Question

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A 7mts flagstaff is fixed on the top of a tower. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 45° and 30° respectively. Find the height of the tower correct to one decimal place.

Answer 1

Hope u like my process
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=> Let the height of tower with flag = y m

=> length of flag (l) = 7 m

=> Height of tower till the bottom of flag = (y - 7)m

Thus.. Now for elevation of 30°
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$= > \blue{ \tan(30)} = \blue{ \frac{(y - 7)}{x} } \\ \\ = > \blue{\frac{1}{ \sqrt{3} } }= \blue{ \frac{y - 7}{x} } \\ \\ = > x = \underline{ \green{\sqrt{3} (y - 7)}}$

Now for the elevation of 45°
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$= > \blue{\tan(45) } = \blue{\frac{y}{x} } \\ \\ = > \blue{x} = \blue y \\ \\ = > \blue{\sqrt{3} y - 7 \sqrt{3}} = \blue{y} \\ \\ = > \blue{ y( \sqrt{3} - 1)} = \blue{ 7 \sqrt{3}} \\ \\ = > \blue{y( \sqrt{3} - 1)( \sqrt{3} + 1)} = \blue{ 7 \sqrt{3} ( \sqrt{3} + 1)} \\ \\ = > \blue{y(( { \sqrt{3}) {}^{2} -( 1) }^{2} ) }= \blue{(7 \times 3) + (7 \times 1.732)} \\ \\ = > \blue{ y }= \blue{\frac{21 + 12.124}{2} } = \blue{16 .5} \\ \\ = > \boxed{y = \underline{\orange{16.5 \: \: m}}}$
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So the required height of tower is( y-7)= (16.5 - 7)m

= 9.5 m
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Hope this is ur required answer

Proud to help you

Answer 2

Here is your answer

Let,
CD be the flagstaff their height = 7m

Bc be the tower height = h

From a point on the ground angle of elevation of top is 45° and bottom of flagstaff is 30°

In triangle ABC

$tan \: 30° = \frac{Bc}{Ab} \\ \frac{1}{ \sqrt{3} } = \frac{h}{Ab } \\ Ab = \sqrt{3}h...............(1)$

In triangle ABD

$tan \: 45° = \frac{Bd}{ab} \\ 1 = \frac{Bc + Cd}{ab} \\ \\ Ab = Bc + Cd \\ Ab = h + 7..............(2)$

On comparing equation (1)and(2)

$\sqrt{3}h = h + 7 \\ \sqrt{3}h - h = 7 \\ h( \sqrt{3} - 1) = 7 \\ h = \frac{7}{ \sqrt{3} - 1 } \times \frac{ \sqrt{3} + 1 }{ \sqrt{3} + 1 } \\ h = \frac{7( \sqrt{3} + 1) }{( \sqrt{3})^{2} - (1) {}^{2} }\\ \\h = \frac{7 \times 2.7 7 }{2} \\ h = 9.56m$
hence the height of tower is 9.56 m

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