8/4/2019 Cones & Spheres AS 12.3.1 12.3.2

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Cones&

Spheres

A S 12.3.1, 12.3.2

8/4/2019 Cones & Spheres AS 12.3.1 12.3.2

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Shape of objectsShape of objectsMost objects are combinations of

shapes or approximations of shapese.g. Fibroblast cells like this

one are cultured on

circular coverslips 10 mmin diameter.

Calculatethe area of

the coverslip.

A = (5)2

78,54mm2

Calculatethe area of

a single

A (45)

(15)= 337 5 m2

How many cellswill cover thecover slip?

5,3371054,78

6x

No =

2,33x105cells

1mm2 = 106 m2

8/4/2019 Cones & Spheres AS 12.3.1 12.3.2

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The ConeA Cone is a three dimensional solid with a

circular base and a curved surface thatgradually narrows to a vertex.

Volume of Cone = 1/3 volume of

cylinder

V = 1/3 r2 h

Find the volume of a

cone with a diameterof5m and a height of

V = (1/3) r2 h

=(1/3)()

(5/2

)2(9)

8/4/2019 Cones & Spheres AS 12.3.1 12.3.2

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Find the area of a closed cone with aradius,

r = 3 m and height, h = 4 m.

l 2 = r2 + h2

l 2 =(3)2 + (4)2

Surface Area of a Cone = r2 +rl

where r = radius ; h = heightl = the slant height

Surface Area of aCone

= r2 + rl

= (3)

2

+

(3)(5) 2

Surf

ace area of an open cone = rl

8/4/2019 Cones & Spheres AS 12.3.1 12.3.2

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TheSphere

Volume of sphere = 4/3

r3

A sphere is a three dimensional circularsolid that is perfectly symmetrical. It hasno edges or vertices. A hemisphere is a sphere.

Calculate the volume ofa

sphere with a radius of6cm.

V = 4/3 r3

= 4/3 (6)3

904,78 cm3

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If the diameter of abubble

is 0,2 inches, what isthe

surface area of thebubble

in mm2?

Area of sphere = 4 r2

A = 4 r2

= 4 (5,08 /2)2

81,07 mm2

1 inch 2,54 cm

0,2 in = 2,54 x 0,2= 0,508cm= 5,08 mm

8/4/2019 Cones & Spheres AS 12.3.1 12.3.2

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A marble paperweight consists

of a hemisphere with a radiusof 4 cm and a cuboid, withdimensions 10 cm x 10 cm x5 cm, as shown.

1.

V = (10 x 10 x 5) + (4/3. . 43)

634,04 cm3

Calculate the volume of the paperweight.

What is the height of the paperweight ?

h = 4 + 5 = 9 cm

ConsolidationConsolidation

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e on o ay n eUSA you see a LCD HD

TV. The dimensions withthe stand are 49.73 x34.22 x 12.17. Showwhether this TV will fitonto your tabletop with

dimensions 130cm x30cm.

metric dimensions= (49,73x2,54) x (34,22x2,54) x (12,17x2,54)

126,31 cm x 86,92 cm x 30,91 cm

2.

Will not fit as the depth of TV is 30,91 cmand the table is only 30 cm deep.1 inch 2,54 cm

8/4/2019 Cones & Spheres AS 12.3.1 12.3.2

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A shop sign is created with semi-circularends. The shop manager wants the signto have a border made from flexible

plastic.

80cm250cm

150

cm

What length of plastic,

in m, will be needed togo

round the sign?

Length = 2 x x 0,82 + 2 x 2,5 7,01 m

If the border costsR19,99 per metre,

howmuch will the total

border cost?

Cost = R19,99 x 7,01 R140,13

3.

8/4/2019 Cones & Spheres AS 12.3.1 12.3.2

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A container is made of sheet metal in theform of an open cone joined to an open-

ended cylinder. The radius of the cylinderand of the base of the cone is 8,6 cm.

Calculate the sheet metal area.4cm

10

,9cmS A open cone = rl

= (8,6)(10,9)= 294,49.. cm2

S A cylinder = 2rh= 2(8,6)(4)= 216,14.. cm2

Area of sheet metal= 294,49.. + 216,14.. 510,63 cm2

4.

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The underneath of a viaduct arch is sold astorage space. Each end of the arch mustbe boarded up before the arch is sold. The

top of the arch forms a semi-circle with thdiameter wider than the base.

Calculate the boarding

area?Area = 2 x (10x8 + 52)

238,54 m2

Calculate the total volume of the archspace.Volume = area base x height

= 238,54 /2 x 16

= 1907,32 m3

5.

r = 15cm - 10cm

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The diagram shows a small hot watercylinder, which is made up from a cylinderwhich is capped with a hemisphere.

Calculate the maximum volume of water, in, that the cylinder can hold.

Vol cylinder = .0,252 x 1

= 0,19634 m3

Vol sphere = x (4/3 0,253)

= 0,032724 m3

1 k = 1 m3

Total volume = 0,196.. + 0,032..= 0,2290649 k

229

6.

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7. A wine glass is in a shape of a cone on astem. A cylindrical tumbler is used to fill

the wine glass. Find the diameter of thetumbler so that it has the same volumeas the wine glass.

12

cm

6cm

10c

mVolume wine glass= 1/3 r

2 h

= 1/3 (6/2)

2 (10)

94,25 cm2 Tumbler:94,25 = r2 (12)

r2 = 94,25 /(.12)

r 1,58 cm

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The diagram shows a section of a rocketfirework. If it is completely filled withgunpowder, what volume of gunpowder

was used?

V powder = r2h + 1/3r2h

= (32) (20) + 1/3 (32)(5,12)

613,74cm3

Find radius: Sin 30o = r/6r = 6 sin30o

= 3cm

Find cone h: h2 = 62 32

h 5,12cm

8.