© t madas. axis vertex generator radius base slant height c y l i n d e r s a n d c o n e s
TRANSCRIPT
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The Volume of a Cylinder
2V r h=
r
h
A cylinder is a P
whose cross section is a c
rism
ircle
V =Base Areax Height
= πr 2 x h
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V =πr 2 h
V =πx 102 x 40
V ≈12566 cm3[n. w. n.]
x
V =x-sectional Area heightx
Calculate the volume of these cylinders
10 cm
40 c
m
V =πr 2 h
V =πx 52x 22
V ≈1728 m3[n. w. n.]
x
V =x-sectional Area heightx
5 m
22 m
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V =πr 2 h
V =πx 122 x 35
V ≈15834 cm3[n. w. n.]
x
Calculate the volume of these cylinders
12 cm
35 c
m
V =πr 2 h
V =πx 42x 18
V ≈905 m3[n. w. n.]
x
4 m
18 m
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S = π r h
S =
S ≈3142 cm2[n. w. n.]
+
Calculate the Surface Area of these Cylinders
10 cm
40 c
m
5 m
22 m
2 π r 22
x πx 10x 40+2 x πx 1022
S ≈2513.27 628.32+
S = π r h
S =
S ≈ 848 m2[n. w. n.]
+2 π r 22
x π x 5x 22+2 x πx 522
S ≈691.15 157.08+
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S = π r h
S =
S ≈6126 cm2[n. w. n.]
+
Calculate the Surface Area of these Cylinders
15 cm
50 c
m
4 m
14 m
2 π r 22
x πx 15x 50+2 x πx 1522
S ≈4712.39 1413.72+
S = π r h
S =
S ≈ 452 m2 [n. w. n.]
+2 π r 22
x π x 4x 14+2 x πx 422
S ≈351.86 100.53+
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The Volume of a Cone
213
V r h= ´ ´h
It can be shown that for a cone:
V = Base Areax Height
r
13
´
213
V r h=
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13
x π x 42x 9
Calculate the volume of these cones
9 c
m
4 cm
8 c
m
6 cm
V = πr 2h13
=
151≈ cm3
13
x π x 62x 8
V = πr 2h13
=
302≈ cm3
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13
x π x 32x 12
Calculate the volume of these cones
12 c
m
3 cm
9 c
m
5 cm
V = πr 2h13
=
113≈ cm3
13
x π x 52x 9
V = πr 2h13
=
236≈ cm3
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h
8.8
4 c
m
A cylinder is shown below.The radius of its base is 6 cm and has a volume of 1000 cm3.Calculate its surface area to 3 significant figures.
6 cmVolume = base area x height
1000 = 6 x 6 x h
1000 ≈ 113.1 x h
h ≈ 1000 ÷ 113.1
h ≈ 8.84 cm
x π
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8.8
4 c
m
A cylinder is shown below.The radius of its base is 6 cm and has a volume of 1000 cm3.Calculate its surface area to 3 significant figures.
Surface Area:
6 x 6 ≈ 226.2 cm2x π6 cm
x 2
8.8
4 c
m
Circumference of circle
2 x 6 ≈ 333.3 cm2x π x 8.84
559.5 cm2
560 cm2 [ 3 s.f.]
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A tank without a lid is in the shape of a cylinder.The radius of its base is 30 cm and has a capacity of 225 litres.Calculate its surface area, to 2 significant figures.
h
79.6
cm
30 cm
Volume = base area x height
225000 = 30 x 30 x h
225000 ≈ 2827 x h
h ≈ 225000 ÷ 2827
h ≈ 79.6 cm
x π
1 litre = 1000 cm3
225 litres = 225000 cm3
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A tank without a lid is in the shape of a cylinder.The radius of its base is 30 cm and has a capacity of 225 litres.Calculate its surface area, to 2 significant figures.
h
79.6
cm
30 cm
79.6
cm
Circumference of circle
Surface Area:
30 x 30 ≈ 2827 cm2x π
2 x 30 ≈ 15004 cm2x π x 79.6
17831 cm2
18000 cm2 [ 2 s.f.]
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A can without a lid is in the shape of a cylinder.The radius of its base is 4 cm and has a capacity of 192π cm3.Calculate its surface area in terms of π.
h
12 c
m4 cm Volume = base area x height
192π = 4 x 4 x h
192π 16πh
h = 12 cm
x π
= 16π16π
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12 c
m
A can without a lid is in the shape of a cylinder.The radius of its base is 4 cm and has a capacity of 192π cm3.Calculate its surface area in terms of π.
4 cmSurface Area:
4 x 4 ≈ 16π cm2x π
2 x 4 ≈ 96π cm2x π x 12
112π cm2
12 c
m
Circumference of circle
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131 2 2r
Using Pythagoras Theorem:
r
212+ 213= Û2r 144+ 169= Û2r 144-169= Û
2r 25= Û=5r
5
Find the volume of this compound shape in terms of π
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131 2
5
213V r h= p
13
The volume of the cone:
´p 25´ 12´ ÛV =
13´p 25´ 12´ ÛV =
100´p ÛV =
=100pV
100V = p
Find the volume of this compound shape in terms of π
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131 2
5
343V r= p
43´
The volume of the semi-sphere:
p 35´12´ ÛV =
46´p 125´ ÛV =5006 ´p ÛV =
= 2503 pV
Volume of a sphere
2503V = p
100V = p
Find the volume of this compound shape in terms of π
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100p131 2
5
2503+ p
Total volume of the object
ÛV =
= 5503 pV250
3V = p
100V = p300
3p 250
3+ p ÛV =
5503p ÛV =
550 5503 3Note that =:p p
Find the volume of this compound shape in terms of π
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´r360
r 2 +h2 = L2
L2´
360=A
=L
r360
Û L2´360
=A L
r360
Û
L2 ´360
1
=A L
r360
Û
L2
L360=A Û
=LrA