Higher Engineering Mathematics

130 GEOMETRY AND TRIGONOMETRY

Figure 12.30

6. A laboratory 9.0 m wide has a span roof which
   slopes at 36◦on one side and 44◦on the other.
   Determine the lengths of the roof slopes.
   [6.35 m, 5.37 m]

12.12 Further practical situations

involving trigonometry

Problem 28. A vertical aerial stands on hori-

zontal ground. A surveyor positioned due east

of the aerial measures the elevation of the top as

48 ◦. He moves due south 30.0 m and measures

the elevation as 44◦. Determine the height of the

aerial.

In Fig. 12.31, DC represents the aerial,Ais the initial

position of the surveyor andBhis final position.

From triangleACD, tan 48◦=

DC

AC

,

from which AC=

DC

tan 48◦

Similarly, from triangleBCD,

BC=

DC

tan 44◦

For triangleABC, using Pythagoras’ theorem:

BC^2 =AB^2 +AC^2

(

DC

tan 44◦

) 2

=(30.0)^2 +

(

DC

tan 48◦

) 2

DC^2

(

1

tan^244 ◦

−

1

tan^248 ◦

)

= 30. 02

DC^2 (1. 072323 − 0 .810727)= 30. 02

DC^2 =

30. 02

0. 261596

= 3440. 4

Figure 12.31

Hence, height of aerial,

DC=

√

3440.4=58.65 m

Problem 29. A crank mechanism of a petrol

engine is shown in Fig. 12.32. ArmOAis 10.0 cm

long and rotates clockwise about O. The con-

necting rodABis 30.0 cm long and endBis

constrained to move horizontally.

Figure 12.32

(a) For the position shown in Fig. 12.32 deter-

mine the angle between the connecting rod

ABand the horizontal and the length ofOB.

(b) How far doesBmove when angleAOB

changes from 50◦to 120◦?

(a) Applying the sine rule:

AB

sin 50◦

=

AO

sinB