Basic Engineering Mathematics, Fifth Edition

Volumes of common solids 251

Curved surface area of cylinder, Q= 2 πrh

= 2 ×π× 3 × 8

= 48 πm^2

The slant height of the cone,l, is obtained by Pythago-
ras’ theorem on triangleABC,i.e.

l=

√(

42 + 32

)

= 5

Curved surface area of cone,

R=πrl=π× 3 × 5 = 15 πm^2

Total surface area of boiler= 18 π+ 48 π+ 15 π

= 81 π=254.5m^2

Now try the following Practice Exercise

PracticeExercise 107 More complex

volumes and surface areas(answers on

page 351)

1. Find the total surface area of a hemisphere of
   diameter 50mm.


2. Find (a) the volume and (b) the total surface
   area of a hemisphere of diameter 6cm.


3. Determine the mass of a hemispherical cop-
   per container whose external and internal
   radii are 12cm and 10cm, assuming that
   1cm^3 of copper weighs 8.9g.


4. A metal plumb bob comprises a hemisphere
   surmounted by a cone. If the diameter of the
   hemisphere and cone are each 4cm and the
   total length is 5cm, find its total volume.


5. A marquee is in the form of a cylinder sur-
   mountedbyacone.Thetotalheightis6mand
   the cylindrical portion has a height of 3.5m
   with adiameter of 15m.Calculatethesurface
   area of material needed to make the marquee
   assuming 12% of the material is wasted in
   the process.


6. Determine (a) the volume and (b) the total
   surface area of the following solids.
   (i) a cone of radius 8.0cm and perpendi-
   cular height 10cm.
   (ii) a sphere of diameter 7.0cm.
   (iii) a hemisphere of radius 3.0cm.

(iv) a 2.5cmby2.5cm square pyramid of

perpendicular height 5.0cm.

(v) a 4.0cm by 6.0cm rectangular pyra-

mid of perpendicular height 12.0cm.

(vi) a 4.2cm by 4.2cm square pyramid

whose sloping edges are each 15.0cm

(vii) a pyramid havingan octagonalbase of

side 5.0cm and perpendicular height

20cm.

7. A metal sphere weighing 24kg is melted
   down and recast into a solid cone of base
   radius 8.0cm. If the density of the metal is
   8000kg/m^3 determine
   (a) the diameter of the metal sphere.
   (b) the perpendicular height of the cone,
   assuming that 15% of the metal is lost
   in the process.


8. Find the volume of a regular hexagonal pyra-
   midiftheperpendicularheightis16.0cmand
   the side of the base is 3.0cm.


9. A buoy consists of a hemisphere surmounted
   by a cone. The diameter of the cone and
   hemisphere is 2.5m and the slant height of
   the cone is 4.0m. Determine the volume and
   surface area of the buoy.


10. A petrol container is in the form of a central
    cylindrical portion 5.0m long with a hemi-
    spherical section surmounted oneach end. If
    the diameters of the hemisphere and cylinder
    are both 1.2m, determine the capacity of the
    tank in litres (1litre=1000cm^3 ).


11. Figure 27.18 shows a metal rod section.
    Determine its volume and total surface area.

1.00 cm

radius 1.00 m

2.50 cm

Figure 27.18