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Volume, SM-Bank 166

The cube and cylinder below both have the same volume.
  

  1. Calculate the volume of the cube in cubic centimetres.  (2 marks)

  2. Calculate the height of the cylinder, \(\large h\), in centimetres. Give your answer correct to 1 decimal place.  (2 marks)
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a.    \(64\ \text{cm}^3\)

b.    \(5.1\ \text{cm (1 d.p.)}\)

Show Worked Solution
 

a.    \(V\) \(=l\times b\times h\)
    \(=4^3\)
    \(=64\)

 
\(\therefore\ \text{The volume of the cube is 64 cm}^3\)
 

b.    \(\text{Diameter = 4 cm}\ \longrightarrow\ \text{Radius = 2 cm}\)

\(V\) \(=\pi r^2h\)
\(64\) \(=\pi\times 2^2\times h\)
\(64\) \(=4\pi h\)
\(\therefore\ h\) \(=\dfrac{64}{4\pi}\)
  \(=5.092\dots\approx 5.1\ \text{(1 d.p.)}\)

 
\(\therefore\ \text{The height of the cylinder is approximately 5.1 cm}\)

Volume, SM-Bank 165

The cylinder and rectangular prism below both have the same volume.
 

  1. Calculate the volume of the cylinder in cubic centimetres, correct to 2 decimal places.  (2 marks)

  2. Calculate the length of the side labelled \(\large x\), in the rectangular prism. Give your answer correct to 1 decimal place.  (2 marks)
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a.    \(314.16\ \text{cm}^3\ \text{(2 d.p.)}\)

b.    \(9.8\ \text{cm (1 d.p.)}\)

Show Worked Solution
 

a.    \(\text{Diameter = 10 cm }\longrightarrow\ \text{Radius = 5 cm}\)

\(V\) \(=\pi r^2h\)
  \(=\pi\times 5^2\times 4\)
  \(=314.159\dots\)
  \(\approx 314.16\ \text{(2 d.p.)}\)

 
\(\therefore\ \text{The volume of the cylinder is approximately 314.16 cm}^3\)
 

b.    \(V\) \(=l\times b\times h\)
  \(314.16\) \(=8\times x\times 4\)
  \(314.16\) \(=32x\)
  \(\therefore\ x\) \(=\dfrac{314.16}{32}\)
    \(=9.8175\)
    \(\approx 9.8\ \text{(1 d.p.)}\)

 
\(\therefore\ \text{The side labelled }x\ \text{is approximately 9.8 cm in length}\)

Volume, SM-Bank 161

A right cylinder has a height of 100 millimetres and a radius of 1.1 millimetres. Calculate the volume of the cylinder, giving your answer as an exact value in terms of \(\pi\).  (2 marks)

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\(121\large\pi\ \)\(\text{mm}^3\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
  \(=\pi\times 1.1^2\times 100\)
  \(=121\large\pi\)

 
\(\therefore\ \text{The exact volume of the cylinder is }121\large\pi\ \)\(\text{mm}^3\)

Volume, SM-Bank 160

A right cylinder has a height of 7 metres and a radius of 4 metres. Calculate the volume of the cylinder, giving your answer as an exact value in terms of \(\pi\).  (2 marks)

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\(112\large\pi\ \)\(\text{m}^3\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
  \(=\pi\times 4^2\times 7\)
  \(=112\large\pi\)

 
\(\therefore\ \text{The exact volume of the cylinder is }112\large\pi\ \)\(\text{m}^3\)

Volume, SM-Bank 159

A right cylinder has a volume of \(11\ 451\) cubic metres. Calculate the radius of the cylinder if the height is 45 metres.

Give your answer to the nearest whole centimetre.  (2 marks)

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\(9\ \text{m}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(11\ 451\) \(=\pi\times r^2\times 45\)
\(11\ 451\) \(=10\pi\times r^2\)
\(r^2\) \(=\dfrac{11\ 451}{45\pi}\)
\(r^2\) \(=80.999\dots\)
\(r\) \(=\sqrt{80.999}=8.999\dots\)
\(r\) \(\approx 9\ \text{m (nearest whole m)}\)

 
\(\therefore\ \text{The radius of the cylinder is approximately 9 m}\)

Volume, SM-Bank 158

A right cylinder has a volume of 22 cubic metres. Calculate the diameter of the cylinder if the height is 7 metres.

Give your answer to the nearest whole metre.  (3 marks)

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\(2\ \text{m}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(22\) \(=\pi\times r^2\times 7\)
\(22\) \(=7\pi\times r^2\)
\(r^2\) \(=\dfrac{22}{7\pi}\)
\(r^2\) \(=1.000\dots\)
\(r\) \(=\sqrt{1.000}=1.000\dots\)
\(r\) \(\approx 1\ \text{m (nearest whole m)}\)

 
\(\therefore\ \text{The diameter of the cylinder is approximately 2 m}\)

Volume, SM-Bank 157

A right cylinder has a volume of 8482.3 cubic millimetres. Calculate the diameter of the cylinder if the height is 12 millimetres.

Give your answer to the nearest whole millimetre.  (3 marks)

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\(30\ \text{mm}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(8482.3\) \(=\pi\times r^2\times 12\)
\(8482.3\) \(=12\pi\times r^2\)
\(r^2\) \(=\dfrac{8482.3}{12\pi}\)
\(r^2\) \(=224.999\dots\)
\(r\) \(=\sqrt{224.999}=14.999\dots\)
\(r\) \(\approx 15\ \text{mm (nearest whole mm)}\)

 
\(\therefore\ \text{The diameter of the cylinder is approximately 30 mm}\)

Volume, SM-Bank 156

A right cylinder has a volume of \(10\ 178.76\) cubic centimetres. Calculate the radius of the cylinder if the height is 10 centimetres.

Give your answer to the nearest whole centimetre.  (2 marks)

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\(18\ \text{cm}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(10\ 178.76\) \(=\pi\times r^2\times 10\)
\(10\ 178.76\) \(=10\pi\times r^2\)
\(r^2\) \(=\dfrac{10\ 178.76}{10\pi}\)
\(r^2\) \(=323.999\dots\)
\(r\) \(=\sqrt{323.9999}=17.999\dots\)
\(r\) \(\approx 18\ \text{cm (nearest whole centimetre)}\)

 
\(\therefore\ \text{The radius of the cylinder is approximately 18 cm}\)

Volume, SM-Bank 155

A right cylinder has a volume of 50.27 cubic millimetres. Calculate the height of the cylinder if the radius is 2 millimetres.

Give your answer to the nearest whole millimetre.  (2 marks)

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\(4\ \text{mm}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(50.27\) \(=\pi\times 2^2\times h\)
\(50.27\) \(=4\pi\times h\)
\(h\) \(=\dfrac{50.27}{4\pi}\)
  \(=4.00\dots\)
  \(\approx 4\ \text{mm (nearest whole millimetre)}\)

 
\(\therefore\ \text{The height of the cylinder is approximately 4 mm}\)

Volume, SM-Bank 154

A right cylinder has a volume of 4021 cubic metres. Calculate the height of the cylinder if the radius is 8 cm.

Give your answer to the nearest whole metre.  (2 marks)

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\(20\ \text{m}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(4021\) \(=\pi\times 8^2\times h\)
\(4021\) \(=64\pi\times h\)
\(h\) \(=\dfrac{4021}{64\pi}\)
  \(=19.998\dots\)
  \(\approx 20\ \text{m (nearest whole metre)}\)

 
\(\therefore\ \text{The height of the cylinder is approximately 20 m}\)

Volume, SM-Bank 153

A right cylinder has a volume of 2827 cubic centimetres. Calculate the height of the cylinder if the radius is 10 cm.

Give your answer to the nearest whole centimetre.  (2 marks)

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\(9\ \text{cm}\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
\(2827\) \(=\pi\times 10^2\times h\)
\(2827\) \(=100\pi\times h\)
\(h\) \(=\dfrac{2827}{100\pi}\)
  \(=8.998\dots\)
  \(\approx 9\ \text{cm (nearest whole centimetre)}\)

 
\(\therefore\ \text{The height of the cylinder is approximately 9 cm}\)

Volume, SM-Bank 142

Calculate the volume of the cylinder below in cubic metres.  Give your answer correct to 2 decimal places.  (2 marks)
 

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\(508\ 938.01\ \text{m}^3\)

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\(\text{diameter =180 m }\rightarrow\text{ radius = 90 m}\)

\(V\) \(=\pi r^2h\)
  \(=\pi\times 90^2\times 20\)
  \(=508\ 938.009\dots\)
  \(\approx 508\ 938.01\ \text{m}^3\ (\text{2 d.p.})\)

Volume, SM-Bank 141

Calculate the volume of the cylinder below in cubic metres.  Give your answer correct to 1 decimal place.  (2 marks)
 

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\(23.5\ \text{m}^3\)

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\(\text{diameter = 2.4 m }\rightarrow\text{ radius = 1.2 m}\)

\(V\) \(=\pi r^2h\)
  \(=\pi\times 1.2^2\times 5.2\)
  \(=23.524\dots\)
  \(\approx 23.5\ \text{m}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 140

Calculate the volume of the cylinder below in cubic centimetres.  Give your answer correct to 2 decimal places.  (2 marks)
 

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\(1.78\ \text{cm}^3\)

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\(\text{diameter = 1.8 cm }\rightarrow\text{ radius = 0.9 cm}\)

\(V\) \(=\pi r^2h\)
  \(=\pi\times 0.9^2\times 0.7\)
  \(=1.781\dots\)
  \(\approx 1.78\ \text{cm}^3\ (\text{2 d.p.})\)

Volume, SM-Bank 139

Calculate the volume of the cylinder below in cubic metres.  Give your answer correct to 2 decimal places.  (2 marks)
 

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\(9424.78\ \text{m}^3\)

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\(V\) \(=\pi r^2h\)
  \(=\pi\times 10^2\times 30\)
  \(=9424.777\dots\)
  \(\approx 9424.78\ \text{m}^3\ (\text{2 d.p.})\)

Volume, SM-Bank 138

Calculate the volume of the cylinder below in cubic metres.  Give your answer correct to the nearest cubic metre.  (2 marks)
 

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\(5\ \text{m}^3\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
  \(=\pi\times 0.6^2\times 4\)
  \(=4.523\dots\)
  \(\approx 5\ \text{m}^3\ (\text{nearest cubic metre})\)

Volume, SM-Bank 137

Calculate the volume of the cylinder below in cubic millimetres.  Give your answer correct to one decimal place.  (2 marks)
 

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\(16\ 964.6\ \text{mm}^3\)

Show Worked Solution
\(V\) \(=\pi r^2h\)
  \(=\pi\times 15^2\times 24\)
  \(=16\ 964.600\dots\)
  \(\approx 16\ 964.6\ \text{mm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 012

Tennis balls are packaged in cylindrical containers.

Frank purchases a container of tennis balls that holds three standard tennis balls, stacked one on top of the other.

This container has a radius of 3.4 cm and a height of 20.4 cm, as shown in the diagram below.
 

  1. Calculate the volume of the cylinder in cubic centimetres, correct to one decimal place.  (2 marks)


  2. If one tennis ball has a volume of 164.6 cm³, how much unused volume, in cubic centimetres, surrounds the tennis balls in this container?
    Round your answer to the nearest whole number.  (1 mark)

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a.    \(740.9\ \text{cm}^3\ \text{(to 1 d.p.)}\)

b.    \(247\ \text{cm}^3\ \text{(nearest cm}^3 \text{)}\)

Show Worked Solution
a.    
\(\text{Volume}\) \(=Ah\)
    \(=\pi\times 3.4^2\times 20.4\)
    \(= 740.86\dots\)
    \(=740.9\ \text{cm}^3\ \text{(to 1 d.p.)}\)

 

b.   
\(\text{Unused volume}\) \(=\text{cylinder volume}-\text{volume of balls}\)
    \(= 740.9-3\times 164.6\)
    \(= 247.1\)
    \(=247\ \text{cm}^3\ \text{(nearest cm}^3 text{)}\)

Volume, SM-Bank 004 MC

GEOMETRY, FUR1 2008 VCAA 4 MC

The solid cylindrical rod shown above has a volume of 490.87 cm3. The length is 25.15 cm.

The radius (in cm) of the cross-section of the rod, correct to one decimal place, is

  1. \(2.5\)
  2. \(5.0\)
  3. \(6.3\)
  4. \(19.6\)
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\(A\)

Show Worked Solution

\(V=\pi r^2h\)

\(\text{Where length} =h = 25.15\ \text{cm,}\ V=490.87\ \text{cm}^3\)

\(\therefore\ 490.87\) \(=\pi\times r^2\times 25.15\)
\(r^2\) \(=\dfrac{490.87}{\pi\times 25.15}\)
  \(= 6.2126\dots\)
\(\therefore\ r\) \(= 2.492\dots\ \text{cm}\)
  \(\approx 2.5\ \text{cm (1 d.p.)}\)

 
\(\Rightarrow A\)