# A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 5 meters. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are respectively ${30^0}$ and ${60^0}$.Find the height of the tower.

Answer

Verified

265.9k+ views

Hint- Draw the diagram of question and use value of trigonometric angles $\tan {30^0} =

\frac{1}{{\sqrt 3 }}$ and $\tan {60^0} = \sqrt 3 $ .

Let QR be the height of the tower (h meters) and RS be the height of flagstaff surmounted on

the tower (RS=5m). Let the point P lie on the horizontal plane at a distance of x meters from

the foot of the tower at point Q (PQ=x meters).

In $\vartriangle PQR$, the angle of elevation of the bottom of the flagstaff is ${30^0}$ .

$

\tan {30^0} = \dfrac{{QR}}{{PQ}} \\

\Rightarrow \dfrac{1}{{\sqrt 3 }} = \frac{h}{x} \\

\Rightarrow x = \sqrt 3 h.........\left( 1 \right) \\

$

In $\vartriangle PQS$, angle of elevation of the top of the flagstaff is ${60^0}$ .

$

\tan {60^0} = \dfrac{{QS}}{{PQ}} = \dfrac{{QR + RS}}{{PQ}} \\

\Rightarrow \sqrt 3 = \dfrac{{h + 5}}{x} \\

\Rightarrow x = \dfrac{{h + 5}}{{\sqrt 3 }}..........\left( 2 \right) \\

$

Eliminating x using (1) and (2) equation

$

\Rightarrow \sqrt 3 h = \dfrac{{h + 5}}{{\sqrt 3 }} \\

\Rightarrow 3h = h + 5 \\

\Rightarrow 2h = 5 \\

\Rightarrow h = 2.5m \\

$

So, the height of tower is 2.5 meters

Note- Whenever we face such types of problems we use some important points. Like draw

the figure of question with notify all points and distances then make the relation between

variables with the help of trigonometric angles then after eliminating x we can get the value

of the height of the tower.

\frac{1}{{\sqrt 3 }}$ and $\tan {60^0} = \sqrt 3 $ .

Let QR be the height of the tower (h meters) and RS be the height of flagstaff surmounted on

the tower (RS=5m). Let the point P lie on the horizontal plane at a distance of x meters from

the foot of the tower at point Q (PQ=x meters).

In $\vartriangle PQR$, the angle of elevation of the bottom of the flagstaff is ${30^0}$ .

$

\tan {30^0} = \dfrac{{QR}}{{PQ}} \\

\Rightarrow \dfrac{1}{{\sqrt 3 }} = \frac{h}{x} \\

\Rightarrow x = \sqrt 3 h.........\left( 1 \right) \\

$

In $\vartriangle PQS$, angle of elevation of the top of the flagstaff is ${60^0}$ .

$

\tan {60^0} = \dfrac{{QS}}{{PQ}} = \dfrac{{QR + RS}}{{PQ}} \\

\Rightarrow \sqrt 3 = \dfrac{{h + 5}}{x} \\

\Rightarrow x = \dfrac{{h + 5}}{{\sqrt 3 }}..........\left( 2 \right) \\

$

Eliminating x using (1) and (2) equation

$

\Rightarrow \sqrt 3 h = \dfrac{{h + 5}}{{\sqrt 3 }} \\

\Rightarrow 3h = h + 5 \\

\Rightarrow 2h = 5 \\

\Rightarrow h = 2.5m \\

$

So, the height of tower is 2.5 meters

Note- Whenever we face such types of problems we use some important points. Like draw

the figure of question with notify all points and distances then make the relation between

variables with the help of trigonometric angles then after eliminating x we can get the value

of the height of the tower.

Last updated date: 29th Sep 2023

â€¢

Total views: 265.9k

â€¢

Views today: 3.65k

Recently Updated Pages

What do you mean by public facilities

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

10 Slogans on Save the Tiger

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is meant by shramdaan AVoluntary contribution class 11 social science CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

An alternating current can be produced by A a transformer class 12 physics CBSE

What is the value of 01+23+45+67++1617+1819+20 class 11 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers