Question

there is a small island in the middle of a 100m wide river and a tall tree stands on the island. p and q are the points directly opp to each other on two banks and in line with the tree. if the angle of elevation of the top of the tree from p and q are 3o and 45 respectively find the height of the tree

Answer 1

TO FIND:

• Height of the tree.

SOLUTION:

Let h be the height of the tree.

In triangle POA

$\sf Tan \: 30 \degree = \frac{AO}{PO}$

$: \implies \: \: \sf \frac{1}{ \sqrt{3} } = \frac{h}{PO}$

$\sf : \implies \: \: PO=h√3......(1)$

$\sf In \: triangle \: AOQ$

$\sf Tan\:45°= \frac{AO }{ OQ}$

$: \implies \: \: \sf1 = \frac{h }{ OQ}$

$: \implies \: \: \sf OQ=h......(2)$

$\sf Add \: (1) \: and \: (2) \: equations.$

$: \implies \: \: \sf PO+OQ=h√3+h$

$: \implies \: \: \sf PQ=h(√3+1)$

$: \implies \: \: \sf 100cm=h(√3+1)$

$: \implies \: \: \sf \frac{100}{ \sqrt{3} + 1 } = h$

$: \implies \: \: \sf \frac{100}{ \sqrt{3} + 1} \times \frac{ \sqrt{3} - 1 }{ \sqrt{3} - 1 } = h$

$\sf : \implies \: \: \frac{100 \sqrt{3} - 100}{ { \sqrt{3} }^{2} - {1}^{2} } = h$

$: \implies \: \: \sf \frac{100( \sqrt{3} - 1)}{3 - 1} = h$

$\sf : \implies \: \: \frac{ \cancel{100}( \sqrt{3} - 1) }{ \cancel2} = h$

$: \implies \: \: \sf50( \sqrt{3} - 1) = h$

So height of the tree is 50(√3-1) metres.

Answer 2

Answer:

there is a small island in the middle of a 100m wide river and a tall tree stands on the island. p and q are the points directly opp to each other on two banks and in line with the tree. if the angle of elevation of the top of the tree from p and q are 3o and 45 respectively find the height of the tree