Further applications of area and volume

2
syllabus reference
Measurement 5 Further applications of area and volume

In this chapter
2A 2B 2C 2D Area of parts of the circle Area of composite shapes Simpsons rule Surface area of cylinders and spheres 2E Volume of composite solids 2F Error in measurement

areyou
2.1
Area of a circle

Are you ready?

Try the questions below. If you have difculty with any of them, extra help can be obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon next to the question on the Maths Quest HSC Course CD-ROM or ask your teacher for a copy. 1 Find the area of a circle with: a radius 4 cm b radius 19.6 cm
Areas of squares, rectangles and triangles

READY?
c diameter 9 cm d diameter 19.7 cm c
3.7 m 7.6 m 13.8 m

2.2

2 Find the area of each of the following. 10.9 m a b

4.5 cm

2.6

Volume of cubes and rectangular prisms (3a, 3b); Volume of triangular prisms (3c)

3 Find the volume of: a

b
11 cm 9 cm

c
8 cm 26 cm 6 cm

2.7
24 cm 18 cm Volume of cylinders (4a); Volume of a sphere (4b)

2.8

4 Find the volume of: a

19 cm 8 cm

b
12 m

2.9

Volume of a pyramid

2.10

Find the volume of:

10 cm

7 cm

2.11

Error in linear measurement

For each of the following linear measurements, state the limits between which the true limits actually lie. a 15 cm (measured correct to the nearest centimetre) b 8.3 m (measured correct to 1 decimal place) c 4800 km (measured correct to the nearest 100 km)

Chapter 2 Further applications of area and volume

43

Area of parts of the circle

From previous work you should know that the area of a circle can be calculated using the formula: A = r2

WORKED Example 1
Calculate the area of a circle with a radius of 7.2 cm. Give your answer correct to 2 decimal places. THINK
1 2 3

WRITE A = r2 A = (7.2)2 A = 162.86 cm2

Write the formula. Substitute for the radius. Calculate the area.

A sector is the part of a circle between two radii as shown on the right. To calculate the area of a sector we nd the fraction of the circle formed by the sector. For example, a semicircle is half of a circle and so the area of a semicircle is half the area of a full circle. A quadrant is a quarter of a circle and so the area is quarter that of a full circle. For other sectors the area is calculated by using the angle between the radii as a fraction of 360 and then multiplying by the area of the full circle. This can be written using the formula:

- 2 A = -------r 360
where is the angle between the two radii.

WORKED Example 2
Calculate the area of the sector drawn on the right. Give your answer correct to 1 decimal place.
5 cm 80

THINK
1 2 3

WRITE A= A=
-------r2 360 80 -------- 360

Write the formula. Substitute for and r. Calculate the area.

52

A = 17.5 cm2

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Maths Quest General Mathematics HSC Course

An annulus is the area between two circles that have the same centre (i.e. concentric circles). The area of an annulus is found by subtracting the area of the smaller circle from the area of the larger circle. This translates to the formula A = (R2 r2) , where R is the radius of the outer circle and r is the radius of the inner circle.

WORKED Example 3
Calculate the area of the annulus on the right. Give your answer correct to 1 decimal place.
5.7 cm 3.2 cm

THINK
1 2 3

WRITE A = (R2 r 2) A = (5.72 3.22) A = 69.9 cm2

Write the formula. Substitute R = 5.7 and r = 3.2. Calculate.

An ellipse is an oval shape and therefore does not have a constant radius. The greatest distance from the centre of the ellipse to the circumference is called the semi-major axis, a, while the smallest distance is called the semi-minor axis, b, as shown in the gure on the right. The area of an ellipse is calculated using the formula, found on the formula sheet: A = ab

semi-minor axis (b) semi-major axis (a)

WORKED Example 4
Calculate the area of the ellipse drawn on the right. Give your answer correct to 2 decimal places.
6.6 m 4.2 m

THINK
1 2 3

WRITE A = ab A = 6.6 4.2 A = 87.08 m2

Write the formula. Substitute for the values of a and b. Calculate the area.

Chapter 2 Further applications of area and volume

45

remember

1. The area of a circle is found using the formula A = r2. 2. The area of a sector can be found by multiplying the area of a full circle by the fraction of the circle given by the angle in the sector. You can use the formula A = ------- r 2. 360 3. An annulus is the area between two concentric circles. The area is found by using the formula A = (R2 r2) , where R is the radius of the outer circle and r is the radius of the inner circle. 4. An ellipse is an oval shape. The area is calculated using the formula A = ab , where a is the length of the semi-major axis and b is the length of the semi-minor axis.

2A
WORKED

Area of parts of the circle

2.1
6.4 cm Area of a circle
SkillS

Example

1 Calculate the area of the circle drawn on the right, correct to 1 decimal place.

HEET

2 Calculate the area of each of the circles drawn below, correct to 2 decimal places. a b c
33 mm 9 cm 7.4 m

f
6.02 m

26.5 cm

3.84 m

3 Calculate the area of a circle that has a diameter of 15 m. Give your answer correct to 1 decimal place.
WORKED

Example

4 Calculate the area of the sector drawn on the right. Give your answer correct to 1 decimal place.
7.2 m

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Maths Quest General Mathematics HSC Course

5 Calculate the area of each of the sectors drawn below. Give each answer correct to 2 decimal places. a b c

5.2 cm 60

135

23 m 74 mm 20

d
9.2 mm 150

e
240 19.5 m

f
39 mm 72

6 Calculate, correct to 1 decimal place, the area of a semicircle with a diameter of 45.9 cm.
WORKED

Example

7 Calculate the area of the annulus shown at right, correct to 1 decimal place.

12 cm

6 cm

8 Calculate the area of each annulus drawn below, correct to 3 signicant gures. a b c
9.7 m 20 cm 18 cm 4.2 m 13 mm 77 mm

9 A circular garden of diameter 5 m is to have concrete laid around it. The concrete is to be 1 m wide. a What is the radius of the garden? b What is the radius of the concrete circle? c Calculate the area of the concrete, correct to 1 decimal place.
WORKED

Example

10 Calculate the area of the ellipse drawn on the right, correct to 4 1 decimal place.
10 cm

6 cm

Chapter 2 Further applications of area and volume

47

11 Calculate the area of each of the ellipses drawn below. Give each answer correct to the nearest whole number. a b c
34 mm 56 mm 13.6 m 7.2 m 21 cm 14 cm

12 multiple choice The area of a circle with a diameter of 4.8 m is closest to: A 15 m2 B 18 m2 C 36 m2 13 multiple choice Which of the following calculations will give the area of the sector shown on the right? D 72 m2

45 8m

1 -8

42

1 -8

82

- 42 C 1 4

- 82 D 1 4

14 multiple choice The area of the ellipse drawn on the right is closest to:
1.2 m 86 cm

A 32 400 cm2

B 324 m2

C 5900 cm2

D 59 m2

15 A circular area is pegged out and has a diameter of 10 m. a Calculate the area of this circle, correct to 1 decimal place. b A garden is to be dug which is 3 m wide around the area that has been pegged out. Calculate the area of the garden to be dug. Give your answer correct to 1 decimal place. c In the garden a sector with an angle of 75 at the centre is to be used to plant roses. Calculate the area of the rose garden, correct to 1 decimal place. 16 A circle has a diameter of 20 cm. a Calculate the area of this circle, correct to 2 decimal places. b An ellipse is drawn such that the radius of the circle forms the semi-major axis. The semi-minor axis is to have a length equal to half the radius of the circle. Calculate the length of the semi-minor axis. c Calculate the area of the ellipse, correct to 2 decimal places.

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Maths Quest General Mathematics HSC Course

Area of composite shapes

A composite shape is a shape that is made up of two or more regular shapes. The area of a composite shape is found by splitting the area into two or more regular shapes and calculating the area of each separately before adding them together. In many cases it will be necessary to calculate the length of a missing side before calculating the area. There will sometimes be more than one way to split the composite shape.

WORKED Example 5
Find the area of the gure at right.
18 cm

6 cm

10 cm 12 cm

THINK
1

WRITE
6 cm A1 8 cm 18 cm A2 12 cm 10 cm

Copy the diagram and divide the shape into two rectangles.

2 3 4 5

Calculate the length of the missing side in rectangle 1. (Write this on the diagram.) Calculate the area of rectangle 1. Calculate the area of rectangle 2. Add together the two areas.

18 10 = 8 cm A1 = 6 8 A1 = 48 cm2 A2 = 10 12 A1 = 120 cm2 Area = 48 + 120 Area = 168 cm2

Composite areas that involve triangles may require you to also make a calculation using Pythagoras theorem.

WORKED Example 6
Find the area of the gure on the right.

13 m

10 m 24 m

THINK
1

WRITE
13 m a 12 m

Draw the triangle at the top and cut the isosceles triangle in half.

Chapter 2 Further applications of area and volume

49

THINK
2

WRITE a2 = c2 b2 = 132 122 = 169 144 = 25 a = 25 =5m A= = 60 m2 A = 24 10 = 240 m2 Area = 60 + 240 Area = 300 m2
1 -2

Calculate the perpendicular height using Pythagoras theorem.

Calculate the area of the triangle. Calculate the area of the rectangle. Add the two areas together.

24 5

4 5

Composite areas can also be calculated by using subtraction rather than addition. In these cases we calculate the larger area and subtract the smaller area in the same way as we did with annuluses in the previous section.

WORKED Example 7
Find the shaded area in the gure on the right.
6 cm 20 cm

30 cm

THINK
1 2 3

WRITE A = 30 20 A = 600 cm2 A = 62 A = 113.1 cm2 Area = 600 113.1 Area = 486.9 cm2

Calculate the area of the rectangle. Calculate the area of the circle. Subtract the areas.

remember
1. To nd the area of any composite gure, divide the shape into smaller regular shapes and calculate each area separately. 2. You may have to use Pythagoras theorem to nd missing pieces of information. 3. Check if the best way to solve the question is by adding two areas or by subtracting one area from the other to nd the remaining area.

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Maths Quest General Mathematics HSC Course

2B
SkillS
HEET

Area of composite shapes

4m

2.2

WORKED

Example

5 Areas of squares, rectangles and triangles

1 Copy the gure on the right into your workbook and calculate its area by dividing it into two rectangles.
18 m

11 m 20 m

2 Find the area of each of the gures below. Where necessary, give your answer correct to 1 decimal place. a
16 cm 25 cm 40 cm 5 cm 22 cm 7 cm b 18 cm 12 cm

5 cm 19 cm 6 cm

4 cm

12 cm 8 cm 16 cm 4 cm

8 cm 4 cm

SkillS

HEET

2.3
Using Pythagoras theorem

3 Look at the triangle on the right. a Use Pythagoras theorem to nd the perpendicular height of the triangle. b Calculate the area of the triangle. 4 Below is an isosceles triangle.

10 cm

17 cm

6 cm

15 cm

Cabr

omet i Ge ry

Pythagoras calculations

8m

12 m

a Use Pythagoras theorem to nd the perpendicular height of the triangle, correct to 1 decimal place. b Calculate the area of the triangle.

Chapter 2 Further applications of area and volume

51
sheet
E

5 Calculate the area of each of the triangles below. Where necessary, give your answer L Spre XCE ad correct to 1 decimal place. Pythagoras a b c 25 cm
26 m 48 cm
GC

am progr C

asio

24 m 124 mm
WORKED

Mensuration

Example

6 Find the area of each of the composite gures drawn below. a b c 25 mm 13 cm 17 m

15 mm 12 cm 30 m 13 m

52 mm 48 mm 54 mm

7 multiple choice The area of the composite gure on the right is closest to: A 139 m2 B 257 m2 2 C 314 m D 414 m2 8 multiple choice The area of the gure drawn on the right is: A 36 m2 B 54 m2 2 C 72 m D 144 m2
10 m

12 m 6m

9 A block of land is in the shape of a square with an equilateral triangle on top. Each side of the block of land is 50 m. a Draw a diagram of the block of land. b Find the perimeter of the block of land. c Find the area of the block of land.
WORKED

Example

10 In each of the following, nd the area of the shaded region. Where necessary, give your answer correct to 1 decimal place. 7 a b c 12 m
10 cm 4 cm 9 cm 16 cm 3 cm 9 cm

8m

e
112 mm

f
1.9 m 7.4 m 40 mm 95 mm 3.1 m

7. 1

cm

36 mm

10 cm

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Maths Quest General Mathematics HSC Course

11 An athletics track consists of a rectangle with two semicircular ends. The dimensions are shown in the diagram on the right. The track is to have a synthetic running surface laid. Calculate the area which is to be laid with the running surface, correct to the nearest square metre.

70 m 90 m

82 m

12 A garden is to have a concrete path laid around it. The garden is rectangular in shape and measures 40 m by 25 m. The path around it is to be 1 m wide. a Draw a diagram of the garden and the path. b Calculate the area of the garden. c Calculate the area of the concrete that needs to be laid. d If the cost of laying concrete is $17.50 per m2, calculate the cost of laying the path.

1
Calculate the area of each of the gures drawn below. Where necessary, give your answer correct to 1 decimal place. 1
12 cm

3
5.8 cm 9.4 cm

6.3 m

4
91 mm 62 mm

10 cm

6
30 cm 20 cm 4 cm 25 cm

25 cm

24 cm

7
20 m 40 m

8
12 cm

76

mm

40 cm

10
6 cm

12

cm

32
15 cm

m m

Chapter 2 Further applications of area and volume

53

Simpsons rule
Simpsons rule is a method used to approximate the area of an irregular gure. Simpsons rule approximates an area by taking a straight boundary and dividing the area into two strips. The height of each strip (h) is measured. Three measurements are then taken perpendicular to the straight boundary, as shown in the gure on the right. The formula for Simpsons rule is: h - (d + 4dm + dl) A -3 f where h = distance between successive measuements df = rst measurement dm = middle measurement dl = last measurement.
dm df h h dl

WORKED Example 8
Use Simpsons rule to approximate the area shown on the right.
30 m 10 m 90 m

THINK
1 2 3 4 5

WRITE h = 90 2 = 45 df = 10, dm = 30, dl = 0 h - (d + 4dm + dl) A -3 f 45 - ( 10 + 4 30 + 0 ) A ----3 = 15 130 1950 m2

Calculate h. Write down the values of df , dm and dl. Write the formula. Substitute. Calculate.

Could Simpsons rule be used to estimate the areas of these irregular shapes from nature?

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Maths Quest General Mathematics HSC Course

Simpsons rule can be used to approximate an irregular area without a straight edge. This is done by constructing a line as in the diagram below and approximating the area of each section separately.

WORKED Example 9
Use Simpsons rule to nd an approximation for the area shown on the right.
30 m 30 m 17 m 30 m 10 m

THINK
1 2

WRITE h = 30 df = 0, dm = 30, dl = 10 h - (d + 4dm + dl) A -3 f 30 - ( 0 + 4 30 + 10 ) A ----3 10 130 1300 m2 df = 0, dm = 17, dl = 0 h - (d + 4dm + dl) A -3 f 30 - ( 0 + 4 17 + 0 ) A ----3 10 68 680 m2 Area 1300 + 680 Area 1980 m2

Write down the value of h. For the top area, write down the values of df , dm and dl . Write the formula. Substitute. Calculate the top area. For the bottom area, write down the values of df , dm and dl . Write down the formula. Substitute. Calculate the bottom area. Add the two areas together.

3 4 5 6

7 8 9 10

Simpsons rule approximates an area, it does not give an exact measurement. To obtain a better approximation, Simpsons rule can be applied several times to the area. This is done by splitting the area in half and applying Simpsons rule separately to each half.

WORKED Example 10
32 m 31 m 24 m 29 m 105 m 30 m

Use two applications of Simpsons rule to approximate the area on the right.

THINK
1 2

WRITE h = 105 4 = 26.25 df = 32, dm = 31, dl = 24

Calculate h by dividing 105 by 4. (We are using 4 sub-intervals.) Apply Simpsons rule to the left half. Write the values of df , dm and dl .

Chapter 2 Further applications of area and volume

55

THINK
3 4 5 6

WRITE h - (d + 4dm + dl) A -3 f 26.25 - ( 32 + 4 31 + 24 ) A -----------3 8.75 180 1575 m2 df = 24, dm = 29, dl = 30 h - (d + 4dm + dl) A -3 f 26.25 - ( 24 + 4 29 + 30 ) A -----------3 8.75 170 1487.5 m2 Area 1575 + 1487.5 Area 3062.5 m2

Write the formula. Substitute. Calculate the approximate area of the left half. Apply Simpsons rule to the left half. Write the values of df , dm and dl . Write the formula. Substitute. Calculate the approximate area of the right half. Add the areas together.

7 8 9 10

remember
1. Simpsons rule is a method of approximating irregular areas. h - ( d + 4 d m + d l ) , where h is the 2. The Simpsons rule formula is A -3 f distance between successive measurements, df is the rst measurement, dm is the middle measurement and dl is the last measurement. 3. A better approximation of an area can be found by using Simpsons rule twice.

2.4

SkillS

HEET

2C
WORKED

Simpsons rule

Substitution into formulas

Example

1 The diagram on the right is of a part of a creek. a State the value of h. b State the value of df , dm and dl . c Use Simpsons rule to approximate the area of this section of the creek.

40 m

9m

60 m

18 m

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Maths Quest General Mathematics HSC Course

2 Use Simpsons rule to approximate each of the areas below. a b c

28 m 12 m 6m 16 m 40 m 22 m 21 m 27 m 5 m 18 m

35 m

10 m

12 m

72 m

0m

54 m

48 m
WORKED

Example

30 m 7m

19 m 11 m A2 33 m 25 m 7m 40 m 31 m

3 The irregular area on the right has been divided into two areas labelled A1 (upper area) and A2 (lower area). a Use Simpsons rule to nd an approximation for Al . b Use Simpsons rule to nd an approximation for A2. c What is the approximate total area of the gure?

A1 30 m

4 Use Simpsons rule to nd an approximation for each of the areas below. a b c

22 m 11 m 17 m 14 m 6 m 27 m 45 m 12 m 45 m 0 16 m 16 m 12 m 10 m

21 m 51 m 90 m 10 m 36 m

5 multiple choice Consider the gure drawn on the right. Simpsons rule gives an approximate area of: A 1200 m2 B 2400 m2 2 C 3495 m D 6990 m2

23 m

27 m

6 multiple choice If we apply Simpsons rule twice, how many measurements from the traverse line need to be taken? A4 B 5 C7 D9
WORKED

Example

50 m 71 m 20 m 44 m

10

7 Use Simpsons rule twice to approximate the area on the right.

45 m 50 m

22 m

18 m 18 m 18 m 18 m

8 Use Simpsons rule twice to approximate each of the areas drawn below. a b c

42 m

102 m

87 m

54 m

63 m

60 m 60 m 60 m 60 m

21 m 21 m 21 m 21 m

10 m 10 m 10 m 10 m

45 m

11 m

Chapter 2 Further applications of area and volume

57

9 The gure on the right is of a cross-section of a waterway. a Use Simpsons rule once to nd an approximate area of this section of land. b Use Simpsons rule twice to obtain a better approximation for this area of land.

32 m 15 m 27 m

10 Apply Simpsons rule four times to approximate the area on the right.
20 m 30 m 35 m 36 m 38 m 41 m 45 m 30 m 24 m
Work
T SHEE

36 m

2.1

9m 9m 9m 9m 9m 9m 9m 9m

Surface area of cylinders and spheres

From earlier work you should remember that surface area is the area of all surfaces of a 3-dimensional shape. Consider a closed cylinder with a radius (r) and a perpendicular height (h). The surface of the cylinder h consists of two circles and a rectangle. Area of top = r 2 r Area of bottom = r 2 The rectangular side of the cylinder will have a length equal to the circumference of the circle (2 r) and a width equal to the height (h) of the cylinder. Area of side = 2 rh The surface area of the closed cylinder can be calculated using the formula: SA = 2 r 2 + 2 rh

WORKED Example 11
Calculate the surface area of the closed cylinder drawn on the right. Give your answer correct to 1 decimal place.
10 cm 9 cm

THINK
1 2 3

WRITE

Write the formula. Substitute the values of r and h. Calculate the surface area.

SA = 2 r 2 + 2 rh SA = 2 92 + 2 9 10 SA = 1074.4 cm2

For cylinders, before calculating the surface area you need to consider whether the cylinder is open or closed. In the case of an open cylinder there is no top and so the formula needs to be written as: SA = r 2 + 2 rh

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Maths Quest General Mathematics HSC Course

Note: On the formula sheet in the exam only the formula for the closed cylinder is provided. You will need to adapt the formula yourself for examples such as this.

WORKED Example 12
Calculate the surface area of an open cylinder with a radius of 6.5 cm and a height of 10.8 cm. Give your answer correct to 1 decimal place. THINK
1 2 3

WRITE SA = r 2 + 2 rh SA = (6.5)2 + 2 6.5 10.8 SA = 573.8 cm2

Write the formula. Substitute the values of r and h. Calculate the surface area.

A sphere is a round 3-dimensional shape, and the only measurement given is the radius (r). The surface area of a sphere can be calculated using the formula: SA = 4 r 2

WORKED Example 13
Calculate the surface area of the sphere drawn on the right. Give the answer correct to 1 decimal place.
2.7 cm

THINK
1 2 3

WRITE SA = 4 r 2 SA = 4 (2.7)2 SA = 91.6 cm2

Write the formula. Substitute the value of r. Calculate the surface area.

remember
1. The surface area of a closed cylinder is found using the formula SA = 2 r 2 + 2 rh . 2. If the cylinder is an open cylinder, the surface area formula becomes SA = r 2 + 2 rh. 3. The surface area of a sphere is found using the formula SA = 4 r 2 .

The Atomium, Brussels

Chapter 2 Further applications of area and volume

59

2D
WORKED

Surface area of cylinders and spheres

2.5
SkillS
HEET

Example

11

1 Calculate the surface area of a closed cylinder with a radius of 5 cm and a height of 11 cm. Give your answer correct to 1 decimal place. Circumference 2 Calculate the surface area of each of the closed cylinders drawn below. Give each answer correct to 1 decimal place. a b c
1.6 m 12 cm 1.1 m 3 cm 20 cm 5 cm of a circle

d
20 cm

e
5.9 cm 5.9 cm r r = 5 cm

1.5 m 2.3 m

3 Calculate the surface area of a closed cylinder with a diameter of 3.4 m and a height of 1.8 m. Give your answer correct to 1 decimal place.
WORKED

Example

12

4 Calculate the surface area of an open cylinder with a radius of 4 cm and a height of 16 cm. Give your answer correct to the nearest whole number. 5 Calculate the surface area of each of the following open cylinders. Give each answer correct to 1 decimal place. a b c
30 cm 20 cm

13.3 cm r r = 4.1 cm 9.6 cm

22 cm

e
50 cm

f
3.2 m 4m 4 cm

23.2 cm 2.4 cm

6 A can of fruit is made of stainless steel. The can has a radius of 3.5 cm and a height of 7 cm. A label is to be wrapped around the can. a Calculate the amount of steel needed to make the can (correct to the nearest whole number). b Calculate the area of the label (correct to the nearest whole number).

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WORKED

Maths Quest General Mathematics HSC Course

Example

13

7 Calculate the surface area of a sphere with a radius of 3 cm. Give your answer correct to the nearest whole number. 8 Calculate the surface area of each of the spheres drawn below. Give each answer correct to 1 decimal place. a b c
2.1 cm 8 cm 14 cm

d
1m

e
3.4 cm

f
1.8 m

9 Calculate the surface area of a sphere with a diameter of 42 cm. Give your answer correct to the nearest whole number. 10 multiple choice An open cylinder has a diameter of 12 cm and a height of 15 cm. Which of the following calculations gives the correct surface area of the cylinder? A 62 + 2 6 15 B 2 62 + 2 6 15 C 122 + 2 12 15 D 2 122 + 2 12 15 11 multiple choice Which of the following gures has the greatest surface area? A A closed cylinder with a radius of 5 cm and a height of 10 cm B An open cylinder with a radius of 6 cm and a height of 10 cm C A cylinder open at both ends with a radius of 7 cm and a height of 10 cm D A sphere with a radius of 6 cm 12 An open cylinder has a diameter and height of 12 cm. a Calculate the surface area of the cylinder (correct to the nearest whole number). b A sphere sits exactly inside this cylinder. Calculate the surface area of this sphere (correct to the nearest whole number). 13 A cylindrical can is to contain three tennis balls each having a diameter of 6 cm. a Calculate the surface area of each ball. b The three balls t exactly inside the can. State the radius and height of the can. c The can is open and made of stainless steel, except the top which will be plastic. Calculate the area of the plastic lid (correct to the nearest whole number). d Calculate the amount of stainless steel in the can (correct to the nearest whole number). e Calculate the area of a paper label that is to be wrapped around the can (correct to the nearest whole number).

Chapter 2 Further applications of area and volume

61
E

Computer Application 1 Minimising surface area

Access the spreadsheet Volume from the Maths Quest General Mathematics HSC Volume Course CD-ROM. A cylindrical drink container is to have a capacity of 1 litre (volume = 1000 cm3). We are going to calculate the most cost efcient dimensions to make the container. To do this, we want to make the container with as little material as possible, in other words we want to minimise the surface area of the cylinder. The spreadsheet should look as shown below. 1. In cell B3 enter the volume of the cylinder, 1000. 2. In cell A6 enter a radius of 1. In cell A7 enter a radius of 2 and so on up to a radius of 20. 3. The formula that has been entered in cell B6 will give the height of the cylinder corresponding to the radius for the given volume. 4. The surface area of each possible cylinder is in column D. Use the charting function on the spreadsheet to graph the surface area against the radius. 5. What are the most cost-efcient dimensions of the drink container?
L Spre XCE ad
sheet

Challenge exercise
Use one of the other worksheets to nd the most efcient dimensions to make a rectangular prism of volume 1000 cm3 and a cone of volume 200 cm3.

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Maths Quest General Mathematics HSC Course

Packaging
A company makes tennis balls that have a diameter of 6.5 cm. The tennis balls are to be sold in packs of four. 1 Calculate the surface area of the packaging needed if the balls are packed in a cylindrical tube that just ts all four balls as shown on the right.

2 Calculate the amount of packaging needed if the balls are packed in a rectangular prism.

3 Calculate the amount of packaging needed if the balls are packed in a 2 2 design as shown on the right.

4 Design the most effective way of packaging nine tennis balls.

Volume of composite solids

Many solid shapes are a composition of two or more regular solids. To calculate the volume of such a gure, we need to determine the best method for each particular part. Many irregular shapes may still be prisms. A prism is a shape in which every cross-section taken parallel to the base shape is equal to that base shape. The formula for the volume of a prism is: V = Ah where A is the area of the base shape and h is the height. Remember that the base of the prism is not necessarily the bottom. The base is the shape that is constant throughout the prism and will usually be drawn as the front of the prism. This means that the height will be drawn perpendicular to the base. To calculate the volume of any prism, we rst calculate the area of the base and then multiply by the height.

Chapter 2 Further applications of area and volume

63

WORKED Example 14
Find the volume of the gure drawn on the right.
12 cm 4 cm 6 cm 10 cm 3 cm

THINK
1

WRITE
4 cm 12 cm A1 A2 10 cm

Divide the front face into two rectangles.

2 3 4 5 6

Calculate the area of each. Add the areas together to nd the value of A. Write the formula. Substitute A = 84 and h = 3. Calculate.

A1 = 4 12 = 48 cm2 A = 48 + 36 = 84 cm2 V=Ah = 84 3 = 252 cm3

6 cm

A2 = 6 6 = 36 cm2

If the shape is not a prism, you may need to divide it into two or more regular 3dimensional shapes. You could then calculate the volume by nding the volume of each shape separately. You will need to use important volume formulas that appear on the formula sheet: 2 3 -- r h -- Ah -- r Cone: V = 1 Cylinder: V = r 2h Pyramid: V = 1 Sphere: V = 4
3 3 3

WORKED Example 15
Calculate the volume of the gure drawn on the right, correct to 2 decimal places.
2.4 cm 1.2 cm

THINK
1 2 3 4

WRITE

The shape is a cylinder with a hemisphere on top. Write down the formula for the volume of a cylinder. Substitute r = 1.2 and h = 2.4. Calculate the volume of the cylinder.

V = r 2h V = (1.2)2 2.4 V = 10.857 cm3

Continued over page

64

Maths Quest General Mathematics HSC Course

THINK
5

WRITE
3 -- r 2 V= 4

Write down the formula for the volume of a hemisphere. (This is the formula for the volume of a sphere divided by 2.) Substitute r = 1.2. Calculate the volume of the hemisphere. Add the two volumes together.

6 7

V=

4 -3

(1.2)3 2

V = 3.619 cm3 Volume = 10.857 + 3.619 Volume = 14.48 cm3

In many cases a volume question may be presented in the form of a practical problem.

WORKED Example 16
A water storage tank is in the shape of a cube of side length 1.8 m, surmounted by a cylinder of diameter 1 m with a height of 0.5 m. Calculate the capacity of the tank, correct to the nearest 100 litres. THINK
1

WRITE
0.5 m 1m

Draw a diagram of the water tank.

1.8 m
2

Calculate the volume of the cube using the formula V = s3. Calculate the volume of the cylinder using the formula V = r 2h. Add the volumes together. Calculate the capacity of the tank using 1 m3 = 1000 L. Give an answer in words.

V = s3 V = 1.83 V = 5.832 m3 V = r 2h V = 0.52 0.5 V = 0.393 m3 Volume = 5.832 + 0.393 Volume = 6.225 m3 Capacity = 6.225 1000 Capacity = 6225 L The capacity of the tank is approximately 6200 litres.

Chapter 2 Further applications of area and volume

65

remember
1. To nd the volume of any prism, use the formula V = A h, where A is the area of the base and h is the height. 2. Important volume formulas: 2 -- r h Cone: V = 1 Cylinder: V = r 2h 3 3 1 - Ah -- r Pyramid: V = -Sphere: V = 4 3 3 where r = radius, h = perpendicular height, A = area of base 3. For other shapes, calculate the volume of each part of the shape separately, then add together each part at the end. 4. Remember to begin a worded or problem question with a diagram and nish with a word answer.

2E
WORKED

Volume of composite solids

6 cm

Example

18 cm

14

1 Look at the gure drawn on the right. a Find the area of the front face. b Use the formula V = A h to calculate the volume of the prism.

2.6
Volume of cubes and rectangular prisms 5 cm

SkillS

HEET

20 cm

4 cm

2 Calculate the volume of each of the gures drawn below. a b c 5 cm

25 cm 12 cm 4 cm 15 cm 10 cm

12 cm

20 cm

5 cm 4 m 0.5 m

20 cm

12 cm 40 cm 3 cm

0.7 m

2.3 m 0.4 m 2.1 m

0.6 m 5m 1m 1.5 m 2m 1.5 m

WORKED

Example

15

3 Consider the gure on the right. The shape consists of a cube with a square pyramid on top. a What is the volume of the cube? b What is the volume of the square pyramid? c What is the total volume of this gure?

2.7
Volume of triangular prisms

SkillS

HEET

2m

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Maths Quest General Mathematics HSC Course

4 The gure on the right is a cylinder with a cone mounted on top. a Calculate the volume of the cylinder, correct to the nearest cm3. b Calculate the volume of the cone, correct to the nearest cm3. c What is the total volume of the gure?
12 cm

40 cm

50 cm

SkillS

HEET

2.8
Volume of a cylinder

5 Calculate the volume of each of the gures drawn below, correct to 1 decimal place. a b c
3 cm 34 cm 5 cm r r =12 cm 50 cm

SkillS

HEET

2.9
Volume of a sphere

6 multiple choice Which of the gures drawn below is not a prism? A B

7 multiple choice The volume of the gure drawn on the right is closest to: A 718 cm3 B 1437 cm3 3 C 2155 cm D 2873 cm3 8 A sh tank is in the shape of a rectangular prism. The base measures 45 cm by 25 cm. The tank is lled to a depth of 15 cm. a Calculate the volume of water in the tank in cm3. b Given that 1 cm3 = 1 mL calculate, in litres, the amount of water in the tank.

14 cm 7 cm

Chapter 2 Further applications of area and volume

67

WORKED

Example

16

9 A hemispherical wine glass of radius 2.5 cm is joined to a cylinder of radius 1 cm and height 5 cm. The glass then rests on a solid base. a Draw a diagram of the wine glass. b Calculate the capacity of the glass, to the nearest 10 mL. c How many glasses of wine can be poured from a 1 litre bottle? 10 The gure on the right is the cross-section of a concrete pipe used as a sewage outlet. a Calculate the area of a cross-section of the pipe, correct to 2 decimal places. b Calculate the amount of concrete needed to make a 10 m length of this pipe.
3m 2.5 m

11 A commemorative cricket ball has a diameter of 7 cm. It is to be preserved in a cubic case that will allow 5 mm on each side of the ball. a What will the side length of the cubic case be? b Calculate the amount of empty space inside the case, to the nearest whole number. c Calculate the percentage of space inside the case occupied by the ball, to the nearest whole number. 12 A diamond is cut into the shape of two square-based pyramids as shown on the right. Each mm3 of the diamond has a mass of 0.04 g. Calculate the mass of the diamond.
6 mm

2.10 SkillS
Volume of a pyramid

HEET

6 mm

Maximising volume
You have been given a piece of sheet metal that is in the shape of a square with a side length of 2 m. The corners are to be cut and 9 cm the sides bent upwards to form a rectangular prism, as shown in the gure on the right. 3 cm 1 If a square of side length 1 cm is cut from each corner, what will be the length and width of the rectangular prism? 2 What will be the volume of this rectangular prism? 3 What will be the volume of the prism if a square of side length 2 cm is cut from each corner? 4 Find the size of the square to be cut from each corner that will make a prism of maximum volume. This exercise can be modelled using a spreadsheet or a graphics calculator.

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Maths Quest General Mathematics HSC Course

2
1 Calculate the area of a circle with a diameter of 8.6 cm, correct to 1 decimal place. 2 Calculate the area of the annulus shown below, correct to 2 decimal places.
9 cm 3 cm

3 Calculate the area of the sector below, correct to 1 decimal place.

4 Calculate the area of the gure below.

10 cm 29 cm

13.2 cm 85 28 cm

5 Calculate the shaded area in the gure drawn below, correct to 2 decimal places.

6 Use Simpsons rule to approximate the area shown below.

43 m

70 m 21 m 4.6 cm 70 m 9.7 cm 32 m

7 Calculate the surface area of a closed cylinder with a radius of 10 cm and a height of 23 cm. Give your answer correct to the nearest whole number. 9 Calculate the volume of the prism drawn below.

8 Calculate the surface area of a sphere with a radius of 1.3 m. Give your answer correct to 3 decimal places. 10 Calculate the volume of the solid below, correct to the nearest whole number.
4 cm

20.3 cm

13.4 cm

9.1 cm 13.7 cm

9 cm 8 cm

Chapter 2 Further applications of area and volume

69

Error in measurement
As we saw in the preliminary course, all measurements are approximations. The degree of accuracy in any measurement is restricted by the accuracy of the measuring device and the degree of practicality. We have previously seen that the maximum error in any measurement is half of the smallest unit of measurement. This error is compounded when further calculations such as surface area or volume are made.

WORKED Example 17
In the rectangular prism on the right, the length, breadth and height have been measured, correct to the nearest centimetre. a Calculate the volume of the rectangular prism. b Calculate the greatest possible error in the volume. THINK a Calculate the volume of the rectangular prism. WRITE a V=lwh = 20 15 8 = 2400 cm3 b Smallest possible dimensions: l = 19.5, w = 14.5, h = 7.5 V=lwh = 19.5 14.5 7.5 = 2120.625 cm3 Largest possible dimensions: l = 20.5, w = 15.5, h = 8.5 V=lwh = 20.5 15.5 8.5 = 2700.875 cm3 Maximum error = 2700.875 2400 Maximum error = 300.875 cm3
8 cm 15 cm 20 cm

1 2

Write the smallest possible dimensions of the prism. Calculate the smallest possible volume. Write the largest possible dimensions of the prism. Calculate the largest possible volume. Calculate the maximum error.

3 4

As can be seen in the above example, a possible error of 0.5 cm in the linear measurement compounds to an error of 300.875 cm3 in the volume measurement. Mismeasurements that are made will compound all further calculations.

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Maths Quest General Mathematics HSC Course

WORKED Example 18
A swimming pool is built in the shape of a rectangular prism with a length of 10.2 m, a width of 7.5 m and a depth of 1.5 m. The oor and the sides of the pool need to be cemented. a Calculate the area that is to be cemented. b The concreter mismeasured the length of the pool as 9.4 m. Calculate the error in the area calculation. c Calculate the percentage error (correct to 1 decimal place) in the area calculation. THINK a
1 2 3 4

WRITE a Area of oor = 10.2 7.5 Area of oor = 76.5 m2 Area of ends = 7.5 1.5 Area of ends = 11.25 m2 Area of sides = 10.2 1.5 Area of sides = 15.3 m2 Total area = 76.5 + 2 11.25 + 2 15.3 Total area = 129.6 m2 b Area of oor = 9.4 7.5 Area of oor = 70.5 m2 Area of ends = 7.5 1.5 Area of ends = 11.25 m2 Area of sides = 9.4 1.5 Area of sides = 14.1 m2 Total area = 70.5 + 2 11.25 + 2 14.1 Total area = 121.2 m2 Error = 129.6 121.2 Error = 8.4 m2 8.4 - 100% c Percentage error = -----------129.6 Percentage error = 6.5%

Calculate the area of the pool oor. Calculate the area of the ends. Calculate the area of the sides. Calculate the total area to be cemented. Use the incorrect measurement to repeat all the above calculations.

Find the difference between the two answers.

c Write the error as a percentage of the correct answer.

remember
1. All measurements are approximations. The accuracy of any measurement is limited by the instrument used and the most practical degree of accuracy. 2. The maximum error in any linear measurement is half the smallest unit used. 3. Any error made in linear measurement will compound when used in further calculations such as those for surface area or volume.

Chapter 2 Further applications of area and volume

71

2F
WORKED

Error in measurement
12 cm

Example

17

1 In the gure on the right each measurement has been taken to the nearest centimetre. a Calculate the volume of the gure. b Calculate the maximum error in the volume calculation.

2.11 SkillS
6 cm 16 cm Error in linear measurement

HEET

2 The radius of a circle is measured as 7.6 cm, correct to 1 decimal place. a What is the maximum possible error in the measurement of the radius? b Calculate the area of the circle. Give your answer correct to 1 decimal place. c Calculate the maximum possible error in the area of the circle. d Calculate the maximum possible error in the area of the circle as a percentage of the area. 3 A cube has a side length of 16 mm, correct to the nearest millimetre. a Calculate the volume of the cube. b Calculate the smallest possible volume of the cube. c Calculate the largest possible volume of the cube. d Calculate the maximum possible percentage error in the volume of the cube. e Calculate the surface area of the cube. f Calculate the smallest possible surface area of the cube. g Calculate the largest possible surface area of the cube. h Calculate the maximum possible percentage error in the surface area of the cube. 4 A cylinder has a radius of 4 cm and a height of 6 cm with each measurement being taken correct to the nearest centimetre. a Calculate the volume of the cylinder (correct to the nearest whole number). b Calculate the smallest possible volume of the cylinder (correct to the nearest whole number). c Calculate the largest possible volume of the cylinder (correct to the nearest whole number). d Calculate the greatest possible percentage error in the volume of the cylinder. 5 For the cylinder in question 4, calculate the greatest possible percentage error in the surface area of the cylinder. 6 The radius of a sphere is 1.4 m with the measurement taken correct to 1 decimal place. a Calculate the volume of the sphere, correct to 1 decimal place. b Calculate the maximum possible error in the volume of the sphere. c Calculate the maximum percentage error in the volume. d Calculate the surface area of the sphere, correct to 1 decimal place. e Calculate the maximum possible error in the surface area of the sphere. f Calculate the maximum percentage error in the surface area.

72
WORKED

Maths Quest General Mathematics HSC Course

Example

18

7 An open cylindrical water tank has a radius of 45 cm and a height of 60 cm. a Calculate the capacity of the tank, in litres (correct to the nearest whole number). b If the tanks radius is given as 50 cm, correct to the nearest 10 cm, calculate the error in the capacity of the tank. c Calculate the percentage error in the capacity of the tank. 8 A rectangular prism has dimensions 56 cm 41 cm 17 cm. a Calculate the volume of the prism. b Calculate the surface area of the prism. c If the dimensions are given to the nearest 10 cm, what will the dimensions of the prism be given as? d Calculate the percentage error in the volume when the dimensions are given to the nearest 10 cm. e Calculate the percentage error in the surface area when the dimensions are given to the nearest 10 cm. 9 The four walls of a room are to be painted. The length of the room is 4.1 m and the width is 3.6 m. Each wall is 1.8 m high. a Calculate the area to be painted. b One litre of paint will paint an area of 2 m2. Each wall will need two coats of paint. Calculate the number of litres of paint required to complete this job. c Karla incorrectly measures the length of the room to be 3.9 m. If Karla does all her calculations using this incorrect measurement, how many litres will she be short of paint at the end of the job? 10 The dimensions of a rectangular house are 16.6 m by 9.8 m. a Simon takes the dimensions of the house to the nearest metre for all his calculations. What dimensions does Simon use? b Simon plans to oor the house in slate tiles. What is the area that needs to be tiled? c The tiles cost $27.50/m2 and Simon buys an extra 10% to allow for cutting and breakage. Calculate the cost of the tiles. d How much extra has Simon spent than would have been necessary had he used the original measurements of the house?

Work

T SHEE

2.2

Chapter 2 Further applications of area and volume

73

summary
Area of parts of the circle
The area of a circle can be calculated using the formula A = r 2. The area of a sector is found by multiplying the area of the full circle by the fraction of the circle occupied by the sector. This is calculated by looking at the angle that the sector makes with the centre. An annulus is the area between two circles. The area is calculated by subtracting the area of the smaller circle from the area of the larger circle or by using the formula A = (R2 r2) , where R is the radius of the large circle and r is the radius of the small circle. The area of an ellipse is calculated using the formula A = ab, where a is the length of the semi-major axis and b is the length of the semi-minor axis.

Area of composite gures

The area of a composite gure is calculated by dividing the gure into two or more regular gures. When calculating the area of a composite gure, some side lengths will need to be calculated using Pythagoras theorem.

Simpsons rule
Simpsons rule is used to nd an approximation for an irregular area. h - (d + 4dm + dl) The formula for Simpsons rule is A -3 f .

To obtain a better approximation for an area, Simpsons rule can be applied twice. This is done by dividing the area in half and applying Simpsons rule separately to each half.

Surface area of cylinders and spheres

The surface area of a closed cylinder is found by using the formula SA = 2 r 2 + 2 rh . If the cylinder is an open cylinder, the surface area is found using SA = r 2 + 2 rh. The surface area of a sphere is calculated using the formula SA = 4 r 2 .

Volume of composite solids

The volume of solid prisms is calculated using the formula V = A h.
2 -- r h The volume of a cone is found using the formula V = 1

. .

The volume of a cylinder is found using the formula V = r 2h

74

Maths Quest General Mathematics HSC Course

3 -- r The volume of a sphere is found using the formula V = 4

. .

-- Ah The volume of a pyramid is found using the formula V = 1 3

Other solids have their volume calculated by dividing the solid into regular solid shapes.

Error in measurement
All measurements are approximations. The maximum error in any measurement is half the smallest unit used. Any error in a measurement will compound when further calculations using the measurement need to be made.

Chapter 2 Further applications of area and volume

75

CHAPTER review
1 Calculate the area of each of the circles below. Give each answer correct to 1 decimal place. a b c
3.7 cm 52 mm 1.7 m

2A

2 Calculate the area of each of the gures below. Give each answer correct to 1 decimal place. a b c
92 mm 237 12.5 cm 4.8 m

2A

30

3 Calculate the area of each of the annuluses below. Give each answer correct to 1 decimal place. a b c
81 mm 94 mm 3.7 m 1.3 m 34 cm 17 cm

2A

4 Calculate the area of each of the ellipses below, correct to 1 decimal place. a b c
30 mm 45 mm 9.2 m 11.4 m 7 cm 3.6 cm

2A

5 Calculate the area of the gure below.

35 cm 15 cm

2B

10 cm

10 cm 12 cm

76
2B

Maths Quest General Mathematics HSC Course

6 Calculate the area of each of the gures below. Where appropriate, give your answer correct to 2 decimal places. a 0.7 m b c
1.5 cm 3 cm 0.9 m 1.5 cm 6 cm 4.1 m

36 cm

3.9 m

2C 2C

50 m

7 Use Simpsons rule to approximate the area on the right.

13 m 42 m 42 m

21 m

8 Use Simpsons rule to nd an approximation for each of the areas below. a b c

36 m 14 m 31 m 96 m 24 m

2m 57 m

29 m

30 m

42 m

2C

27 m

9 By dividing the area shown on the right into two sections, use Simpsons rule to nd an approximation for the area.

50 m 30 m 30 m 25 m 19 m 11 m

2C

33 m

10 Use Simpsons rule twice to nd an approximation for the area on the right.

62 m

38 m

9m 15 m 15 m 15 m 15 m

2D

11 Calculate the surface area of each of the closed cylinders drawn below, correct to 1 decimal place. a b c
10 cm 7 cm 4 cm 25 cm 60 cm 1.1 m

44 m

23 m

57 m

62 m

Chapter 2 Further applications of area and volume

77
2D 2D 2E

12 Calculate the surface area of an open cylinder with a diameter of 9 cm and a height of 15 cm. Give your answer correct to the nearest whole number. 13 Calculate the surface area of a sphere with: a a radius of 5 cm b a radius of 2.4 m Give each answer correct to the nearest whole number. 14 Calculate the volume of the solid drawn on the right.
3.1 m 0.5 m

c a diameter of 156 mm.

2.7 m

0.6 m

15 Calculate the volume of each of the solids drawn below. Where necessary, give your answer correct to the nearest whole number. a b c 12 cm
3 cm 19 cm 12 cm 20 cm 9 cm 15 cm 3 cm 3 cm 10 cm

1.9 m

2E

17 cm

22 cm

40 cm

10 cm

16 Calculate the volume of the gure drawn on the right, correct to 2 decimal places.
15 cm 9 cm

2E

17 A sphere has a diameter of 16 cm when measured to the nearest centimetre. a State the maximum error made in the measurement of the radius. b Calculate the volume of the sphere. Answer correct to the nearest whole number. c Calculate the maximum percentage error in the volume of the sphere. 18 An aluminium soft drink can has a diameter of 8 cm and a height of 10 cm. a Calculate the capacity of the can, in millilitres, correct to the nearest 10 millilitres. b The machine that cuts the aluminium for the can is mistakenly set to 12 cm. Calculate the percentage error in the capacity of the can (correct to the nearest whole number).

2F 2F

78

Maths Quest General Mathematics HSC Course

Practice examination questions

1 multiple choice Which of the following calculations will correctly give the area of the ellipse drawn on the right? A 6.22 B 8.52 C 10.82 D 10.8 6.2 2 multiple choice The eld drawn on the right is to have its area approximated by two applications of Simpsons rule. The value of h is: A 16 B 20 C 40 D 80 3 multiple choice The gure drawn on the right is an open cylinder. Which of the calculations below will correctly give the surface area of the cylinder? A 52 + 2 5 20 B 2 52 + 2 5 20 C 102 + 2 10 20 D 2 102 + 2 10 20 4 multiple choice A closed cylinder is measured as having a radius of 1.2 m and a height of 1.4 m. The maximum error in the calculation of the surface area is: A 1.2 m2 B 1.5 m2 C 1.6 m2 D 19.6 m2 5 The gure on the right shows a section of a concrete drainage pipe. a Calculate the area of the annulus, correct to 1 decimal 2.5 m place. b Calculate the volume of concrete needed to make a 1.5 m 5 m length of this pipe (correct to 1 decimal place). c Calculate the volume of water that will ow through the 5 m length of the pipe (in litres, to the nearest 100 L). d Calculate the surface area of a 5 m section of pipe (correct to the nearest m2). (Hint: Include the area of the inside of the pipe.) 6 The diagram on the right shows the cross-section of a river. a Use two applications of Simpsons rule to nd the approximate area of the rivers cross-section. b If the river ows with this cross-section for approximately 800 m, calculate the volume of the river. c The length of the river has been approximated to the nearest 100 m. Calculate the maximum percentage error in calculating this volume.
60 m 5.1 m 9.2 m 4.9 m 6.2 mm 10.8 mm

30 m

25 m 20 cm

10 m

80 m

10 cm

CHAPTER

test yourself

15 m

20 m