第1章：有向數 - wikispaces · pdf filethe base of a pyramid is an isosceles right-angled...

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4.11

[ In this exercise, give your answers correct to 3 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 8 cm. The height

VE of the pyramid is 15 cm. Find the volume of the pyramid.

2. 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.

3. In the figure, VABC is a pyramid. ABC is an isosceles

right-angled triangle. If cm 40 ACAB and cm 50 VCVB ,

find the volume of pyramid VABC.

4. In the figure, ABCD is a trapezium where cm 16 AB ,

cm 10 AD and cm 20 CD .

(a) Find the area of trapezium ABCD.

(b) If trapezium ABCD is a base of a pyramid with a height of

20 cm, find the volume of the pyramid.

5. The height and volume of a pyramid are 12 cm and 3cm 120 respectively. Its base is a rectangle with

dimensions cm cm 6 x . Find x.

6. The height and volume of a pyramid are 12 cm and 3cm 100 respectively. If the base of the pyramid

is a square, find the length of each side of the square base.

15 cm

8 cm

8 cm

V

A

B

C

E

V

A

B

C

A B

CD

10 cm

20 cm

16 cm


4.12

7. In the figure, VABCD is a right pyramid. Its base ABCD is a

square with sides of 6 cm each. E is a point on BC such

that BCVE and cm 10 VE . Find the total surface area of

the pyramid.

8. In the figure, VABCD is a right pyramid. The base ABCD is a

square with sides of 5 cm each. The slant edge is 8 cm long.

(a) Find the height VO of the pyramid.

(b) Find the volume of the pyramid.

9. The base of a right pyramid is a square with an area of

2cm 81 .

The height is 15 cm. Find the length of the slant edge of the

pyramid.

10. In the figure, VABC is a pyramid. ABC is an isosceles

right-angled triangular base where cm 30 ACAB . The

height VA of the pyramid is 20 cm.

(a) Find the area of VBC.

(b) Find the total surface area of the pyramid.

11. In the figure, VABCD is a right pyramid where the base is a

rectangle with dimensions cm 10cm 24 . The slant edge is

30 cm long.

(a) Find the height VE of the pyramid.

(b) Find the volume of the pyramid.

(c) Find the total surface area of the pyramid.

12. The figure shows a frustum with right-angled triangular bases

where cm 15 AC , cm 12 AB , cm 12 PQ and cm 10 AP .

(a) By using similar triangles VPQ and VAC, find VA.

(b) By using similar triangles VPR and VAB, find PR.

(c) Hence find the volume of frustum ABCQPR.

6 cm

V

A B

C

ED

10 cm

5 cm

V

A B

CD

8 cm

O

30 cm

V

A

B

C

30 cm

20 cm

V

A

B

C

P Q

R

V

30 cm

D

B

10 cm

C

E

A 24 cm


4.13

13. In the figure, ABCDEFGH is a solid cuboid with the

height of 50 cm. Its base is a square with dimensions

cm 20cm 20 . VEFGH is a right pyramid with the same

height as the cuboid.

(a) Find the total surface area of pyramid VEFGH.

(b) If pyramid VEFGH is removed from the cuboid, find

the total surface area of the remaining solid.

14. If a solid square-based metal pyramid is melted and recast

to form another square-based pyramid which is 21%

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

cm 20 AB , cm 12 PQ and cm 8 PA .

(a) By using similar triangles VPQ and VAB, find VA and

the height of pyramid VABCD.

(b) Hence find the volume of frustum PQRSDABC.

(c) Find the height of VAB from point V.

(d) Hence find the total surface area of frustum

PQRSDABC.


16. Find the volume of each of the following right circular cones. (Express your an swers in terms of .)

(a)

4 cm

9 cm

(b) 24 cm

10 cm

(c)

15 cm

25 cm

V

A D

50 cm

20 cm

G

E

H

B

20 cm

C

F

V

B

DC

A

P Q

RS


4.14

17. Find the curved surface area of each of the following right circular cones. (Express your answers in

terms of .)

(a)

24 cm

5 cm (b)

20 cm16 cm

(c)

4 m

0.9 m

18. Find the volume and total surface area of each of the following rig ht circular cones. (Express your

answers in terms of .)

(a)

12 cm

16 cm

(b)

17 cm

16 cm (c)

24 cm25 cm

19. The figure shows an inverted right conical paper cup. The capacity

of the paper cup is

3cm 180 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 .

20. The slant height of a right circular cone is 18 cm and the height is half of the slant height .

(a) Find the volume of the cone in terms of .

(b) Find the total surface area of the cone.

4 cm


4.15

21. 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 25 cm and

the perimeter of the base is 18 cm.

(a) Find the area of the paper sector in terms of .

(b) If the cost of paper for making the hat is 2m/10$ , find the cost

of paper for making 50 conical hats.

22. (a) A metal right cylinder with both its base radius and height of 10 cm is melted.

3

1 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.

23. The figure shows an ice-cream cone where the volume of the

ice-cream is

3cm 400 . The height of the cone is 12 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 5:3 , find the base radius of the

ice-cream cone.

24. The figure shows a chocolate in the shape of a right circular

frustum. The upper and lower base diameters are 2 cm and 3 cm

respectively.

(a) Find the volume of the chocolate.

(b) It is given that every

3cm of chocolate weighs 3 g. How many

chocolates as shown in the figure can be produced from 1 kg of

chocolate?

25. The figure shows a paper sector with an area of

2cm 120 . 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.

25 cm

120cm2

12 cm

2 cm

3 cm

1 cm


4.16

A B

CD

9 cm

15 cm

6 cm26. A right circular frustum is formed by rotating trapezium ABCD

360 about the axis AD. It is given that cm 6 AB , cm 9 AD and

cm 15 DC .

(a) Find the volume of the frustum in terms of .

(b) Find the total surface area of the frustum.

27. 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. After a while, the water level drops to 8 cm.

(a) Find the radius of the water surface in the figure .

(b) What percentage of water is dripped from the funnel?

28. The figure shows an inverted right conical paper cup

containing

3cm 8 of water. The diameter of the water surface is

4 cm and the water surface is 2 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.

29. 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 .

(b) Find the volume of the rocket model in terms of .

(c) Find the total surface area of the rocket model .

8 cm

16 cm

5 cm

4 cm

2 cm

15 cm

I

II

III

15 cm

15 cm

12 cm

8 cm


4.17

15 m

[ In this exercise, express your answers in terms of if necessary. ]

30. The radius of a sphere is 1.5 cm.

(a) Find the volume of the sphere.

(b) Find the surface area of the sphere.

31. The diameter of a sphere is 8 cm.



32. The diameter of a sphere is 15 m.



33. Find the volume and total surface area of each of the following solids .

(a)

The radius of the sphere

is 8 cm.

(b)

The diameter of the hemisphere

is 10 cm.

(c)

The circumference of the base of

the hemisphere is 20 cm.

1.5 cm

8 cm


4.18

34. If the volume of a sphere is

3cm 10 , find the radius of the sphere. (Give your answer correct to

3 significant figures.)


3cm 100 , find the diameter of the sphere. (Give your answer correct to

3 significant figures.)


3cm 3

4 , find the surface area of the sphere.

37. A hemispherical pudding with the volume of 3cm 144 is

shown in the figure. Find its total surface area .

38. If the surface area of a crystal ball is

2cm 144 , find the volume

of the crystal ball.

39. If the surface area of a sphere is

2cm 600 3 , find the volume of

the sphere.

40. In the figure, O is the centre of the circle, the circumference is

36 cm. A sphere is formed by rotating the circle 360 about

diameter AOB.

(a) Find the surface area of the sphere.

(b) Find the volume of the sphere.

41. A metal hemisphere with the radius of 4 cm is melted and recast to form a metal sphere .

(a) Determine 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 3 significant figures.)

O

B

A


4.19

42. If the outer diameter of a hollow metal sphere is 12 mm and the thickness is 2 mm, find the volume of

metal required to form the metal sphere.

43. A few years ago, the standard diameter of a table tennis ball for competition changed from 38 mm to

40 mm. Find the percentage increase in the surface area of a table tennis ball for competition. (Give

your answer correct to 3 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 cm 12cm 15 . 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 2.4 cm

each are put into the container. Assume 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 3 significant figures.)

45. Figure A shows a hemisphere with the radius of r cm.

Figure B 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.

(a) Show that the volumes of solids in Figures A and

B are equal.

(b) Figures C and D show the cross-sections at z cm

from the bases of Figures A and B respectively.

Show that the areas of the two cross-sections are

equal.

12 cm

15 cm

z cm

Figure D

z cm

Figure C

Figure BFigure A


4.20


46. A and B 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.

47. In the figure, A and B are two similar solids. If the area

of the cross-section of solid A is

2cm 84 , find the area

of the cross-section of solid B. (Express your answer in

terms of .)

48. In the figure, A and B are two similar solids . If the

volume of solid B is

3cm 6 , find the volume of solid A.

49. The volume of a figure of Bruce Lee with height equals

8

1 of his real height is

3cm 600 . If a similar

bronze statue is produced with its height equals 1.5 times the real height of Bruce Lee, what is its

volume?

A

B

10 cm

4 cm

B

Perimeter 28 cm

A

Perimeter 21 cm

AB

12 cm 8 cm


4.21

50. (a) According to the given ratios of the volumes of the similar solids 21 : VV , find the ratios of their

corresponding lengths 21 : and the ratios of their total surface areas 21

: AA .

(i) 512:125 : 21 VV

(ii) 27:64 : 21 VV

(b) According to the given ratios of the total surface areas of the similar solids 21

: AA , find the ratios

of their corresponding lengths 21 : and the ratios of their volumes 21 : VV .

(i) 25:4: 21

AA

(ii) 169:121: 21

AA

51. In the figure, the volume of the solid is

3cm 792 . If a similar

solid is produced such that the area of the top is 2.25 times of

the given one, find the volume of the new solid.

52. The figure shows an inverted right conical paper cup with

water. The depth of water is 5 cm. After drinking half of the

water, what is the depth of water?

53. 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. A and B are two similar sectors with the areas of

2cm 36 and

2cm 25 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.

5 cm

A

B


4.22

55. A and B are two similar bottles with the capacities of

750 mL and 1 200 mL respectively.

(a) Find the ratio of the height of bottle A to that of

bottle B in the form of k:1 .

(b) Find the ratio of the base area of bottle A to that of

bottle B in the form of k:1 .

56. Two different sizes of ice-cream served in two similar

ice-cream cones A and B are sold in a convenience store .

The selling price of a small ice-cream is $2 and that of

the big one is $4. Given that the ratio of the heights of

the two cones is 3:2 , which size of ice-cream is more

economical? Explain briefly.

57. The ratio of the radius of metal ball A to that of metal

ball B is 3:2 . After melting the two metal balls, all the

metal is used to recast into metal ball C.

(a) Find the ratio of the volume of metal ball B to that of

metal ball C.

(b) If the volume of metal ball B is

3cm 10

441

, find the

radius of metal ball C.

58. A cone is divided into 3 portions A, B and C by planes

parallel to the base. The ratio of the slant heights of

portions A, B and C is 1:2:1 .

(a) Find the ratio of the curved surface areas of portions

A, B and C.

(b) Find the ratio of the volumes of portions A, B and C.

AB

AB

A

B C

A

B

C

第1章：有向數 - wikispaces · pdf filethe base of a pyramid is an isosceles right-angled...

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