The angle of elevation of the top P of a vert | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) In $	riangle PQA$,$\angle PAQ = 45^{\circ}$ and $PQ = 10$,so $QA = \frac{PQ}{	an 45^{\circ}} = 10$.In $	riangle BRA$,$RA = d \cos 30^{\circ} =

Vedclass · Accepted Answer

$(10(\sqrt{3}-1), 25)$

Vedclass · Answer

(A) In $	riangle PQA$,$\angle PAQ = 45^{\circ}$ and $PQ = 10$,so $QA = \frac{PQ}{	an 45^{\circ}} = 10$.In $	riangle BRA$,$RA = d \cos 30^{\circ} = \frac{\sqrt{3}d}{2}$ and $BR = d \sin 30^{\circ} = \frac{d}{2}$.Since $R$ is on $AQ$,$QR = QA - RA = 10 - \frac{\sqrt{3}d}{2}$.In $	riangle PRB$,the angle of elevation of $P$ from $B$ is $60^{\circ}$,so $	an 60^{\circ} = \frac{PQ - BR}{QR} = \frac{10 - d/2}{10 - \sqrt{3}d/2}$.$\sqrt{3} = \frac{20 - d}{20 - \sqrt{3}d} \implies 20\sqrt{3} - 3d = 20 - d \implies 2d = 20(\sqrt{3} - 1) \implies d = 10(\sqrt{3} - 1)$.The area of trapezium $PQRB$ is $\alpha = \frac{1}{2}(PQ + BR) \cdot QR = \frac{1}{2}(10 + d/2)(10 - \sqrt{3}d/2)$.Substituting $d = 10(\sqrt{3} - 1)$,we get $BR = 5(\sqrt{3} - 1)$ and $QR = 10 - 5\sqrt{3}(\sqrt{3} - 1) = 10 - 15 + 5\sqrt{3} = 5\sqrt{3} - 5$.$\alpha = \frac{1}{2}(10 + 5\sqrt{3} - 5)(5\sqrt{3} - 5) = \frac{1}{2}(5\sqrt{3} + 5)(5\sqrt{3} - 5) = \frac{1}{2}(75 - 25) = 25$.Thus,$(d, \alpha) = (10(\sqrt{3} - 1), 25)$.

Similar Questions

If the angle of elevation of the top of a tower at a distance of $500 \, m$ from its foot is $30^\circ$,then the height of the tower is:

From the top of a hill $h$ metres high, the angles of depression of the top and the bottom of a pillar are $\alpha$ and $\beta$ respectively. The height (in metres) of the pillar is

Consider a triangular plot $ABC$ with sides $AB = 7 \ m$,$BC = 5 \ m$,and $CA = 6 \ m$. $A$ vertical lamp-post at the midpoint $D$ of $AC$ subtends an angle $30^o$ at $B$. The height (in $m$) of the lamp-post is

$A$ tower, of $x$ metres high, has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant $y$ metres from the foot of the tower. Then, the length of the flagstaff (in metres) is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module