sf\:CD\:=\:\frac{16}{\sqrt{3}}\:m\:}}\\\\\sf\:AC\:=\:CD\:\:\:-\:-\:[\:From\:(\:1\:)\:]\\\\\implies\boxed{\red{\sf\:AC\:=\:\frac{16}{\sqrt{3}}\:m\:}}$

$\sf\:AB\:=\:AC\:+\:CB\:\:\:[\:A\:-\:C\:-\:B\:]\\\\\implies\sf\:AB\:=\:\frac{16}{\sqrt{3}}\:+\:\frac{8}{\sqrt{3}}\\\\\implies\sf\:AB\:=\:\dfrac{16\:+\:8}{\sqrt{3}}\\\\\implies\sf\:AB\:=\:\frac{24}{\sqrt{3}}\\\\\implies\sf\:AB\:=\:\dfrac{24\:\times\:\sqrt{3}}{\sqrt{3}\:\times\:\sqrt{3}}\:\:\:-\:-\:-\:[\:Rationalising\:the\:denominator\:]\\\\\implies\sf\:AB\:=\:\dfrac{24\:\sqrt{3}}{3}\\\\\implies\sf\:AB\:=\:\dfrac{\cancel{24}\:\times\:1.73}{\cancel3}\:\:\:-\:-\:-\:[\:\because\:\sqrt{3}\:\approx\:1.73\:]\\\\\implies\sf\:AB\:=\:8\:\times\:1.73\\\\\implies\boxed{\red{\sf\:AB\:=\:13.84\:m\:}}$

The height of the tree is 13.84 m.