Basic Engineering Mathematics, Fifth Edition

234 Basic Engineering Mathematics

(b) Length of major arc=(circumference−minor

arc)= 2 π( 8. 4 )− 18. 3 =34.5cm, correct to 3

significant figures.

(Alternatively, major arc=rθ

= 8. 4 ( 360 − 125 )

( π

180

)

=34.5cm.)

Problem 14. Determine the angle, in degrees and

minutes, subtended at the centre of a circle of

diameter 42mm by an arc of length 36mm.

Calculate also the area of the minor sector formed

Since length of arc,s=rθthenθ=

s

r

Radius, r=

diameter

2

=

42

2

=21mm,

hence θ=

s

r

=

36

21

= 1 .7143 radians.

1.7143rad= 1. 7143 ×

(

180

π

)◦

= 98. 22 ◦= 98 ◦ 13 ′=

angle subtended at centre of circle.

From equation (2),

area of sector=

1

2

r^2 θ=

1

2

( 21 )^2 ( 1. 7143 )

=378mm^2.

Problem 15. A football stadium floodlight can

spread its illumination over an angle of 45◦to a

distance of 55m. Determine the maximum area that

is floodlit.

Floodlit area=area of sector=

1

2

r^2 θ

=

1

2

( 55 )^2

(

45 ×

π

180

)

=1188m^2

Problem 16. An automatic garden sprayer

produces spray to a distance of 1.8m and revolves

through an angleαwhich may be varied. If the

desired spray catchment area is to be 2.5m^2 ,towhat

should angleαbe set, correct to the nearest degree?

Area of sector=

1

2

r^2 θ,hence 2. 5 =

1

2

( 1. 8 )^2 α

from which, α=

2. 5 × 2

1. 82

= 1 .5432 radians

1 .5432rad=

(

1. 5432 ×

180

π

)◦

= 88. 42 ◦

Hence,angleα= 88 ◦, correct to the nearest degree.

Problem 17. The angle of a tapered groove is

checked using a 20mm diameter roller as shown in

Figure 26.8. If the roller lies 2.12mm below the top

of the groove, determine the value of angleθ

2.12mm

20mm

30mm



Figure 26.8

In Figure 26.9, triangleABCis right-angled atC(see

property (g) in Section 26.2).

2.12mm

2

B10mm

A

C

30mm



Figure 26.9

LengthBC=10mm (i.e. the radius of the circle), and

AB= 30 − 10 − 2. 12 = 17 .88mm, from Figure 26.9.

Hence, sin

θ

2

=

10

17. 88

and

θ

2

=sin−^1

(

10

17. 88

)

= 34 ◦

andangleθ= 68 ◦.

Now try the following Practice Exercise

PracticeExercise 103 Arc length and area

of circles and sectors(answers on page 351)

1. Calculate the area of a circle of radius 6.0cm,
   correct to the nearest square centimetre.


2. The diameter of a circle is 55.0mm. Deter-
   mine its area, correct to the nearest square
   millimetre.