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