Higher Engineering Mathematics, Sixth Edition

126 Higher Engineering Mathematics

Floodlit area=area of sector

=

1

2

r^2 θ=

1

2

( 55 )^2

(

45 ×

π

180

)

=1188m^2

Problem 14. An automatic garden spray produces

a 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 , to what should

angleαbe set, correct to the nearest degree.

Area of sector=^12 r^2 θ, hence 2. 5 =^12 ( 1. 8 )^2 α

from which,α=

2. 5 × 2

1. 82

= 1 .5432rad

1 .5432rad=

(

1. 5432 ×

180

π

◦)

= 88. 42 ◦

Henceangleα= 88 ◦, correct to the nearest degree.

Problem 15. The angle of a tapered groove is

checked using a 20mm diameter roller as shown in

Fig. 13.8. If the roller lies 2.12mm below the top of

the groove, determine the value of angleθ.

30 mm



20 mm

2.12 mm

Figure 13.8

In Fig. 13.9, triangleABCis right-angled atC(see

Section 13.2 (vii)).

30 mm

10 mm

C

2

A

B

2.12 mm



Figure 13.9

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

AB= 30 − 10 − 2. 12 = 17 .88mm from Fig. 13.9.

Hence, sin

θ

2

=

10

17. 88

and

θ

2

=sin−^1

(

10

17. 88

)

= 34 ◦

andangleθ= 68 ◦

Now try the following exercise

Exercise 57 Further problems on arc

length and area of circles and sectors

1. Calculate the area of a circle of radius 6.0cm,
   correct to the nearest square centimetre.
   [113cm^2 ]


2. Thediameterofacircleis55.0mm.Determine
   its area, correct to the nearest square
   millimetre. [2376mm^2 ]


3. The perimeter of a circle is 150mm. Find its
   area, correct to the nearest square millimetre.
   [1790mm^2 ]


4. Find the area of the sector, correct to the
   nearest square millimetre, of a circle having
   a radius of 35mm, with angle subtended at
   centre of 75◦. [802mm^2 ]


5. An annulus has an outside diameter of
   49.0mm and an inside diameter of 15.0mm.
   Find its area correct to 4 significant figures.
   [1709mm^2 ]


6. Find the area, correct to the nearest square
   metre, of a 2m wide path surrounding a
   circular plot of land 200m in diameter.
   [1269m^2 ]


7. A rectangular park measures 50m by 40m. A
   3m flower bed is made round the two longer
   sides and one short side. A circular fish pond
   of diameter 8.0m is constructed in the centre
   of the park. It isplanned to grass the remaining
   area. Find, correct to the nearest square metre,
   the area of grass. [1548m^2 ]


8. Find the length of an arc of a circle of radius
   8.32cm when the angle subtended at the centre
   is 2.14rad. Calculate also the area of the minor
   sector formed.
   [17.80cm, 74.07cm^2 ]


9. If the angle subtended at the centre of a circle
   of diameter 82mm is 1.46rad, find the lengths
   of the (a) minor arc (b) major arc.
   [(a) 59.86mm (b) 197.8mm]


10. A pendulum of length 1.5m swings through
    an angle of 10◦in a single swing. Find, in
    centimetres, the length of the arc traced by the
    pendulum bob. [26.2cm]