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WorkSHEET 6.3 Volume Name: ___________________________ 1 A sphere has a radius of 4.5 cm. Find, correct

to 1 decimal place: (a) the volume (b) the surface area.

(a) V =

V = ´ p (4.5)3

V = 381.7 cm3 (b) SA = 4pr2

SA = 4 ´ p ´ (4.5)2 SA = 254.5 cm2

2 How many litres of water could a cube of side length 10 m hold?

Side of cube is 1000 cm Volume of cube = l3 = 1 0003 = 1 000 000 000 cm3 1 cm3 is 1 mL 1000 mL in 1 litre So 1 000 000 000 cm3 is equivalent to 1 000 000 L of water.

3 Find the volumes of the following: (a) a cube of side length 75 cm (b) a rectangular prism 14 cm by 18 cm by

22 cm.

(a) V = l 3 V = 753 V = 421 875 cm3

(b) V = lwh V = 14 ´ 18 ´ 22 V = 5544 cm3

4 Find the volumes of the following (correct to the nearest whole number): (a) a sphere of diameter 15 cm (b) a cylinder of diameter 4 cm and height

8 cm.

(a) V = pr3

V = ´ p ´

V = 1767 cm3 (b) V = pr2h V = p ´ 22 ´ 8 V = 101 cm3

334 rp

34

4343

3152

æ öç ÷è ø

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5 Find the volumes of the following (correct to the nearest whole number): (a) a square-based pyramid with base length

12 m and vertical height 16 m (b) a cone with diameter of 6 cm and vertical

height of 7 cm.

(a) V = Ah

A = l 2 A = 122 A = 144 m2 V = ´ 144 ´ 16

A = 768 m3

(b) V = pr2h

A = ´ p ´ 32 ´ 7

A = 66 cm3

6 Four balls, each with diameter 7 cm, are placed in a cylinder. What is the smallest volume the cylinder could be to hold all four balls? How much unused space is in the cylinder?

Dimensions of cylinder: Diameter = 7 cm Length = 4 ´ 7 = 28 cm

Volume of cylinder = pr2h Volume of cylinder = p ´ 3.52 ´ 28 Volume of cylinder = 1078 cm3 Volume of space = pr2h - 4 ´ pr3

Volume of space = 1078 - ´ p ´ 3.53

Volume of space = 1078 - 718 Volume of space = 360 cm3

7 How much water is required to fill the swimming pool depicted in the figure below?

The pool is a prism so V = A ´ h The base is two semicircles (which make a whole circle) and a rectangle. A = pr2 + lw A = p ´ 32 + 10 ´ 6 A = 88.3 m2 V = A ´ h A = 88.3 ´ 1.5 A = 132.45 m3 Capacity = 132.45 ´ 1000 = 132 450 L

31

31

31

31

34

316

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8 A sphere has a volume of 500 cm3. What is the radius of the sphere?

Volume of sphere =

The radius of the sphere is about 4.9 cm.

9 A block of ice in the shape of a cube is made from 1 L of water. What is the length of the side of the ice cube?

Volume of cube = side3 1 L of water is equivalent to 1000 mL or 1000 cm3.

The ice cube has a side length of 10 cm.

10 A large water pipe is needed at a dam and is to be made out of concrete. The pipe needs to be 5 metres long, with an internal diameter of 2 metres. The concrete is to have a thickness of 0.05 metres. What volume of concrete is needed to make the pipe?

Recognise the shape as a “Pipe”, which is a Prism with an Annulus on each end. Inside annulus has a radius of 1 metre Outside annulus has a radius of 1.05 metre

𝑉!"#$% = 𝐴&'$( × 𝐻

𝑉!#)( = 𝐴*++,-,$ × 𝐻

𝑉!#)( = &𝐴&#. − 𝐴/%'--( × 𝐻

𝑉!#)( = &𝜋𝑟&#.0 − 𝜋𝑟/%'--0 ( × 𝐻

𝑉!#)( = {𝜋 × 1.050 − 𝜋 × 10} × 5

𝑉!#)( = 1.61𝑚1

343rp

3

3

3

4 5003

50043375

375

4.9 cm

r

r

r

p

p

p

p

=

=

=

=

=

3

3

1000

100010 cm

l

l

=

==

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11 Find the EXACT area of the figure below:

Use Pythagoras’ theorem to find the length of the hypotenuse of the right-angled triangle. This will then be the diameter of the semi-circle.

Total area = 16 cm2 + 10 cm2

12 A square and a circle have the same area. If the circle has a radius of 3, what is the side length of the square?

𝐴/2,'"( = 𝐴3#"4-(

𝑠0 = 𝜋𝑟0

𝑠0 = 𝜋 × 30

𝑠0 = 28.27

𝑠 = 5.32

13 A square and a circle have the same area. If the circle has a radius of 3, what is the Exact side length of the square?

𝐴/2,'"( = 𝐴3#"4-(

𝑠0 = 𝜋𝑟0

𝑠0 = 𝜋 × 30

𝑠0 = 9𝜋

𝑠 = 3√𝜋

Hypot2 = 82 + 42

= 64+16= 80

Hypot = 80 = 4 5

2

1Area of triangle = 8 42

16 cm

´ ´

=

Area of semi-circle = 12×π ×

4 52

"

#$$

%

&''

2

=12×π ×

16×54

=10π cm2

π

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14 A square and a circle have the same area. What is the Exact side length of the square in terms of the radius of the circle?

𝐴/2,'"( = 𝐴3#"4-(

𝑠0 = 𝜋𝑟0

𝑠 = :𝜋𝑟0

𝑠 = √𝜋 × :𝑟0

𝑠 = √𝜋𝑟

15 Who cut the cheese?

If the piece of cheese has a diameter of 12cm and a height of 4cm and the section cut out has an angle of 205, what is the Volume of the piece of cut cheese?

𝑉!"#$% = 𝐴&'$( × 𝐻

=𝜃360𝜋𝑟

0 × 𝐻

=20360 × 𝜋 × 6

0 × 4

= 25.13𝑐𝑚1

16 Who cut the cheese? Now calculate the EXACT volume of the piece of cut cheese? ** Note the use of the word EXACT: that means you need to give your answer in terms of 𝜋!

𝑉!"#$% = 𝐴&'$( × 𝐻

=𝜃360𝜋𝑟

0 × 𝐻

=20360 × 𝜋 × 6

0 × 4

= 8𝜋𝑐𝑚1

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17 Cheesecakes come in two sizes;

Each cheesecake is the same thickness. The smaller cake will serve eight people. Determine how many people the larger cheesecake will serve.

You eat cheesecake by Volume:

𝑉 = 𝜋𝑟0ℎ Small cheesecake:

𝑉 = 𝜋 × 100ℎ = 100𝜋ℎ

Big cheesecake:

𝑉 = 𝜋 × 150ℎ = 225𝜋ℎ

Apply finding x ratios: If 100𝜋ℎ serves 8 people, then 225𝜋ℎ serves ?

100𝜋ℎ: 8 = 225𝜋ℎ: 𝑥

8100𝜋ℎ =

𝑥225𝜋ℎ

8

100𝜋ℎ ×225𝜋ℎ1 =

𝑥225𝜋ℎ ×

225𝜋ℎ1

18 = 𝑥

So, the larger cheesecake will serve 18 people.