chapter 4 surface area and volume
DESCRIPTION
This is chapter 4 Surface Area and Volume for year 10 and it has exercises 4 U.TRANSCRIPT
-
ihg
7.2
5.6
3.5
12.83.5 2.4
2.85.5
11.3
7.7
2 Rice crackers of diameter 4 cm are packed in acardboard box of height 20 cm. Calculate, correctto one decimal place: WAFERS
20 cm
4 cm
a the volume of the crackers in the boxb the volume of the boxc the percentage of the box that is empty space.
3 This swimming pool is 25 m long and10 m wide. The depth of the waterranges from 1 m to 3 m. Calculatethe capacity of this pool in kilolitres. 3 m
10 m
25 m1 m
4 A wedding cake with three tiers rests on a table. Eachtier is 6 cm high. The layers have radii of 20 cm, 15 cmand 10 cm respectively. Find the total volume of thecake, correct to the nearest cm3.
620
615
610
5 A fish tank that is 60 cm long, 30 cm wide and 40 cm high is filled with water to 5 cm belowthe top. Calculate the volume of the water in litres.
6 Find, correct to two decimal places, the volume of each solid. All lengths shown are in centimetres.
cba
fed
1648
8
12
20
40
10 10
radius of circle = 4 cm
50
35
15
5
5
510 45
15 5 5
1012
Shut
ters
tock
.com
/Joh
nW
ollw
erth
See Example 14
1279780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
lkj
36
8 625
15
8
560
5 14
100
ihg11.3
7.2
19.6
12.73.2
14
10
25
3.6
4.8 6.4
8.3
7 a Find, correct to two decimal places, the volume ofthis greenhouse.
b If this greenhouse costs 0.5c per m3 per hour to heat,how much is this per day, correct to the nearest cent?
3 m
4 m 10 m
Technology Approximating the volume of
a pyramid
In this activity, we will use a spreadsheet toapproximate the volume of a rectangularpyramid by slicing it into tiny layers ofrectangular prisms of equal thickness. 6
8
10
Let L 8 be the length of the prism, W 6 be the width and H 10 be the height.The thickness, T, of each layer is given by the formula T H
number of layers).
Starting at the bottom, the length and width of each layer are decreased by the amountsL
number of layersand W
number of layerswith each step.
1 Set up your spreadsheet as shown.A B C D E F
12 Number of
layers 3 H L W Thickness of
layersVolume oflayer
Sum ofvolumes
4 10 8 6 $A$4/$D$2 B4*C4*D4 E45 B4-$B$4/$D$2 C4-$C$4/$D$2 E5F4...
13
Stage 5.3
128 9780170194662
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Surface area and volume
-
2 Let the number of layers be 10. Enter 10 in cell D2.3 Copy each formula down to row 13.4 Explain the results in cells E13 and F13.5 How accurate was your result in F13? Explain.6 Print out your spreadsheet and paste it into your book.7 Enter 40 (layers) in cell D2 and copy each formula down to row 43.8 In one or two sentences compare your result in F43 with the previous result in F13 from
question 4.9 Enter each value in cell D2 and copy down the formulas as requested.
a 100 (copy down to row 103) b 200 (copy down to row 203)c 400 (copy down to row 403)
10 Use the formula V 13
Ah to calculate the exact volume of the pyramid.
11 Write a brief report about your results in questions 9 and 10.
4-07Volumes of pyramids, cones andspheres
Volume of a pyramid
Summary
Volume of a pyramid
h
A
V 13
Ah
where A area of the base and h perpendicular height.
Example 15
Find the volume of each pyramid.
ba
27 mm 32 mm
36 mm
8 m
10 m
14 m
Stage 5.3
Technology worksheet
Drawing pyramids andcones
MAT10MGCT10006
Technology worksheet
Measuring pyramids
MAT10MGCT10002
Worksheet
Back-to-front problems(Advanced)
MAT10MGWK10206
1299780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
Solutiona A 27 3 32
864b A 1
23 8 3 14
56
V 13
Ah
13
3 864 3 36
10 368 mm3
V 13
Ah
13
3 56 3 10
186 23
m3
Example 16
Find the volume of a square pyramid with base length 48 mm and slant height 51 mm.
SolutionFirst find h, the perpendicular height of the pyramid.
48 mm
h
51 mmh2 512 242
2025
h ffiffiffiffiffiffiffiffiffiffi
2025p
45 mm
A 48 3 48 2304
V 13
3 2304 3 45
34 560 mm3
Volume of a coneA cone is like a circular pyramid so:
V 13
Ah 13
3 pr2 3 h 13
pr2h
Summary
Volume of a cone
r
h
V 13
pr2h
where r radius of the base and h perpendicular height.
Stage 5.3
Technology worksheet
Approximating thevolume of a cone
MAT10MGCT10003
Video tutorial
Area and volume
MAT10MGVT00004
130 9780170194662
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Surface area and volume
-
Example 17
Find, correct to the nearest cubic millimetre, thevolume of this cone.
25 mm
28 mm
Solution
V 13
pr2h
13
3 p 3 12:52 3 28
4581:4892 . . . 4581 mm3
Example 18
A cone has a base radius of 14 cm and a slant height of 50 cm. Find its volume, correct to twosignificant figures.
Solution
First find the height, h.
h
14 cm
50 cm
h2 502 142
2304
h ffiffiffiffiffiffiffiffiffiffi
2304p
48 cm
V 13
3 p 3 142 3 48
9852:0345 . . . 9900 cm3
Volume of a sphere
Summary
Volume of a sphererV 4
3pr3
where r radius of the sphere.
Stage 5.3
1319780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
Example 19
Find, correct to two significant figures, the volume of each solid.ba
18 cm
1.3 m
Solution
a V 43
pr3
43
3 p 3 93 r 12
3 18 9
3053:6280 . . . 3100 cm3
b V 12
343
pr3
23
pr3
23
3 p 3 1:33
4:6013 . . . 4:6 m3
Exercise 4-07 Volumes of pyramids, cones andspheres
1 Find the volume of each pyramid.
cba
fed
8 cm
9 cm
10 cm
10 cm
6 cm
8 cm
12 cm
5 cm
14 m
18 m
8 m
20 cm
12 cm
15 cm
5 m
8 m
6 m
Stage 5.3
See Example 15
132 9780170194662
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Surface area and volume
-
2 For each pyramid, find correct to one decimal place:i its perpendicular height ii its volume
cba
fed
18 cm
18 cm
15 cmh
h
60 m
41 m18 m
50 m25 mm 25 mm
14 mm14
mm
68 mm 61 mm
11 mm
11 mm32 mm 32 mm
8.5 m 8.5 m
3.6 m 3.6 m3.6 m 3.6 m
160 cm
126 cm
116 cm
105 cm
3 Find, correct to the nearest whole number, the volume of each cone.
cba
9 m
4 m
10 cm
12 cm
17 mm
20 mm
fed
12 cm7 cm 10 cm
15 cm
30 mm
18 mm
4 For each cone, find correct to one decimal place:i its perpendicular height ii its volume
cba
fed
7 cm
3 cm
4.4 m
4.5 m
10 cm
8 cm
0.8 m
3.6 m
68 m
247 m83 cm
83 cm
Stage 5.3
See Example 16
See Example 17
See Example 18
1339780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
5 For each solid, find correct to the nearest whole number:i its volume ii its capacity
cba
fed
15 mm 11 m10.8 cm
24 m8 cm
16 mm
6 The Earth has a radius of approximately 6400 km. Calculate its volume in scientific notationcorrect to two significant figures.
7 A grain hopper is in the shape of a square pyramid. 4.5 m
5 m
4.5 m
a Find the volume of grain that it holds when full.b If there are 750 kg of wheat per m 3, find the mass of
grain in the hopper when it is filled to three-quarters ofcapacity. Give your answer correct to the nearest tonne.
8 A pyramid has a volume of 360 m3 and a base area of 48 m2.Calculate its perpendicular height.
9 A square pyramid has a volume of 800 cm3 and a perpendicular height of 12 cm. Calculate,correct to one decimal place, the length of its base.
10 A cone has a volume of 600 m3 and a base radius of 10 m. Calculate, correct to one decimalplace, its perpendicular height.
11 A cone has a volume of 160 cm3 and a perpendicular height of 20 cm. Calculate, correct toone decimal place, its radius.
12 Calculate, correct to one decimal place, the radius of a sphere with a volume of 81 585 mm3.
4-08 Volumes of composite solids
Summary
PrismV Ah A
h
CylinderSA 2pr2 2prh
V pr2hh
r
PyramidV 1
3Ah
h
A
ConeSA prl pr2
V 13
pr2h
lh
r
SphereSA 4pr2
V 43
pr3r
Stage 5.3
See Example 19
Worksheet
A page of solid shapes
MAT10MGWK10205
Worksheet
Volume and capacity
MAT10MGPS00046
134 9780170194662
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Surface area and volume
-
Note that the formulas for surface area involve two dimensions, for example, r2 or rh, while theformulas for volume involve three dimensions, for example, lwh, r2h or r3.
Example 20
a Find, correct to the nearest cubic centimetre, the volume of this solid.b Find, correct to the nearest litre, the capacity of this solid.
20 cm
35 cm
Solutiona Volume volume of cylinder volume of hemisphere
pr2h 12
343
pr3
pr2h 23
pr3
p 3 102 3 35 23
3 p 3 103
13 089:9693 . . . 13 090 cm3
r 12
3 20 10
b Capacity 13 090 mL 13:09 L 13 L
Exercise 4-08 Volumes of composite solids1 The storage tank shown is completely filled with water.
4 m
2 m
4 m
a Calculate, correct to the nearest cubic metre, the volume ofthe tank.
b Find the capacity of the tank, correct to the nearest kilolitre.
2 Find the volume of each solid. All measurements are in centimetres.
ba c
4
7
7
910
10 6
12
12
12
Stage 5.3
See Example 20
1359780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
fed
20
15
12
25
18 24
21
30
20
10
15
3 For each solid, find:i the volume (to the nearest cm3)ii the capacity (in litres, correct to three decimal places).
All measurements are in centimetres.
cba
40
15
2014
24
5
56
12
4 A conical tank (A) and a hemispherical tank (B) have measurements as shown. How muchmore does tank B hold compared to tank A? Answer correct to two decimal places.
3 m
3 m BA
3 m
3 m
5 Spherical balls of diameter 10 cm are stacked inside a box inthe shape of a rectangular prism, as shown.
30 cm40 cm
50 cm
a How many balls will fit in the bottom layer?b If the balls are stacked in the same manner as in the bottom
layer until the box is full, how many balls will fit in the box?
c Calculate, correct to the nearest cubic centimetre, the volumeof the space occupied by the balls when the box is full.
d What percentage of the box is empty space? Give your answercorrect to the nearest whole percentage.
Stage 5.3
136 9780170194662
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Surface area and volume
-
6 The sand in this hourglass takes up three-quarters of thevolume of the bottom cone.
20 cm
50 cm
a Calculate, correct to the nearest cubic centimetre, the volumeof sand in the hourglass.
b If the sand takes one hour to fall from the top cone to thebottom cone, at what rate is it falling? Give your answer incm3/s, correct to two significant figures.
7 a Calculate the volume of this swimming pool.
10 m
1 m
20 m
10 m
2 m
b Calculate the capacity of the pool if it isfilled to a depth of 20 cm from the top.
c If water costs $1.98/kL, find the cost offilling the pool.
4-09 Areas of similar figures
Summary
Areas of similar figuresIf the matching sides of two similar figures are in the ratio 1 : k, then their areas are in theratio 1 : k2.If the matching sides are in the ratio m : n, then their areas are in the ratio m2 : n2.
A1 : A2 m2 : n2 orA1A2 m
n22
Example 21
What is the ratio of the areas of the similar rectangles shown?
B
14 mm
8 mm
20 mm
35 mm
ASolutionRatio of matching sides A to B 35 : 14
5 : 2
Ratio of areas 52 : 22
25 : 4
Stage 5.3
Technology worksheet
Excel worksheet: Areaof similar shapes
MAT10MGCT00013
Technology worksheet
Excel spreadsheet:Area of similar shapes
MAT10MGCT00043
1379780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
Example 22
Two similar pentagons have areas in the ratio 144 : 169. Find the ratio of the lengths of theirmatching sides.
SolutionRatio of areas m2 : n2 144 : 169
) Ratio of sides m : n ffiffiffiffiffiffiffiffi
144p
:ffiffiffiffiffiffiffiffi
169p
12 : 13
Example 23
Two similar triangles have matching sides in the ratio 3 : 5. If the area of the larger triangle is225 cm2, find the area of the smaller triangle.
SolutionLet the area of the smaller figure be A.
A3 5
225 cm2Ratio of matching sides 3 : 5Ratio of areas 32 : 52 9 : 25
) A225 9
25
A 925
3 225
81 cm2
The area of the smaller triangle is 81 cm2.
Exercise 4-09 Areas of similar figures1 For each pair of similar figures, find the ratio of their areas.
ba
dc
1 cm3 cm
1.5 m
2.5 m
9 cm 5 cm 4 cm 6 cm
Stage 5.3
See Example 21
138 9780170194662
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Surface area and volume
-
2 For each ratio of the areas of two similar figures, find the ratio of the lengths of their matchingsides.
a 9 : 25 b 1 : 100 c 64 : 25 d 16 : 813 Find x if these triangles are similar.
12
x
A1 = 144
A1 = 108A2 = x
A = xA = 3
A2 = 324
7.85.2
2.80.8
Area = 12 cm2
Area = 3 cm2
7 cma b
c d
x cm
4 Two circles have radii in the ratio 3 : 5. If the larger area is 150 cm2, find the area of thesmaller circle.
5 Similar squares have sides in the ratio 7 : 4. If the area of the smaller square is 14.4 cm2, findthe area of the larger square.
6 Two similar triangles have areas in the ratio 4 : 9. If the length of the base of the smallertriangle is 5 cm, find the length of the base of the larger triangle.
7 Two similar rectangles have their areas in the ratio 36 : 121. If the width of the smallerrectangle is 84 cm, find the width of the larger rectangle.
8 If the radius of a circle is doubled, how has its area changed?
9 If the area of a square is divided by 9, how have the sides changed?
10 If the sides of a triangle are increased by 2.5, how has its area changed?
11 If the area of a trapezium is decreased by 1100
, how have the sides changed?
Investigation: Surface areas and volumes of similar solids
1 a Calculate the volume of this rectangular prism.2 cm
6 cm
8 cm
b Calculate the surface area of the rectangular prism.c If the length, width and height are all doubled, what
happens to:i the volume? ii the surface area?
d Copy and complete:If the length, width and height are all doubled, the volume is increased ______ times andthe surface area is increased ______ times.
Stage 5.3
See Example 22
See Example 23
1399780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
4-10Surface areas and volumes of similarsolids
Summary
Surface areas and volumes of similar solidsIf the matching sides of two similar solids are in the ratio 1 : k, then their surface areas are inthe ratio 1 : k2 and their volumes are in the ratio 1 : k3.If the matching sides are in the ratio m : n, then their surface areas are in the ratio m2 : n2
and their volumes are in the ratio m3 : n3.
SA1SA2 m
2
n2and
V1V2 m
3
n3
2 a Explain why these rectangular prisms are similar solids.
2 cm
1 cm3 cm
2 cm
6 cm
4 cmb What is the ratio of their matching sides?c What is the ratio of their surface areas?d What is the ratio of their volumes?
3 For the spheres A and B, find the ratio of:a their radiib their surface areasc their volumes
9 cm
3 cm
A
B
4 How is the ratio of the surface areas of similar solids related to the ratio of matchingsides?
5 How is the ratio of the volumes of similar solids related to the ratio of their matchingsides?
Stage 5.3
NSW
Worksheet
Areas and volumes ofsimilar figures
MAT10MGWK10207
140 9780170194662
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Surface area and volume
-
Example 24
For these two similar triangular prisms, find the ratio of their:
a surface areasb volumes
2.2 cm2.4 cm
3 cmX3.3 cm
3.6 cm
4.5 cmY
Solutiona Ratio of sides X to Y 3 : 4:5 or 2:2 : 3:3 or 2:4 : 3:6
6 : 9 2 : 3
Ratio of surface areas 22 : 32
4 : 9
b Ratio of volumes 23 : 33
8 : 27
Example 25
Two similar cylinders have their surface areas in the ratio 25 : 36. If the volume of the smallercylinder is 250 cm3, find the volume of the larger solid.
SolutionRatio of surface areas 25 : 36
) Ratio of matching sides ffiffiffiffiffi
25p
:ffiffiffiffiffi
36p
5 : 6
) Ratio of volumes 53 : 63
125 : 216
Let the volume of the larger cylinder be V.
V
250 216
125
V 216125
3 250
432
[ The volume of the larger cylinder is 432 cm3.
Stage 5.3
1419780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
Exercise 4-10 Surface areas and volumes of similarsolids
1 For each pair of similar solids, find the ratio of:i the smaller surface area to the larger surface areaii the smaller volume to the larger volume
3 cm
a b
c d
5 cm
3.6 m 2.4 m
12 cm15 cm
22.5 m
9
2 Two similar pyramids have surface areas of 81 cm2 and 100 cm2. Find the ratio of their:a matching side lengths b volumes.
3 Two similar prisms have volumes of 125 cm3 and 343 cm3. Find the ratio of their:a matching sides b surface areas.
4 Blocks of chocolate are sold in the shape of similar triangular prisms. The areas of thetriangular faces of two prisms are 6400 mm2 and 1600 mm2. If the volume of the smallerprism is 9600 mm3, find the volume of the larger prism.
5 There are two similar cylindrical drink cans. The larger can is 15 cm high and contains 350 mLof drink. If the smaller can is 9 cm high, how much drink does it contain?
6 A box of washing powder is 12 cm tall and contains 750 g of washing powder. A similar box is18 cm tall. How much washing powder does it contain?
7 A large fish tank has a capacity of 624 L. A smaller, similar fish tank has half the length, widthand depth of the large tank. Find the capacity of the smaller tank.
8 A cylinder has its height and radius increased 1.5 times. By what factor has its:a surface area increased? b volume increased?
9 A spherical balloon has a radius of 8 cm. By what factor is the volume decreased if the radiuschanges to 6 cm?
Stage 5.3
See Example 24
See Example 25
142 9780170194662
Chapter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Surface area and volume
-
Power plus
1 A square prism and square pyramid have the same base and the same surface area. Showthat the slant height, l, of the pyramid is l 5
2s where s is the length of the base.
2 A cylinder with diameter and height 2r has the same surface area as a sphere of radius R.
Show that R ffiffiffiffiffiffi
32
r
r
.
R
2r
2r
3 A sphere and a cone have the same radius and volume. Show that the cones height, h, isfour times the radius, r.
r
h
r
4 A sphere and a cone fit inside identical cylinders with the same base diameter and height.
2r
2r
2r
2r
a Find the ratio Volume of cone : Volume of sphere : Volume of cylinderb Show that Volume of cone Volume of sphere Volume of cylinder
5 A cube is divided into six identical square pyramids as shown, each with a perpendicularheight that is half the length of the base edge. Show that the volume of each pyramid isone-third the volume of a square prism with the same base edge and perpendicularheight.
2s
2s2s
s
2s
1439780170194662
NEW CENTURY MATHS ADVANCEDfor the A u s t r a l i a n C u r r i c u l u m1010A
-
Chapter 4 review
n Language of maths
apex base capacity circle
cone cross-section cubic curved surface
cylinder diameter hemisphere kilolitre
litre perpendicular height pyramid radius
ratio sector similar figures similar solids
slant height sphere surface area volume
1 Which word means a slice of a prism or cylinder?
2 Name three solids that have a curved surface area.
3 What is the formula for the curved surface area of a cone?
4 Explain the difference between the perpendicular height and the slant height of a pyramid.
5 What is the formula V 13pr2h used for?
6 Describe the relationship between the volumes of similar solids.
n Topic overview
Copy and complete the table below.
The best part of this chapter was
The worst part was
New work
I need help with
Puzzle sheet
Surface area andvolume crossword
(Advanced)
MAT10MGPS10208
Quiz
Area and volume
MAT10MGQZ00004
144 9780170194662
-
Copy and complete this mind map of the topic, adding detail to its branches and usingpictures, symbols and colour where needed. Ask your teacher to check your work.
Compositesolids Prisms
Cylinder Cone Sphere Pyramids
SURFACEAREA
Similar solids ratio of areas :
VOLUME Similar solids ratio of volumes :
1459780170194662
Chapter 4 review
-
1 Find the surface area of each prism.
cba
fed
0.4 m
0.5 m
0.8 m0.3 m
45 mm
15 mm7 cm
48 cm50 cm
3.6 m
12 m
3 m
8 m
6 cm
4 mm
5 mm24 mm
2 Calculate, correct to one decimal place, the surface area of each solid.
cba
21
35
23
15
4.8Cylinder,open atone end
2.7
fed
50 cm
50 cm
20 cm5 cm 5 cm
15 cm
30 cm
30 cm30 cm
18 cm 34 cm
25 cm
3 Find the surface area of each pyramid.
cba
16 cm16 cm
22 cm
54 cm
36 cm
30 cm
14 cm
25 cm
See Exercise 4-01
See Exercise 4-02
Stage 5.3
See Exercise 4-03
146 9780170194662
Chapter 4 revision
-
4 Find, correct to the nearest square metre, the surface area of each solid. All measurementsare in metres.
cba
fed
8
20
closed
48
40
open
11
60
closed
6 m
17 m
25 m
5 Find, correct to the nearest square centimetre, the surface area of each solid. All measurementsare in centimetres.
fed
30
16
18
12
25
25
cba
18
16
282
45
12 4
20
18
127
6 Calculate, correct to nearest cubic metre, the volume of each solid. All measurements are inmetres.
a5025
25
b
24
42
28
18
c
20
23
15
Stage 5.3
See Exercise 4-04
See Exercise 4-05
Stage 5.3
See Exercise 4-06
1479780170194662
Chapter 4 revision
-
7 Find, correct to two decimal places (where necessary), the volume of each solid.
b ca
11 m
11 m
8 m
15 cm 18 cm
25 m
m
14 mm 14 mm
12 cm
ed
8 cm
20 cm
28 mm
50 mm
f
6 m
8 Find, correct to the nearest whole number, the volume of each solid.
cba
fed
80 mm
45 mm
80 mm
45 mm
45 mm
45 mm
6 cm
8 cm
8 cm
8 cm
4.5 m
4.5 m
4.5 m
18 cm
24 cm
12 cm
24 m
44 m
9 a Two similar circles have radii in the ratio 4 : 5. If the smaller area is 150cm2, find the areaof the larger circle.
b The radius of a circle is increased by a factor of 2 12. By what factor has the area increased?
10 a The areas of the bases of two similar rectangular prisms are in the ratio of 25 : 64. If thevolume of the larger prism is 1024 cm2, find the volume of the smaller prism.
b Two similar pyramids have volumes of 216 cm3 and 343 cm3. Find the ratio of theirsurface areas.
Stage 5.3
See Exercise 4-07
See Exercise 4-08
See Exercise 4-09
See Exercise 4-10
148 9780170194662
Chapter 4 revision