Basic Engineering Mathematics, Fifth Edition

Volumes of common solids 255

r= 5 .0cm,R= 10 .0cm

and l=

√[

25. 02 + 5. 02

]

= 25 .50cm

from Pythagoras’ theorem.
Hence, curved surface area
=π( 25. 50 )( 10. 0 + 5. 0 )= 1201 .7cm^2

i.e., the area of material needed to form the lampshade
is1200cm^2 , correct to 3 significant figures.

Problem 29. A cooling tower is in the form of a

cylinder surmounted by a frustum of a cone, as

shown in Figure 27.26. Determine the volume of air

space in the tower if 40% of the space is used for

pipes and other structures

12.0m

25.0m

12.0m

30.0m

Figure 27.26

Volume of cylindrical portion=πr^2 h

=π

(

25. 0

2

) 2

( 12. 0 )

=5890m^3

Volume of frustum of cone =

1

3

πh(R^2 +Rr+r^2 )

where h= 30. 0 − 12. 0 = 18 .0m,

R= 25. 0 ÷ 2 = 12 .5m

and r= 12. 0 ÷ 2 = 6 .0m.

Hence, volume of frustum of cone

=

1

3

π( 18. 0 )

[

( 12. 5 )^2 +( 12. 5 )( 6. 0 )+( 6. 0 )^2

]

=5038m^3

Total volume of cooling tower= 5890 + 5038

=10 928m^3

If 40% of space is occupied then

volume of air space= 0. 6 × 10928 =6557m^3

Now try the following Practice Exercise

PracticeExercise 108 Volumes and surface

areasof frusta of pyramidsand cones

(answers on page 352)

1. The radii of the faces of a frustum of a cone
   are 2.0cmand4.0cm and the thickness of the
   frustum is 5.0cm. Determine its volume and
   total surface area.


2. A frustum of a pyramid has square ends,
   the squares having sides 9.0cm and 5.0cm,
   respectively. Calculate the volume and total
   surface area of the frustum if the perpendicular
   distance between its ends is 8.0cm.


3. A cooling tower is in the form of a frustum of
   a cone. The base has a diameter of 32.0m,the
   top has a diameter of 14.0m and the vertical
   height is 24.0m. Calculate the volume of the
   tower and the curved surface area.


4. A loudspeaker diaphragm is in the form of a
   frustum of a cone. If the end diameters are
   28 .0cmand6.00cm and the vertical distance
   between the ends is 30.0cm, find the area of
   material needed to cover the curved surface of
   the speaker.


5. A rectangular prism of metal having dimen-
   sions 4.3cm by 7.2cm by 12.4cm is melted
   down and recast into a frustum of a square
   pyramid, 10% of the metal being lost in the
   process. If the ends of the frustum are squares
   of side 3cm and 8cm respectively, find the
   thickness of the frustum.


6. Determine the volume and total surface area
   of a bucket consisting of an inverted frustum
   of a cone, of slant height 36.0cm and end
   diameters 55.0cm and 35.0cm.


7. A cylindrical tank of diameter 2.0m and per-
   pendicular height 3.0m is to be replaced by
   a tank of the same capacity but in the form
   of a frustum of a cone. If the diameters of
   the ends of the frustum are 1.0m and 2.0m,
   respectively, determine the vertical height
   required.