# The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is ${60^ \circ }$. From a point Y, 40m vertically above X, the angle of elevation of the top Q of the tower is ${45^ \circ }$. Find the height of the tower PQ and the distance PX. (Use $\sqrt 3 = 1.73$)

Last updated date: 19th Mar 2023

•

Total views: 206.1k

•

Views today: 1.84k

Answer

Verified

206.1k+ views

Hint: In this question, we are supposed to find the length of tower and distance between point P and point X. We are going to consider the $\Delta YRQ$ and find the value of YR and XP from there and similarly, find the value of PX from it.

In $\Delta YRQ$, we have,

$\tan {45^ \circ } = \dfrac{{QR}}{{YR}}$

$1 = \dfrac{x}{{YR}}$

$ \Rightarrow YR = x$

Or $XP = x\left[ {As\,YR = XP} \right]$ …..(1)

Now, In $\Delta XPQ$ , we have,

$\tan {60^ \circ } = \dfrac{{PQ}}{{PX}}$

$\sqrt 3 = \dfrac{{x + 40}}{x}$ [Using (1)]

$\sqrt 3 x = x + 40$

$XQ = 109.3m$

$x = \dfrac{{40}}{{\sqrt 3 - 1}}$

On rationalising the denominator, we get,

$x = \dfrac{{40}}{{\sqrt 3 - 1}} \times \dfrac{{\sqrt 3 + 1}}{{\sqrt 3 + 1}}$

$x = \dfrac{{40\sqrt 3 + 1}}{{3 - 1}}$

$x = \dfrac{{40\sqrt 3 + 1}}{2}$

$x = 20\left( {\sqrt 3 + 1} \right) = 54.64m$

So, height of the tower, PQ= x+40 = 54.64 + 40 = 94.64m

Now in $\Delta XPQ$, we have,

$\sin {60^ \circ } = \dfrac{{PQ}}{{XQ}}$

$\dfrac{{\sqrt 3 }}{2} = \dfrac{{94.64}}{{XQ}}$

$XQ = \dfrac{{94.64 \times 2}}{{\sqrt 3 }}$

$XQ = \dfrac{{94.64 \times 2 \times \sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }}$

Note: In these questions, make sure you take the correct trigonometric angle.

Answer = $XQ = 109.3m$

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE