126 Higher Engineering Mathematics
Floodlit area = area of sector
=
1
2
r
2
θ =
1
2
(55)
2
45 ×
π
180
= 1188 m
2
Problem 14. An automatic garden spray produces
a spray to a distance of 1.8 m and revolves through
an angle α which may be varied. If the desired
spray catchment area is to be 2.5 m 2 , to what should
angle α be 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
1.8 2 = 1.5432 rad
1.5432 rad =
1.5432 ×
180
π
◦
= 88.42 ◦
Hence angle α = 88 ◦ , correct to the nearest degree.
Problem 15. The angle of a tapered groove is
checked using a 20 mm diameter roller as shown in
Fig. 13.8. If the roller lies 2.12 mm below the top of
the groove, determine the value of angle θ.
30 mm
20 mm
2.12 mm
Figure 13.8
In Fig. 13.9, triangle ABC is right-angled at C (see
Section 13.2 (vii)).
30 mm
1 0 m m
C
2
A
B
2.12 mm
Figure 13.9
Length BC = 10 mm (i.e. the radius of the circle), and
AB = 30 −10 −2.12 = 17.88 mm from Fig. 13.9.
Hence, sin
θ
2
=
10
17.88
and
θ
2
= sin −1
10
17.88
= 34 ◦
and angle θ = 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 6.0 cm,
correct to the nearest square centimetre.
[113 cm 2 ]
2. The diameter of a circle is 55.0 mm. Determine
its area, correct to the nearest square
millimetre.
[2376 mm 2 ]
3. The perimeter of a circle is 150 mm. Find its
area, correct to the nearest square millimetre.
[1790 mm 2 ]
4. Find the area of the sector, correct to the
nearest square millimetre, of a circle having
a radius of 35 mm, with angle subtended at
centre of 75 ◦ .
[802 mm 2 ]
5. An annulus has an outside diameter of
49.0 mm and an inside diameter of 15.0 mm.
Find its area correct to 4 significant figures.
[1709 mm 2 ]
6. Find the area, correct to the nearest square
metre, of a 2 m wide path surrounding a
circular plot of land 200 m in diameter.
[1269 m 2 ]
7. A rectangular park measures 50 m by 40 m. A
3 m flower bed is made round the two longer
sides and one short side. A circular fish pond
of diameter 8.0 m is constructed in the centre
of the park. It is planned to grass the remaining
area. Find, correct to the nearest square metre,
the area of grass.
[1548 m 2 ]
8. Find the length of an arc of a circle of radius
8.32 cm when the angle subtended at the centre
is 2.14 rad. Calculate also the area of the minor
sector formed.
[17.80 cm, 74.07 cm 2 ]
9. If the angle subtended at the centre of a circle
of diameter 82 mm is 1.46 rad, find the lengths
of the (a) minor arc (b) major arc.
[(a) 59.86 mm (b) 197.8 mm]
10. A pendulum of length 1.5 m swings through
an angle of 10 ◦ in a single swing. Find, in
centimetres, the length of the arc traced by the
pendulum bob.
[26.2 cm]
Floodlit area = area of sector
=
1
2
r
2
θ =
1
2
(55)
2
45 ×
π
180
= 1188 m
2
Problem 14. An automatic garden spray produces
a spray to a distance of 1.8 m and revolves through
an angle α which may be varied. If the desired
spray catchment area is to be 2.5 m 2 , to what should
angle α be 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
1.8 2 = 1.5432 rad
1.5432 rad =
1.5432 ×
180
π
◦
= 88.42 ◦
Hence angle α = 88 ◦ , correct to the nearest degree.
Problem 15. The angle of a tapered groove is
checked using a 20 mm diameter roller as shown in
Fig. 13.8. If the roller lies 2.12 mm below the top of
the groove, determine the value of angle θ.
30 mm
20 mm
2.12 mm
Figure 13.8
In Fig. 13.9, triangle ABC is right-angled at C (see
Section 13.2 (vii)).
30 mm
1 0 m m
C
2
A
B
2.12 mm
Figure 13.9
Length BC = 10 mm (i.e. the radius of the circle), and
AB = 30 −10 −2.12 = 17.88 mm from Fig. 13.9.
Hence, sin
θ
2
=
10
17.88
and
θ
2
= sin −1
10
17.88
= 34 ◦
and angle θ = 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 6.0 cm,
correct to the nearest square centimetre.
[113 cm 2 ]
2. The diameter of a circle is 55.0 mm. Determine
its area, correct to the nearest square
millimetre.
[2376 mm 2 ]
3. The perimeter of a circle is 150 mm. Find its
area, correct to the nearest square millimetre.
[1790 mm 2 ]
4. Find the area of the sector, correct to the
nearest square millimetre, of a circle having
a radius of 35 mm, with angle subtended at
centre of 75 ◦ .
[802 mm 2 ]
5. An annulus has an outside diameter of
49.0 mm and an inside diameter of 15.0 mm.
Find its area correct to 4 significant figures.
[1709 mm 2 ]
6. Find the area, correct to the nearest square
metre, of a 2 m wide path surrounding a
circular plot of land 200 m in diameter.
[1269 m 2 ]
7. A rectangular park measures 50 m by 40 m. A
3 m flower bed is made round the two longer
sides and one short side. A circular fish pond
of diameter 8.0 m is constructed in the centre
of the park. It is planned to grass the remaining
area. Find, correct to the nearest square metre,
the area of grass.
[1548 m 2 ]
8. Find the length of an arc of a circle of radius
8.32 cm when the angle subtended at the centre
is 2.14 rad. Calculate also the area of the minor
sector formed.
[17.80 cm, 74.07 cm 2 ]
9. If the angle subtended at the centre of a circle
of diameter 82 mm is 1.46 rad, find the lengths
of the (a) minor arc (b) major arc.
[(a) 59.86 mm (b) 197.8 mm]
10. A pendulum of length 1.5 m swings through
an angle of 10 ◦ in a single swing. Find, in
centimetres, the length of the arc traced by the
pendulum bob.
[26.2 cm]