Volumes of common solids
Volume of cylindrical portion 25.0 2 = πr 2 h = π (12.0) = 5890 m3 2 Volume of frustum of cone =
1 π h(R2 + Rr + r 2 ) 3
where h = 30.0 − 12.0 = 18.0 m, R = 25.0/2 = 12.5 m and r = 12.0/2 = 6.0 m. Hence volume of frustum of cone 1 = π (18.0) (12.5)2 + (12.5)(6.0) + (6.0)2 = 5038 m3 3
189
7. A cylindrical tank of diameter 2.0 m and perpendicular height 3.0 m 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 1.0 m and 2.0 m, respectively, determine the vertical height required.
24.5 Volumes of similar shapes The volumes of similar bodies are proportional to the cubes of corresponding linear dimensions. For example, Fig. 24.18 shows two cubes, one of which has sides three times as long as those of the other.
Total volume of cooling tower = 5890 + 5038 = 10 928 m3 . If 40% of space is occupied then volume of air space = 0.6 × 10 928 = 6557 m3 3x
Now try the following exercise 3x
x
Exercise 89 Further problems on volumes and surface areas of frustra of pyramids and cones (Answers on page 280) 1. The radii of the faces of a frustum of a cone are 2.0 cm and 4.0 cm and the thickness of the frustum is 5.0 cm. Determine its volume and total surface area. 2. A frustum of a pyramid has square ends, the squares having sides 9.0 cm and 5.0 cm, respectively. Calculate the volume and total surface area of the frustum if the perpendicular distance between its ends is 8.0 cm. 3. A cooling tower is in the form of a frustum of a cone. The base has a diameter of 32.0 m, the top has a diameter of 14.0 m and the vertical height is 24.0 m. 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 28.0 cm and 6.00 cm and the vertical distance between the ends is 30.0 cm, find the area of material needed to cover the curved surface of the speaker. 5. A rectangular prism of metal having dimensions 4.3 cm by 7.2 cm by 12.4 cm 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 3 cm and 8 cm 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 36.0 cm and end diameters 55.0 cm and 35.0 cm.
x x
3x
(a)
(b)
Fig. 24.18
Volume of Fig. 24.18(a) = (x)(x)(x) = x3 Volume of Fig. 24.18(b) = (3x)(3x)(3x) = 27x3 Hence Fig. 24.18(b) has a volume (3)3 , i.e. 27 times the volume of Fig. 24.18(a). Problem 22. A car has a mass of 1000 kg. 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 then: Mass of model = Mass of car
1 50
3