Basic Engineering Mathematics, Fifth Edition

Volumes of common solids 249

Total surface area of pyramid

= 2 ( 26. 80 )+ 2 ( 39. 83 )+( 3. 60 )( 5. 40 )

= 53. 60 + 79. 66 + 19. 44

=152.7cm^2

Problem 18. Calculate the volume and total

surface area of a hemisphere of diameter 5.0cm

Volume of hemisphere=

1

2

(volume of sphere)

=

2

3

πr^3 =

2

3

π

(

5. 0

2

) 3

=32.7cm^3

Total surface area

=curved surface area+area of circle

=

1

2

(surface area of sphere)+πr^2

=

1

2

( 4 πr^2 )+πr^2

= 2 πr^2 +πr^2 = 3 πr^2 = 3 π

(

5. 0

2

) 2

=58.9cm^2

Problem 19. A rectangular piece of metal having

dimensions 4cm by 3cm by 12cm is melted down

and recast into a pyramid having a rectangular base

measuring 2.5cm by 5cm. Calculate the

perpendicular height of the pyramid

Volume of rectangular prism of metal=^4 ×^3 ×^12

=144cm^3

Volume of pyramid

=

1

3

(area of base)(perpendicular height)

Assuming no waste of metal,

144 =

1

3

( 2. 5 × 5 )(height)

i.e.perpendicular height of pyramid=

144 × 3

2. 5 × 5

=34.56cm

Problem 20. A rivet consists of a cylindrical

head, of diameter 1cm and depth 2mm, and a shaft

of diameter 2mm and length 1.5cm.Determinethe

volume of metal in 2000 such rivets

Radius of cylindrical head=

1

2

cm= 0 .5cmand

height of cylindrical head=2mm= 0 .2cm.

Hence, volume of cylindrical head

=πr^2 h=π( 0. 5 )^2 ( 0. 2 )= 0 .1571cm^3

Volume of cylindrical shaft

=πr^2 h=π

(

0. 2

2

) 2

( 1. 5 )= 0 .0471cm^3

Total volume of 1 rivet= 0. 1571 + 0. 0471

= 0 .2042cm^3

Volume of metal in 2000 such rivets

= 2000 × 0. 2042 =408.4cm^3

Problem 21. A solid metal cylinder of radius

6cm and height 15cm is melted down and recast

into a shape comprising a hemisphere surmounted

by a cone. Assuming that 8% of the metal is wasted

in the process, determine the height of the conical

portion if its diameter is to be 12cm

Volume of cylinder=πr^2 h=π×^62 ×^15

= 540 πcm^3

If 8% of metal is lost then 92% of 540π gives the

volume of the new shape, shown in Figure 27.16.

h

r

12cm

Figure 27.16