Basic Engineering Mathematics, Fifth Edition

256 Basic Engineering Mathematics

27.6 Volumes of similar shapes

Figure 27.27 shows two cubes, one of which has sides

three times as long as those of the other.

3 x

x

x

x

3 x

3 x

(a) (b)

Figure 27.27

Volume of Figure 27.27(a)=(x)(x)(x)=x^3

Volume of Figure 27.27(b)=( 3 x)( 3 x)( 3 x)= 27 x^3

Hence, Figure 27.27(b) has a volume( 3 )^3 ,i.e.27,times

the volume of Figure 27.27(a).

Summarizing, the volumes of similar bodies are

proportional to the cubes of corresponding linear

dimensions.

Problem 30. A car has a mass of 1000kg.

A model of the car is made to a scale of 1 to 50.

Determine the mass of the model if the car and its

model are made of the same material

Volume of model

Volume of car

=

(

1

50

) 3

since the volume of similar bodies are proportional to

the cube of corresponding dimensions.

Mass=density×volume and, since both car and model

are made of the same material,

Mass of model

Mass of car

=

(

1

50

) 3

Hence, mass of model

=(mass of car)

(

1

50

) 3

=

1000

503

=0.008kgor8g

Now try the following Practice Exercise

PracticeExercise 109 Volumes of similar

shapes (answers on page 352)

1. The diameter of two spherical bearings are in
   the ratio 2:5. What is the ratio of their vol-
   umes?


2. An engineering component has a mass of
   400g. If each of its dimensions are reduced
   by 30%, determine its new mass.