Basic Engineering Mathematics, Fifth Edition

192 Basic Engineering Mathematics

From equation (2), height of building,

h= 1. 0724 x= 1. 0724 ( 56. 74 )=60.85m.

Problem 25. The angle of depression of a ship

viewed at a particular instant from the top of a 75m

vertical cliff is 30◦. Find the distance of the ship

from the base of the cliff at this instant. The ship is

sailing away from the cliff at constant speed and 1

minute later its angle of depression from the top of

the cliff is 20◦. Determine the speed of the ship in

km/h

Figure 21.31 shows the cliffAB, the initial position of

the ship atCand the final positionatD. Since the angle

of depression is initially 30◦,∠AC B= 30 ◦(alternate

angles between parallel lines).

x

75m

308

208

308

208

A

B C D

Figure 21.31

tan30◦=

AB

BC

=

75

BC

hence,

BC=

75

tan30◦

= 129 .9m=initial position

of ship from base of cliff

In triangleABD,

tan20◦=

AB

BD

=

75

BC+CD

=

75

129. 9 +x

Hence, 129. 9 +x=

75

tan20◦

= 206 .06m

from which x= 206. 06 − 129. 9 = 76 .16m

Thus, the ship sails 76.16m in 1 minute; i.e., 60s,

Hence,speed of ship=

distance

time

=

76. 16

60

m/s

=

76. 16 × 60 × 60

60 × 1000

km/h=4.57km/h.

Now try the following Practice Exercise

PracticeExercise 86 Angles of elevation

and depression (answers on page 349)

1. A vertical tower stands on level ground. At
   a point 105m from the foot of the tower the
   angle of elevation of the top is 19◦.Findthe
   height of the tower.


2. If the angle of elevation of the top of a vertical
   30m high aerial is 32◦,howfarisittothe
   aerial?


3. From the top of a vertical cliff 90.0m high
   the angle of depression of a boat is 19◦ 50 ′.
   Determine the distance of the boat from the
   cliff.


4. From the top of a vertical cliff 80.0m high the
   angles of depression of two buoys lying due
   west of the cliff are 23◦and 15◦, respectively.
   How far apart are the buoys?


5. From a point on horizontal ground a surveyor
   measures the angle of elevation of the top of
   a flagpole as 18◦ 40 ′. He moves 50m nearer
   to the flagpole and measures the angle of ele-
   vation as 26◦ 22 ′. Determine the height of the
   flagpole.


6. A flagpole stands on the edge of the top of a
   building. At a point 200m from the building
   the angles of elevation of the top and bot-
   tom of the pole are 32◦and 30◦respectively.
   Calculate the height of the flagpole.


7. From a ship at sea, the angles of elevation of
   the top and bottom of a vertical lighthouse
   standing on the edge of a vertical cliff are
   31 ◦and 26◦, respectively. If the lighthouse is
   25.0m high, calculate the height of the cliff.


8. From a window 4.2m above horizontal ground
   the angle of depression of the foot of a building
   acrosstheroadis24◦andtheangleofelevation
   of the top of the building is 34◦. Determine,
   correct to the nearest centimetre, the width of
   the road and the height of the building.


9. The elevation of a tower from two points, one
   due west of the tower and the other due east
   of it are 20◦and 24◦, respectively, and the two
   points of observation are 300m apart. Find the
   height of the tower to the nearest metre.