VOLUME OF SOLIDS WITH UNIFORM CROSS SECTION

If we can take any slice of a solid parallel to its end and find the exposed surface is exactly the same shape and size as the end, then the solid is said to have uniform cross-section.

For any solid of uniform cross-section 

Volume  =  area of end x length

Find the volumes of the following solids:

Example 1 :

Solution :

Given :

Area of end  =  9 m²

Height of the solid  =  2 m

Volume  =  9 (2)

=  18 m³

Example 2 :

Solution :

Given :

Area of end  =  28 m²

Height of the solid  =  9 m

Volume  =  28 (9)

=  252 m³

Example 3 :

Solution :

Given :

Area of end  =  50 cm²

Height of the solid  =  12 cm

Volume  =  50 (12)

=  600 cm³

Example 4 :

Solution :

Given :

Area of end  =  6 mm²

Height of the solid  =  6 mm

Volume  =  6(6)

=  36 mm³

Example 5 :

Solution :

Given :

Area of end  =  12 m²

Height of the solid  =  7.5 m

Volume  =  12(7.5)

=  90 m³

Example 6 :

Solution :

Area of base triangle  =  (1/2) ⋅ base ⋅ height

=  (1/2) ⋅ 4 ⋅ 3

=  6 m²

Height  =  6 m

Volume  =  Base area x height

=  6 (6)

=  36 m³

Example 7 :

An empty garage has floor area 80 m² and a roof height of 4 m. Find the volume of air in the garage

Solution :

Area of base  =  80 m²

height  =  4 m

Volume  =  Area of base x height

=  80 (4)

=  320 m³

Example 8 :

Each month a rectangular swimming pool 6 m by 5 m by 2 m deep costs $0.50 per cubic meter of water to maintain. How much will it cost to maintain the pool for one year?

Solution :

Volume of rectangular swimming pool 

=  Base area x height

Area of rectangular base with length 6 m and width = 5 m.

=  6(5)

=  30 m²

Height  =  2 m

Volume  =  30(2)

=  60 m³

Cost of maintaining  =  0.50 per cubic meter

=  60 (0.50)

=  $30 (per month)

Cost of maintaining per year  =  =  30(12)

=  $360

Example 9 :

Concrete costs $128 per cubic meter. What will it cost to concrete a driveway 20 m long and 3 m wide to a depth of 12 cm?

Solution :

Cost of concrete  =  $128 per cubic meter

length  =  20 m, width  =  3 m and height  =  0.12 m

Volume of concrete  =  20 x 3 x 0.12

=  7.2

Required cost  =  7.2(128)

= $921.6

So, the cost required is $921.6.

Example 10 :

64 cartons of paper are delivered to your school. Each carton measures 40 cm by 30 cm by 25 cm. Is it possible to fit all the cartons into a storage cupboard 1 m by 1 m by 2 m? Explain your reasoning, using a diagram if you wish.

Solution :

length  =  40 cm  ==>  0.40 m

width  =  30 cm ==>  0.30 m

and height  =  25 cm  ==>  0.25 m

Volume of 1 carton  =  0.40 x 0.30 x 0.25

=  0.03 m³

Volume of 64 cartons  =  64(0.03)

=  1.92 m³

Volume of cupboard  =  1 x 1 x 2

=  2 m³

So, it is not possible to fit all in the cupboard.

Example 11 :

The triangular prism has volume 504 cm³. Work out its length.

Solution :

To find volume of any 3D shape, we have to use the formula base area x height

Volume of triangular prism

= (1/2) x base x height x length

(1/2) x 9 x 4 x l = 504

9 x 2 x l = 504

l = 504/18

l = 28

So, the missing length of the prism is 28 cm.

Example 12 :

Solution :

Volume of rectangular prism = length x width x height

Length = 9 cm, width = 8 cm and height = 5.5 cm

Applying all the values in the formula, we get

= 9 x 8 x 5.5

= 396 cm³

So, volume of the rectangular prism is 396 cm³.

Example 13 :

Solution :

Measures of rectangular prism at the bottom :

Length = 1 m ==> 100 cm

Width = 30 cm and height = 40 - 25 ==> 15 cm

Measures of rectangular prism at the top :

Length = 100 - 60 ==> 40 cm

width = 30 cm and height = 25 cm

Volume of bottom = 100 x 30 x 15

= 45000 cm³ -----(1)

Volume of top = 40 x 30 x 25

= 30000 cm³ -----(2)

(1) + (2)

= 45000 + 30000

= 75000 cm³

So, volume of prism given is 75000 cm³.

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