Question

A person, standing on the bank of a river, observes that the angle subtended by a tree on the opposite bank is 60°. When he retreates 20 m from the bank, he finds the angle to be 30°. Find the height of the tree and the breadth of the river? What skill is used by the person.
Value Based Question

Answer 1

SOLUTION :-

see the abvove attachment given below :-

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and the skill used by the person is Knowledge of lengths and angles.

Answer 2

Breadth of the river = 10 meter  

Height of the tree =$\bold{10\sqrt{3}}$

Solution:

A person standing on bank sees the branch of the tree of the opposite to be 60 degree  

If the person retreats 20m back he then face the branch at 30 degree as we can see in the diagram attached.

Height of branch from ground = h

Let x be the length of river at which tree forms 60 degree.

To find: Height and breadth of the branch and river.  

$\tan\ 60^{\circ}=\frac{h}{x}$

If the person backs 20 m it becomes $\tan\ 30^{\circ}=\frac{h}{x+20}$

$\begin{array}{l}{\tan 60^{\circ}=\sqrt{3}} \\\\ {\tan 30^{\circ}=\frac{1}{\sqrt{3}}}\end{array}$

Substituting the values we get

$\begin{array}{l}{\sqrt{3}=\frac{h}{x}} \\ \\{\frac{1}{\sqrt{3}}=\frac{h}{x+20}} \\ \\{h=\sqrt{3} x} \\ \\{x+20=\sqrt{3} h}\end{array}$

Substituting the value of h in $x+20=\sqrt{3} h$

$x+20=\sqrt{3} \sqrt{3} x$

3x = x + 20

2x = 20

x = 10  

$\bold{h=10 \sqrt{3}}$