120 cm respectively. Its base is a rectangle with
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![[ In this exercise, give your answers correct to significant figures if necessary. ] 1. The base of a pyramid is an isosceles right-angled triangle where the lengths of the two equal sides are.](/docs-images/81/82764736/images/1-0.jpg "The height E of the pyramid is. Find the volume of the pyramid.. It is given that the base of a pyramid is a triangle with base a cm and height b cm.")
1 [ In this exercise, give your answers correct to significant figures if necessary. ] 1. The base of a pyramid is an isosceles right-angled triangle where the lengths of the two equal sides are. The height E of the pyramid is. Find the volume of the pyramid.. It is given that the base of a pyramid is a triangle with base a cm and height b cm. If the height of the pyramid is h cm, express the volume of the pyramid in terms of a, b and h. E. In the figure, is a pyramid. is an isosceles right-angled triangle. If = = 40 cm and = = 50 cm, find the volume of pyramid. 4. In the figure, is a trapezium where = 16 cm, = 10 cm and = 0 cm. (a) Find the area of trapezium. (b) If trapezium is a base of a pyramid with a height of 0 cm, find the volume of the pyramid. 10 cm 16 cm 0 cm 5. The height and volume of a pyramid are 1 cm and dimensions 6 cm x cm. Find x. 10 cm respectively. Its base is a rectangle with 6. The height and volume of a pyramid are 1 cm and 100 cm respectively. If the base of the pyramid is a square, find the length of each side of the square base. 010 hung Tai Educational Press. ll rights reserved. 4.11
2 7. In the figure, is a right pyramid. Its base is a square with sides of 6 cm each. E is a point on such that E and E = 10 cm. Find the total surface area of the pyramid. 10 cm E 6 cm 8. In the figure, is a right pyramid. The base is a square with sides of 5 cm each. The slant edge is long. (a) Find the height O of the pyramid. (b) Find the volume of the pyramid. O 5 cm 9. The base of a right pyramid is a square with an area of 81 cm. The height is. Find the length of the slant edge of the pyramid. 10. In the figure, is a pyramid. is an isosceles right-angled triangular base where = = 0 cm. The height of the pyramid is 0 cm. (a) Find the area of Δ. (b) Find the total surface area of the pyramid. 0 cm 0 cm 0 cm 11. In the figure, is a right pyramid where the base is a rectangle with dimensions 4 cm 10 cm. The slant edge is 0 cm long. (a) Find the height E of the pyramid. (b) Find the volume of the pyramid. (c) Find the total surface area of the pyramid. E 4 cm 0 cm 10 cm 1. The figure shows a frustum with right-angled triangular bases where =, = 1 cm, PQ =1 cm and P = 10 cm. (a) y using similar triangles PQ and, find. (b) y using similar triangles PR and, find PR. R (c) Hence find the volume of frustum QPR. P Q hung Tai Educational Press. ll rights reserved.
3 1. In the figure, EFGH is a solid cuboid with the height of 50 cm. Its base is a square with dimensions 0 cm 0 cm. EFGH is a right pyramid with the same height as the cuboid. 50 cm (a) Find the total surface area of pyramid EFGH. (b) If pyramid EFGH is removed from the cuboid, find the total surface area of the remaining solid. G F 0 cm E 0 cm H 14. If a solid square-based metal pyramid is melted and recast to form another square-based pyramid which is 1% higher than the original pyramid, find the percentage decrease in the length of each side of the square base. 15. The figure shows a square-based right frustum, where = 0 cm, PQ =1 cm and P =. (a) y using similar triangles PQ and, find and the height of pyramid. (b) Hence find the volume of frustum PQRS. (c) Find the height of Δ from point. (d) Hence find the total surface area of frustum PQRS. P S Q R [ In this exercise, give your answers correct to significant figures if necessary. ] 16. Find the volume of each of the following right circular cones. (Express your answers in terms of π.) (a) 9 cm (b) 4 cm (c) 5 cm 10 cm 4 cm 010 hung Tai Educational Press. ll rights reserved. 4.1
4 17. Find the curved surface area of each of the following right circular cones. (Express your answers in terms of π.) (a) 5 cm (b) (c) 0.9 m 16 cm 0 cm 4 cm 4 m 18. Find the volume and total surface area of each of the following right circular cones. (Express your answers in terms of π.) (a) 16 cm (b) 16 cm (c) 4 cm 5 cm 1 cm 17 cm 19. The figure shows an inverted right conical paper cup. The capacity of the paper cup is 180 cm and the base radius is 4 cm. (a) Find the height of the paper cup. (b) If the cup is filled with water, find the area of the wet surface. 4 cm 0. The slant height of a right circular cone is 1 and the height is half of the slant height. (a) Find the volume of the cone in terms of π. (b) Find the total surface area of the cone hung Tai Educational Press. ll rights reserved.
5 1. The figure shows a right circular conical hat formed by rolling up a paper sector. It is given that the slant height of the hat is 5 cm and the perimeter of the base is 18π cm. (a) Find the area of the paper sector in terms of π. (b) If the cost of paper for making the hat is $ 10/m, find the cost of paper for making 50 conical hats. 5 cm 1. (a) metal right cylinder with both its base radius and height of 10 cm is melted. of the metal is recast to form a right circular cone with the base same as the original cylinder. Find the height of the cone. (b) The rest of the metal is recast to form another right circular cone with the base same as the original cylinder. Find the total surface area of this cone.. The figure shows an ice-cream cone where the volume of the ice-cream is 400 cm. The height of the cone is 1 cm and it is filled with ice-cream. The ratio of the volume of ice-cream outside the cone to that inside the cone is : 5, find the base radius of the ice-cream cone. 1 cm 4. The figure shows a chocolate in the shape of a right circular frustum. The upper and lower base diameters are cm and cm respectively. (a) Find the volume of the chocolate. (b) It is given that every cm of chocolate weighs g. How many chocolates as shown in the figure can be produced from 1 kg of chocolate? 1 cm cm cm 5. The figure shows a paper sector with an area of 10 π cm. If a right circular cone is formed by rolling up the paper sector, (a) find the base radius of the cone. (b) find the height of the cone. (c) find the volume of the cone. 10 π cm 010 hung Tai Educational Press. ll rights reserved. 4.15
6 6. right circular frustum is formed by rotating trapezium 60 about the axis. It is given that = 6 cm, = 9 cm and =. (a) Find the volume of the frustum in terms of π. 9 cm 6 cm (b) Find the total surface area of the frustum. 7. The figure shows an inverted right conical funnel with the base radius of 5 cm and height of 16 cm. Initially, the funnel is filled with water. fter a while, the water level drops to. (a) Find the radius of the water surface in the figure. (b) What percentage of water is dripped from the funnel? 16 cm 5 cm 8. The figure shows an inverted right conical paper cup containing 8 π cm of water. The diameter of the water surface is 4 cm and the water surface is cm below the rim of the cup. (a) Find the depth of water. (b) Find the area of the wet surface. (c) Find the capacity of the cup. cm 4 cm 9. The figure shows a rocket model made up of three parts. Solid I is a right circular cone. Solid II is a right cylinder. Solid III is a right circular frustum. (a) Find the volume of solid III in terms of π. (b) Find the volume of the rocket model in terms of π. (c) Find the total surface area of the rocket model. I II III 1 cm hung Tai Educational Press. ll rights reserved.
7 [ In this exercise, express your answers in terms of π if necessary. ] 0. The radius of a sphere is 1.5 cm. (a) Find the volume of the sphere. (b) Find the surface area of the sphere. 1.5 cm 1. The diameter of a sphere is. (a) Find the volume of the sphere. (b) Find the surface area of the sphere.. The diameter of a sphere is 15 m. (a) Find the volume of the sphere. (b) Find the surface area of the sphere. 15 m. Find the volume and total surface area of each of the following solids. (a) (b) (c) The radius of the sphere is. The diameter of the hemisphere is 10 cm. The circumference of the base of the hemisphere is 0 π cm. 010 hung Tai Educational Press. ll rights reserved. 4.17
8 4. If the volume of a sphere is significant figures.) 10 π cm, find the radius of the sphere. (Give your answer correct to 5. If the volume of a sphere is 100 cm, find the diameter of the sphere. (Give your answer correct to significant figures.) 6. If the volume of a sphere is 4 π, find the surface area of the sphere. cm 7. hemispherical pudding with the volume of 144 π cm is shown in the figure. Find its total surface area. 8. If the surface area of a crystal ball is 144 π cm, find the volume of the crystal ball. 9. If the surface area of a sphere is the sphere. 600 π cm, find the volume of 40. In the figure, O is the centre of the circle, the circumference is 6π cm. sphere is formed by rotating the circle 60 about diameter O. (a) Find the surface area of the sphere. (b) Find the volume of the sphere. O 41. metal hemisphere with the radius of 4 cm is melted and recast to form a metal sphere. (a) etermine whether the total surface area of the solid increases or decreases. (b) Find the percentage increase / percentage decrease in the total surface area of the solid. (Give your answer correct to significant figures.) hung Tai Educational Press. ll rights reserved.
9 4. If the outer diameter of a hollow metal sphere is 1 mm and the thickness is mm, find the volume of metal required to form the metal sphere. 4. few years ago, the standard diameter of a table tennis ball for competition changed from 8 mm to 40 mm. Find the percentage increase in the surface area of a table tennis ball for competition. (Give your answer correct to significant figures.) 44. In the figure, there is a metal ball with the radius of 5 cm inside a container in the shape of right prism. The base of the container is a rectangle of dimensions 1 cm. Water is poured into the container until the metal ball is just covered by water. (a) Find the volume of water. (b) Now, 10 more metal balls with diameters of.4 cm each are put into the container. ssume that the metal balls are fully immersed in water and water does not overflow, how much does the water level rise? (Give your answers correct to significant figures.) 1 cm 45. Figure shows a hemisphere with the radius of r cm. Figure shows a solid which is formed by removing an inverted right circular cone from a right circular cylinder. The base radii and heights of the cone and cylinder are all r cm. z cm Figure z cm Figure (a) Show that the volumes of solids in Figures and are equal. (b) Figures and show the cross-sections at z cm from the bases of Figures and respectively. Show that the areas of the two cross-sections are equal. Figure Figure hung Tai Educational Press. ll rights reserved.
10 [ In this exercise, give your answers correct to significant figures if necessary. ] 46. and are the uniform cross-sections of two similar prisms. (a) Find the ratio of the total surface area of the larger prism to that of the smaller one. (b) Find the ratio of the volume of the larger prism to that of the smaller one. Perimeter = Perimeter = 1 cm 47. In the figure, and are two similar solids. If the area of the cross-section of solid is 84 π cm, find the area of the cross-section of solid. (Express your answer in terms of π.) 1 cm 48. In the figure, and are two similar solids. If the volume of solid is 6 cm, find the volume of solid. 10 cm 4 cm The volume of a figure of ruce Lee with height equals of his real height is 600 cm. If a similar 8 bronze statue is produced with its height equals 1.5 times the real height of ruce Lee, what is its volume? hung Tai Educational Press. ll rights reserved.
11 50. (a) ccording to the given ratios of the volumes of the similar solids 1 :, find the ratios of their corresponding lengths 1 : and the ratios of their total surface areas 1 :. (i) 15 : 51 1 : = (ii) 64 : 7 1 : = (b) ccording to the given ratios of the total surface areas of the similar solids 1 :, find the ratios of their corresponding lengths 1 : and the ratios of their volumes 1 :. (i) 4 : 5 1 : = (ii) 11: : = 51. In the figure, the volume of the solid is 79 cm. If a similar solid is produced such that the area of the top is.5 times of the given one, find the volume of the new solid. 5. The figure shows an inverted right conical paper cup with water. The depth of water is 5 cm. fter drinking half of the water, what is the depth of water? 5 cm 5. When a metal rod is heated, the length of the rod increases by 8%. Find the percentage increase in the volume of the metal rod. 54. and are two similar sectors with the areas of 6 π cm and 5 π cm respectively. Two right circular cones are formed by rolling up the two sectors. (a) Find the ratio of the base radius of the larger cone to that of the smaller one. (b) Find the ratio of the volume of the larger cone to that of the smaller one. 010 hung Tai Educational Press. ll rights reserved. 4.1
12 55. and are two similar bottles with the capacities of 750 ml and 1 00 ml respectively. (a) Find the ratio of the height of bottle to that of bottle in the form of 1 : k. (b) Find the ratio of the base area of bottle to that of bottle in the form of 1 : k. 56. Two different sizes of ice-cream served in two similar ice-cream cones and are sold in a convenience store. The selling price of a small ice-cream is $ and that of the big one is $4. Given that the ratio of the heights of the two cones is :, which size of ice-cream is more economical? Explain briefly. 57. The ratio of the radius of metal ball to that of metal ball is :. fter melting the two metal balls, all the metal is used to recast into metal ball. + (a) Find the ratio of the volume of metal ball to that of metal ball. 441π (b) If the volume of metal ball is cm, find the 10 radius of metal ball. 58. cone is divided into portions, and by planes parallel to the base. The ratio of the slant heights of portions, and is 1 : : 1. (a) Find the ratio of the curved surface areas of portions, and. (b) Find the ratio of the volumes of portions, and hung Tai Educational Press. ll rights reserved.
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