Question 1: In the figure, given below, it is given that is perpendicular to and is of length meters. and Without using tables, find .

Answer:

From

Similarly, from

Substituting from above

Question 2: Find the height of a tree when it is found that on walking away from it ? in a horizontal line through its base, the elevation of its top changes from to .

Answer:

From

Similarly, from

Substituting from above

Question 3: Find the height of a building, when it is found that on walking towards it in a horizontal line through its base the angular elevation of its top changes from to .

Answer:

From

Similarly, from

Substituting from above

Question 4: From the top of a light house high, the angles of depression of two ships are observed as and respectively. Find the distance between the two ships (in the nearest meter) if:

(i) the ships are on the same side of the light house,

(ii) the ships are on the opposite sides of the light house. **[2010]**

Answer:

(i) From

Similarly, from

Therefore, the distance between the ships is =

(ii) If the ships were on the opposite sides, then the distance between the ships is =

Question 5: Two pillars of equal heights stand on either side of a roadway, which is wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are and ; find the height position of the pillar and the position of the point.

Answer:

From

Similarly, from

Hence

Question 6: From the figure, given below, calculate the length of .

Answer:

From

Similarly, from

Therefore

Question 7: The angle of elevation of the top of a tower is observed to be . At a Point, vertically above the first point of observation, the elevation is found to be Find: (i) the height of the tower, (ii) its horizontal distance from the points of Observation

Answer:

From

Similarly, from

Therefore

Given

Therefore

Question 8: From the top of a cliff, high, the angles of depression of the top and bottom of a tower are observed to be and . Find the height of the tower.

Answer:

From

Similarly, from

Therefore

Question 9: A man on a cliff observes a boat. at an angle of depression , which is sailing towards the shore to the point immediately beneath him. Three minutes later, the angle of depression of the boat is found to be . Assuming that the boat sails at a uniform speed, determine: (i) how much more time it will take to reach the shore. (ii) the speed of the boat in meter per second if the height of the cliff is .

Answer:

From

Similarly, from

Also

Therefore the time that the boat will take to reach the shore

Speed of the boat

Question 10: A man in a boat rowing away from a lighthouse high, takes minutes to change the angle of elevation of the top of the lighthouse from to . Find the speed of the boat.

Answer:

From

Similarly, from

Question 11: A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is . When he moves away from the bank, he finds the angle of elevation to be . Find: (i) the height of the free, correct to decimals places, (ii) the width of the river.

Answer:

From

Similarly, from

Therefore

Question 12: The horizontal distance between two towers is and the angular depression of the top of the first tower as seen from the top of the second. which is high, is . Find the height of the first tower.

Answer:

From

Question 13: The length of the shadow of a tower standing on level plane is found to be meters longer when the sun’s altitude is then when it is . Prove that the height of the tower is meters.

Answer:

From

Similarly, from

Substituting

Multiplying both numerator and denominator by we get

Question 14: After seconds, its elevation is observed to be ; find the uniform speed of the Aeroplane in km per hour.

Answer:

From

Similarly, from

Question 15: From the top of a hill, the angles of depression of two consecutive kilometer stones, due east, are found to be and respectively. Find the distances of the two stones from the foot of the hill. **[2007]**

Answer:

From

Similarly, from

Therefore