Question 1: The height of a tree is times the length of its shadow. Find the angle of elevation of the sun.

Answer:

Let the length of the shadow

Therefore the height of the tree

Therefore

Question 2: The angle of elevation of the top of a tower, from a point on the ground and at a distance of from its foot, is found to be .Find the height of the tower.

Answer:

Let the height of the tower

Therefore

Question 3: A ladder is placed along a wall such that its upper end is resting against a vertical wall. The foot of the ladder is from the wall and the ladder is making an angle of with the ground. Find the height, up to which the ladder reaches.

Answer:

Let the height to which the ladder reaches

Distance from the base of the wall

Therefore

Question 4: Two persons are standing on the opposite sides of a tower. They observe the angles of elevation of the top of the tower to be and respectively. Find the distance between them, if the height of the tower is .

Answer:

Let the distance of the first person from the tower

Let the distance of the second person from the tower

Therefore

Similarly

Therefore the distance between the two persons

Question 5: A kite is attached to a sling. Find the length of the string, when the height of the kite is and the string make an angle with the ground.

Answer:

Height of the kite

Let the length of the string

Therefore

Question 6: A boy, tall, is away from a tower and observes the angle of elevation of the top of the tower to be (i) (ii) . Find the height of the tower in each case.

Answer:

Case 1: Angle of elevation

Distance of the boy from the tower

Let the height of the tower

Now

Hence the height of the tower

Case 2: Angle of elevation

Distance of the boy from the tower

Let the height of the tower

Now

Hence the height of the tower

Question 7: The upper part of a tree, broken over by the wind, makes an angle of with the ground; and the distance from the root to the point where the top of the tree touches the ground, is . What was the height of the tree before it was broken?

Answer:

Let the broken part of the tree is and the part of the tree still standing upright be

Therefore

Similarly,

Hence the height of the tree before it was broken

Question 8: The angle of elevation of the top of an unfinished tower at a point distance from its base is . How much higher must the tower be raised so that its angle of elevation at the same point may be ?

Answer:

Let the height of the unfinished structure

Therefore

Let the tower be raised

Therefore

Question 9: At a particular time when sun’s altitude is , the length of the shadow of altitude is a vertical tower is . Calculate:

(i) the height of the tower,

(ii) the length of the shadow of the tower, when the sun’s altitude is same (b) .

Answer:

Let the height of the tower

(i) Therefore

(ii) Let the shadow be when sun’s altitude is same and when sun’s altitude is .

Therefore

Similarly,

Question 10: The vertical poles are on either side of a road. A long ladder is placed between the two poles. When the ladder rests against one pole, it makes angle with the pole and when it is turned to rest against another pole, it makes angle with the road. Calculate the width of the road.

Answer:

Let the distance of the foot of the ladder from the towers be and

For first tower, the angle of elevation

Therefore

For first tower, the angle of elevation

Therefore

Therefore the width of the road

Question 11: Two climbers are at points and on a vertical cliff face. To an observer , from the foot of the cliff, on the level ground, is at an elevation of and of . What is the distance between the climbers?

Answer:

Let the climber be at height and climber be at height

Therefore

Similarly

Therefore the distance between the two climbers

Question 12: A man stands away from a flag-pole. He observes that angle of elevation of the top of the pole is and the angle of depression of the bottom of the pole is . Calculate the height of the pole.

Answer:

Let the height of the top of the pole

Let the height of the bottom of the pole

Therefore

The angle of elevation of the bottom of the pole = angle of elevation of the bottom of the pole

Therefore

Hence the height of the pole

Question 13: From the top of a cliff height, the angle of depression of a buoy is . Calculate, to the nearest meter, the distance of the buoy from the foot of the cliff.

Answer:

Angle of depression = angle of elevation

Let the distance of the buoy from the foot of the cliff

Therefore