Chapter 8

On level ground the angle of elevation of the top of the tower is \(30^{\circ} .\) On moving 20 meters near then the angle of elevation is \(60^{\circ} .\) The height of the tower is (a) \(20 \sqrt{3} \mathrm{~m}\) (b) \(10 \sqrt{3} \mathrm{~m}\) (c) \(10(\sqrt{3}-1) m\) (d) None

A vertical tower subtends an angle of \(60^{\circ}\) at a point on the same level as the foot of the tower. On moving 100 \(\mathrm{m}\) further from the first point in line with the tower, it subtends an angle of \(30^{\circ}\) at the point. Find the height of the tower.

From the top of a light house 60 meters high with its base at sea level, the angle of depression is \(15^{\circ}\). The distance of the boat from the foot of the light house is (a) \(60 \times\left(\frac{\sqrt{3}-1}{\sqrt{3}+1}\right) m\) (b) \(60 \times\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) m\) (c) \(\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right) m\) (d) None

The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of \(100 \mathrm{~m}\) from its base is \(\frac{\pi}{4}\). If the angle of elevation of the top of the complete pillar at the same point is to be \(\frac{\pi}{3}\) such that the height of the pillar is increased by \(h \mathrm{~m}\), then find \(h\).

Find the height of a tower when it is found that on walking \(80 \mathrm{~m}\) towards it along a horizontal line through its base, the angular elevation of its top changes from \(30^{\circ}\) and \(60^{\circ}\).

Three poles whose feet \(A, B, C\) lie on a circle subtend angles \(\alpha, \beta, \gamma\) and respectively, at the centre of the circle. If the height of the poles are in A.P, then \(\cot (\alpha), \cot (\beta), \cot (\gamma)\) are in (a) \(\mathrm{A} . \mathrm{P}\) (b) G.P (c) H.P (d) None

A person walking along a straight road observes that at two points \(1000 \mathrm{~m}\) apart, then angle of elevation of a vertical tower in front of him are \(\frac{\pi}{6}\) and \(\frac{5 \pi}{12}\). Find the height of the tower.

A vertical pole on one side of a street subtends a right angle at a window exactly on the opposite side. If the angle of elevation of the window from the base of the pole be \(60^{\circ}\) and the width of the street be \(30 \mathrm{~m}\), find the heights of the window and top of the pole.

A ladder \(20 \mathrm{ft}\) long reaches a point \(20 \mathrm{ft}\) below the top of a flag. The angle of elevation of the top of the flag at the foot of the ladder is \(60^{\circ} .\) Then the height of the flag is (a) \(25 \mathrm{ft}\) (b) \(30 \mathrm{ft}\) (c) \(35 \mathrm{ft}\) (d) \(40 \mathrm{ft}\)

A man in a boat rowing uniformly away from a cliff \(150 \mathrm{~m}\) high takes 2 minutes to change the angle of elevation of the top of the hill from \(\frac{\pi}{3}\) to \(\frac{\pi}{4}\). Find the speed of the boat.

An object is observed from three points \(A, B, C\) lying in a horizontal straight line which passes directly underneath the object. The angular elevation at \(B\) is twice that at \(A\) and at \(C\) three times at \(A\). If \(A B=a, B C=b\), find the height of the object.

From an aeroplane vertically over a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aeroplane are observed to \(45^{\circ}\) and \(60^{\circ}\). Then the height in miles of aeroplane above the road is (a) \(\frac{\sqrt{3}}{\sqrt{3}+1}\) (b) \(\frac{\sqrt{3}}{\sqrt{3}-1}\)

A flagstaff of \(5 \mathrm{~m}\) high stands on a building of \(25 \mathrm{~m}\) high. The flagstaff and the building subtends equal angles at a point \(P, 30 \mathrm{~m}\) high above the ground. Find the distance of \(P\) from the top of flagstaff.

A man notices two objects in a striaght line due west of him. After walking a distance \(c\) due north he observes that the objects subtend an angle \(\alpha\) at his eye and after walking a further distance \(c\) due north, an angle \(\beta\). Find the distance between the objects.

\(A B\) is a vertical tower ' \(A\) ' being its foot standing on a horizontal ground. ' \(C\) ' is the mid-point of \(A B\). Portion \(C B\) subtends an angle \(\theta\) at the point \(P\) on the ground. If \(A P=2 A B\), then find \(\tan (\theta)\).

The angle of elevation of an aeroplane from a point 200 meters above a lake is \(45^{\circ}\) and the angle of depression of its replection is \(75^{\circ} .\) Find the height of the aeroplane above the surface of the lake.

At the foot of a mountain the elevation of its peak is found to be \(\frac{\pi}{4}\), after ascending \(10 \mathrm{~m}\) toward the mountain up a slope of \(\frac{\pi}{6}\) inclination, the elevation is found to be \(\frac{\pi}{3}\). Find the height of the mountain.

From the bottom of a pole of height \(h\), the angle of elevation of the top of a tower is \(\alpha .\) The pole subtends an angle \(\beta\) at the top of the tower. Find the height of the tower.

A man finds that at a point due south of a vertical tower the angle of elevation of the tower is \(\frac{\pi}{3}\). He then walks due west \(10 \sqrt{6} \mathrm{~m}\) on the horizontal plane and find the angle of elevation of the tower to be \(\frac{\pi}{6}\). Find the original distance of the man from the tower.

The angle of elevation of the top of vertical tower from a point \(A\) on the horizontal ground is found tobe \(\frac{\pi}{4}\). From ' \(A\) ' a man walks \(10 \mathrm{~m}\) up a path sloping at an angle \(\pi / 6\). After this the slope becomes steeper and after walking up another \(10 \mathrm{~m}\), the man reaches the top of the tower. Find the distance of ' \(A\) ' from the foot of the tower.

A train is moving at a constant speed at an angle \(\theta\) East of North. Observations of the train are made from a fixed point. It is due north at some instant. Ten minutes earlier its bearing was \(\alpha\) West of North, where as 10 minutes afterwards its bearing is \(\beta\) East of North. Find \(\tan (\theta)\).

A vertical tower erected at the focus of the parabola \(y^{2}=40 x\) subtends an angle \(\frac{\pi}{3}\) at the vertex of the parabola. If the tower subtends an angle \(\frac{\pi}{6}\) at a point \(P\) lying on the parabola. Then find the possible co-ordinates of point \(P\).(c) \(\frac{\sqrt{3}+1}{\sqrt{3}-1}\) (d) \(\frac{\sqrt{3}-1}{\sqrt{3}+1}\)