Height and Distance

From the top \(\mathrm{A}\) of a vertical wall \(\mathrm{AB}\) of height \(30 \mathrm{~m}\), the angles of depression of the top \(\mathrm{P}\) and bottom \(\mathrm{Q}\) of a vertical tower \(\mathrm{PQ}\) are \(15^{\circ}\) and $$60^{\circ...

The angle of elevation of the top \(\mathrm{P}\) of a tower from the feet of one person standing due South of the tower is \(45^{\circ}\) and from the feet of another person standing due west of the tower is \(30^{\circ}\). If the height of...

From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60\(^\circ\). The pole subtends an angle 30\(^\circ\) at the top of the tower. Then the height of the tower is :

The angle of elevation of the top of a tower from a point A due north of it is \(\alpha\) and from a point B at a distance of 9 units due west of A is \(\cos ^{-1}\left(\frac{3}{\sqrt{13}}\right)\). If the distance of the point B from the t...

Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let \({\pi \over 8}\) and \(\theta\) be the angles of elevation from C to P and A, respectively. If ...

A horizontal park is in the shape of a triangle \(\mathrm{OAB}\) with \(\mathrm{AB}=16\). A vertical lamp post \(\mathrm{OP}\) is erected at the point \(\mathrm{O}\) such that \(\angle \mathrm{PAO}=\angle \mathrm{PBO}=15^{\circ}\) and $$\an...

Let a vertical tower \(A B\) of height \(2 h\) stands on a horizontal ground. Let from a point \(P%\) on the ground a man can see upto height \(h\) of the tower with an angle of elevation \(2 \alpha\). When from \(P\), he moves a distance $...

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is \(45^{\circ}\). Let R be a point on AQ and from a point B, vertically above \(\mathrm{R}\), the angle of elevation of $$\math...

A tower PQ stands on a horizontal ground with base \(Q\) on the ground. The point \(R\) divides the tower in two parts such that \(Q R=15 \mathrm{~m}\). If from a point \(A\) on the ground the angle of elevation of \(R\) is \(60^{\circ}\) a...

A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, th...

Two poles, AB of length a metres and CD of length a + b (b \(\ne\) a) metres are erected at the same horizontal level with bases at B and D. If BD = x and tan\(\angle\)ACB = \({1 \over 2}\), then :

A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that PC = \(\sqrt 5\) inches and \(\angle\)PCB = tan-1(2). The acute angle through which the pencil must be rotated about C s...

A spherical gas balloon of radius 16 meter subtends an angle 60\(^\circ\) at the eye of the observer A while the angle of elevation of its center from the eye of A is 75\(^\circ\). Then the height (in meter) of the top most point of the bal...

A man is observing, from the top of a tower, a boat speeding towards the lower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is 30\(^\circ\) (Ignore man's height). After sailin...

Two vertical poles are 150 m apart and the height of one is three times that of the other. If
from the middle point of the line joining their feet, an observer finds the angles of elevation of
their tops to be complementary, then the height...

The angle of elevation of a jet plane from a point A on the ground is 60\(^\circ\). After a flight of 20 seconds at the speed of 432 km/hour, the angle of elevation changes to 30\(^\circ\). If the jet plane is flying at a constant height, t...

Let in a right angled triangle, the smallest angle be \(\theta\). If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then sin\(\theta\) is equal to :

A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be \({\pi \over 3}\). If the radius of the circumcircle of \(\Delta\)ABC is 2, then the height of the pol...

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the
foot of the hill is found to be 45o. After walking a distance of 80 meters towards the top, up a slope
inclined at an angle of 30o to the h...

Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If
AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point
A such that MD2 + MC2 is minimum is ______.

The angle of elevation of the summit of a
mountain from a point on the ground is 45°.
After climbing up one km towards the summit
at an inclination of 30° from the ground, the
angle of elevation of the summit is found to be
60°. Then the he...

The angle of elevation of a cloud C from a point P, 200 m above a still lake is 30°. If the angle of depression of the image of C in the lake from the point P is 60°,then PC (in m) is equal to :

Two poles standing on a horizontal ground are of
heights 5m and 10 m respectively. The line joining
their tops makes an angle of 15º with ground. Then
the distance (in m) between the poles, is :-

Two vertical poles of heights, 20 m and 80 m stand
a part on a horizontal plane. The height (in meters)
of the point of intersection of the lines joining the
top of each pole to the foot of the other, from this
horizontal plane is :

If the angle of elevation of a cloud from a point P which is 25 m above a lake be 30o and the angle of depression of reflection of the cloud in the lake from P be 60o, then the height of the cloud (in meters) from
the surface of the lake is...

The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be 45o from
a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the
top of the tower ...

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

ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid-point of BC. If
the angles of elevation of the top of the tower at A and B are cot–1
(3\(\sqrt 2\) ) and cosec–1
(2\(\sqrt 2\) ) respectively,
th...

A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min. for the angle of depression of the car to change from 30o to 45o ; then after this, the time taken (in...

An aeroplane flying at a constant speed, parallel to the horizontal ground, \(\sqrt 3\) kmabove it, is obsered at an elevation of \({60^o}\) from a point on the ground. If, after five seconds, its elevation from the same point, is $${30^o...

A tower T1 of height 60 m is located exactly opposite to a tower T2 of height 80 m on a straight road. Fromthe top of T1, if the angle of depression of the foot of T2 is twice the angle of elevation of the top of T2, then the width (in m) o...

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles
of elevation of the top of the tower at P, Q and R are respectively 45\(^\circ\), 30\(^\circ\) and 30\(^\circ\), then the height of ...

Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point
on the ground such that AP = 2AB. If \(\angle\)BPC = \(\beta\), then tan\(\beta\) is equal to:

The angle of elevation of the top of a vertical tower from a point A, due east of it is 45o. The angle of elevation of the top of the same tower from a point B, due south of A is 30o. If the distance between A and B is \(54\sqrt 2 \,m,\) th...

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30o. After walking for 10 minutes from A in the same di...

If the angles of elevation of the top of a tower from three collinear points \(A, B\) and \(C,\) on a line leading to the foot of the tower, are \({30^ \circ }\), \({45^ \circ }\) and \({60^ \circ }\) respectively, then the ratio, $$AB:BC,$...

A bird is sitting on the top of a vertical pole \(20\) m high and its elevation from a point \(O\) on the ground is \({45^ \circ }\). It files off horizontally straight away from the point \(O\). After one second, the elevation of the bird ...

\(ABCD\) is a trapezium such that \(AB\) and \(CD\) are parallel and \(BC \bot CD.\) If \(\angle ADB = \theta ,\,BC = p\) and \(CD = q,\) then AB is equal to:

\(AB\) is a vertical pole with \(B\) at the ground level and \(A\) at the top. \(A\) man finds that the angle of elevation of the point \(A\) from a certain point \(C\) on the ground is \({60^ \circ }\). He moves away from the pole along th...

A tower stands at the centre of a circular park. \(A\) and \(B\) are two points on the boundary of the park such that \(AB(=a)\) subtends an angle of \({60^ \circ }\) at the foot of the tower, and the angle of elevation of the top of the to...

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is \({60^ \circ }\) and when he retires \(40\) meters away from the tree the angle of elevation becomes $${...