FBE05 – Mathematics and Statistics

1. FBE05 – Mathematics and Statistics Lecture 4 - Discovering Volume

2. Introduction • Volume may be defined as the space in a three-dimensional object. This is different from area, which is applicable to two-dimensional shapes. The most basic shapes in volume calculations are the cube and the cuboid. • A cube is a three-dimensional figure which has six square faces. This means that the length, width and height are equal in a cube. • In a cuboid, at least one of the sides will be different from the others. Typical example of a cuboid is an aerated concrete block.

3. Volume • To understand the concept of volume, divide a cuboid into smaller units, as shown in below. Each small unit, a cube, measures 1 cm x 1 cm x 1 cm

4. Volume of prisms, cylinders, pyramids and cones

5. Volume of prisms, cylinders, pyramids and cones

7. Find the volume of the solids shown in Figure below.

8. Find the volume of the solids shown in Figure below.

9. Volume • The dimensions of a concrete pipe are shown in Figure below. Find the volume of concrete used in its manufacture.

10. Volume • The roof of a building has been designed as a pyramid. Find the volume of the space enclosed by the building.

11. Volume • The cross-section of an embankment is trapezoidal (trapezium) in shape. The measurements at the top and the base of the embankment are 10.0 m and 16.0 m respectively. The height of the embankment is 2.0 m and the length 200 m. If the cross-section remains constant throughout the length of the embankment, find the volume of the soil: • a) in its loose state (the soil bulks @ 15%) • b) in compact state, after the construction of the embankment

12. Volume • A water storage cistern measuring 2.5 m x 2.0 m x 1.2 m high is to be provided in a building: • (a) Calculate the volume of water when the cistern is filled to 15cm from the top rim • (b) Use the volume of water calculated in part (a) to determine the mass of water. The density of water is 1000 kg/m3 • (c) Find the load (force) on the joists supporting the cistern. 1kg = 9.8 newton (N)

13. Concrete mix and its constituents • Concrete is one of the important materials used in the construction of buildings and civil engineering structures. • The constituents of a concrete mix are cement, fine aggregates (sand), coarse aggregates (gravel, blast-furnace slag etc.) and water. Water is added for the chemical reaction between cement and water to take place and produce a solid concrete from a semi-fluid state. • The amounts of these constituents may be determined by considering either their volume or mass. If the proportions of cement, sand and gravel are measured by volume, the concrete mix is known as a nominal mix. Mixing by volume does not take into account the moisture content of the aggregates, therefore nominal mixes are used only for minor work. • Typical examples of nominal mixes are 1:2:4 concrete and 1:3:6 concrete. The 1:2:4 concrete means that it is prepared by mixing 1 part cement, 2 parts fine aggregates and 4 parts coarse aggregates. The volume of water is about 50–60% of the volume of cement. In this section only nominal mixes are considered.

14. Volume • If we want to prepare 1 m3 of 1:2:4 concrete mix, the quantities of dry materials may be calculated as shown below. The volume of cement in a 25 kg bag is 0.0166 m3. • Assuming the volume of water to be 55% of the volume of cement, the proportions of the concrete mix may be written as 1:2:4:0.55. The total of these proportions is 7.55:

15. Volume • A concrete drive to a garage is to be 10 m long, 3 m wide and 150 mm thick. Calculate: • (a) The volume of concrete required to construct the drive • (b) The quantities of cement, sand and gravel required, if 1:2:4 concrete is to be used

16. Volume • Figure 3.17 shows the plan of the proposed extension to a building. Find:(a) The volume of the earth to be excavated(b) The volume of concrete required to the construct the deep strip foundation

17. Figure 3.17