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Classifying Data, SM-Bank 015 MC

Organisers were choosing relay teams for the regional athletics carnival.

They asked the following question.

'What were the times for the 15 Years boys relay teams at zone athletics carnivals?'

How would the responses be classified?

  1. Categorical, ordinal
  2. Categorical, nominal
  3. Numerical, discrete
  4. Numerical, continuous
Show Answers Only

\(D\)

Show Worked Solution

\(\text{The times recorded are measurements}\)

\(\text{so all numbers on the scale are possible}\)

\(\longrightarrow\ \text{Numerical, continuous.}\) 

\(\Rightarrow D\)

Classifying Data, SM-Bank 014 MC

Which of the following is an example of categorical nominal data?

  1. Gold, Silver, Bronze medals
  2. Small, Medium, Large
  3. French Bulldog, Poodle, Cavoodle
  4. First place, Second place
Show Answers Only

\(C\)

Show Worked Solution

\(\text{French Bulldog, Poodle and Cavoodle are types of dogs (order not important) }\)

\(\longrightarrow\ \text{categorical nominal}\)

\(\Rightarrow C\)

Classifying Data, SM-Bank 013

Which of the following is an example of categorical ordinal data?

  1. Pink, Blue, Mauve
  2. Small, Medium, Large
  3. Apple, Pear, Orange
  4. Labrador, Beagle, Poodle
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Small, Medium, Large are categories with an order }\longrightarrow\ \text{categorical ordinal}\)

\(\Rightarrow B\)

Classifying Data, SM-Bank 012

Which of the following is an example of numerical continuous data?

  1. The weights of babies born in a hospital during the month of May
  2. The types of cars sold at a car yard
  3. The number of funnel-web spiders collected by the Australian Reptile Park in 2023
  4. The colours used in the American flag
Show Answers Only

\(A\)

Show Worked Solution

\(\text{Weights are measurements (quantitative) }\longrightarrow\ \text{numerical continuous}\)

\(\Rightarrow A\)

Classifying Data, SM-Bank 011 MC

Which of the following is an example of numerical discrete data?

  1. The minimum temperature in each capital city on Monday morning
  2. The number of competitors in a triathlon
  3. The different types of snakes in a reptile display
  4. The lengths of earth worms in a garden
Show Answers Only

\(B\)

Show Worked Solution

\(\text{The number of competitors in a triathlon can be counted }\longrightarrow\ \text{numerical discrete}\)

\(\Rightarrow B\)

Classifying Data, SM-Bank 010

State whether the following categorical data is nominal or ordinal.

  1. Rating exam questions as easy or hard.  (1 mark)

  2. Listing the colours of cars passing the bus stop during period 3.  (1 mark)

  3. Recording dietary preferences such as vegan, gluten-free, vegetarian.  (1 mark)

  4. Receiving an A, B, C, D or E on a report card.  (1 mark)

Show Answers Only

a.    \(\text{Ordinal}\)

b.    \(\text{Nominal}\)

c.    \(\text{Nominal}\)

d.    \(\text{Ordinal}\)

Show Worked Solution

a.    \(\text{The data is ranked}\ \longrightarrow\ \text{Ordinal}\)

b.    \(\text{Colours are the names assigned}\ \longrightarrow\ \text{Nominal}\)

c.    \(\text{Preferences are the names assigned}\ \longrightarrow\ \text{Nominal}\)

d.    \(\text{Grades rank the data}\ \longrightarrow\ \text{Ordinal}\)

Classifying Data, SM-Bank 009

State whether the following data is categorical or numerical.  If numerical, state whether discrete or continuous.

  1. The distance thrown by the 15 years javelin champion.  (1 mark)

  2. The types of fruit used in a fruit salad.  (1 mark)

  3. The number of students travelling to the Year 7 camp by bus.  (1 mark)

Show Answers Only

a.    \(\text{Numerical continuous}\)

b.    \(\text{Categorical}\)

c.    \(\text{Numerical discrete}\)

Show Worked Solution

a.    \(\text{Data is a measurement}\ \longrightarrow\ \text{Numerical continuous}\)

b.    \(\text{Fruit is grouped into categories or types}\ \longrightarrow\ \text{Categorical}\)

c.    \(\text{Students can be counted}\ \longrightarrow\ \text{Numerical discrete}\)

Classifying Data, SM-Bank 008

Match each variable with its classification on the right.  (2 marks)

\begin{array} {ll} \text{A.  Hair colour of students in Year 7} &\text{1.  Numerical discrete} \\\text{B.  Heights of players in the Boomers basketball team} & \text{2.  Categorical ordinal} \\\text{C.  The number of people living in each household in NSW}\  & \text{3.  Numerical continous}\\\text{D.  A, B, C, D, E grades on a report card}  & \text{4.  Categorical nominal}\end{array}

Show Answers Only

\(\text{A}\longrightarrow 4,\ \text{B}\longrightarrow 3,\ \text{C}\longrightarrow 1,\ \text{D}\longrightarrow 2\)

Show Worked Solution

\(\text{A. Hair colour is data grouped in categories with no order}\)

\(\therefore\ \text{4. Categorical nominal}\)

\(\text{B. Heights are measurements which are numerical and continuous}\)

\(\therefore\ \text{3. Numerical continous}\)

\(\text{C. The people living in each household are counted}\)

\(\therefore\ \text{1. Numerical discrete}\)

\(\text{D. Grades on a report card are categorical but the order is important}\)

\(\therefore\ \text{2. Categorical ordinal}\)

Classifying Data, SM-Bank 007

Classify the following as either categorical or numerical data. 

  1. The colours of the cars entering a shopping centre.  (1 mark)

  2. The temperatures recorded in Brisbane over a 2 week period.  (1 mark)

Show Answers Only

a.    \(\text{categorical}\)

b.    \(\text{numerical}\)

Show Worked Solution

a.    \(\text{Colours of cars are words therefore categorical.}\)

b.    \(\text{Temperatures are measurements therefore numerical.}\)

Data Analysis, SM-Bank 050

Brandon made a dot plot to show the hours he worked over the last 16 weeks.
 

 
What is the mean number of hours that Brandon worked over that last 16 weeks?  (2 marks)

Show Answers Only

\(15.75\ \text{hours}\)

Show Worked Solution
\(\text{Mean}\ \) \(=\dfrac{\text{Sum of the scores}}{\text{Number of scores}}\)
  \(=\dfrac{2\times 13+3\times 14+3\times 15+3\times 16+2\times 17+3\times 19}{16}\)
  \(=\dfrac{252}{16}\)
  \(=15.75\ \text{hours}\)

Data Analysis, SM-Bank 049

Evie made a dot plot to show the distances she has swum in her training for a long distance ocean swim.
 

 
What is the mean distance that Evie has swum? Give your answer correct to 1 decimal place.  (2 marks)

Show Answers Only

\(21.6\ \text{km}\)

Show Worked Solution
\(\text{Mean}\ \) \(=\dfrac{\text{Sum of the scores}}{\text{Number of scores}}\)
  \(=\dfrac{18+19+2\times 20+2\times 21+22+3\times 24+25}{11}\)
  \(=\dfrac{238}{11}=21.636\dots\)
  \(\approx 21.6\ \text{km (1 d.p.)}\)

Data Analysis, SM-Bank 048 MC

Dante made a dot plot to show the distances he has run in his training for a half-marathon.
 

 
What is the median of the distances Dante has run?

  1. \(2\)
  2. \(7\)
  3. \(21\)
  4. \(24\)
Show Answers Only

\(C\)

Show Worked Solution
\(\text{Median}\ \) \(=\ \text{6th score}\)
  \(=21\)

\(\Rightarrow C\)

Data Analysis, SM-Bank 047 MC

Angelica made a dot plot to show the distances she has run in her training for a marathon.
 

 
What is the range of the distances Angelica has run?

  1. \(3\)
  2. \(7\)
  3. \(21\)
  4. \(24\)
Show Answers Only

\(B\)

Show Worked Solution
\(\text{Range}\ \) \(=\ \text{high – low}\)
  \(=25-1\)
  \(=7\)

\(\Rightarrow B\)

Data Analysis, SM-Bank 046

The back-to-back ordered stem-and-leaf plot below shows the female and male smoking rates, expressed as a percentage, in 18 countries.
 

  1. For the 18 countries listed, what is the range of the male smoking rates?  (1 mark)
  2. For the 18 countries listed, what is the mode of the female smoking rates?  (1 mark)
  3. For the 18 countries listed, what is the difference between the medians of the female and male smoking rates?  (2 marks)
Show Answers Only

a.     \(30\%\)

b.     \(25\%\)

c.     \(5.5\%\)

Show Worked Solution

a.    \(\text{Range}=47-17=30\%\)

b.    \(\text{Female mode}=25\%\)

c.     \(\text{Female Median }\) \(=\ \text{average of 9th and 10th scores}\)
    \(=\dfrac{21+22}{2}=21.5\%\)

 

\(\text{Male Median }\) \(=\ \text{average of 9th and 10th scores}\)
  \(=27\%\)

  
\(\therefore\ \text{The difference in medians}\)

\(=27-21.5=5.5\%\)

Classifying Data, SM-Bank 004 MC

The variables age (under 55 years, 55 years and over) and preferred travel destination (domestic, international) are

  1. both categorical variables.
  2. both numerical variables.
  3. a numerical variable and a categorical variable respectively.
  4. a categorical variable and a numerical variable respectively.
Show Answers Only

\(A\)

Show Worked Solution

\(\text{Age is a categorical ordinal variable}\)

\(\text{because it is categorical data that can}\)

\(\text{have an order.}\)

\(\text{Preferred travel destination is a categorical}\)

\(\text{nominal value because the data has a name.}\)
 

\(\Rightarrow A\)

Classifying Data, SM-Bank 003

The variables blood pressure (low, normal, high) and age (under 50 years, 50 years or over) are

  1. both nominal variables.
  2. both ordinal variables.
  3. a nominal variable and an ordinal variable respectively.
  4. an ordinal variable and a nominal variable respectively.
Show Answers Only

\(B\)

Show Worked Solution

\(\text{Blood pressure is an ordinal variable}\)

\(\text{because it is categorical data that can}\)

\(\text{have an order.}\)

\(\text{Under 50 and over 50, likewise, is an}\)

\(\text{ordinal variable.}\)
 

\(\Rightarrow B\)

Classifying Data, SM-Bank 002 MC

The variables recovery time after exercise (in minutes) and fitness level (below average, average, above average) are

  1. both numerical.
  2. an ordinal variable and a nominal variable respectively.
  3. a numerical variable and a nominal variable respectively.
  4. a numerical variable and an ordinal variable respectively.
Show Answers Only

\(D\)

Show Worked Solution

\(\text{Recovery time in minutes → numerical variable}\)

\(\text{Fitness level → ordinal (categories that can be ordered)}\)

\(\Rightarrow D\)

Data Analysis, SM-Bank 044 MC

The total birth weight of a sample of 12 babies is 39.0 kg.

The mean birth weight of these babies, in kilograms, is

  1.  2.50
  2.  2.75
  3.  3.00
  4.  3.25
Show Answers Only

\(D\)

Show Worked Solution
\(\text{Mean}\) \(=\dfrac{\text{Total birth weight}}{\text{# babies}}\)
  \(=\dfrac{39.0}{12}\)
  \(=3.25\ \text{kg}\)

 
\(\Rightarrow D\)

Data Analysis, SM-Bank 043 MC

The total weight of nine oranges is 1.53 kg.

Using this information, the mean weight of an orange would be calculated to be closest to

  1. 115 g
  2. 153 g
  3. 162 g
  4. 170 g
Show Answers Only

\(D\)

Show Worked Solution
\(\text{Mean Weight}\) \(=\dfrac{\text{Total weight}}{\text{# Oranges}}\)
  \(=\dfrac{1.53}{9}\)
  \(= 0.17\ \text{kg}\)
  \(= 170\ \text{g}\)

 
\(\Rightarrow D\)

Classifying Data, SM-Bank 001 MC

The variables

region (city, urban, rural)

population density (number of people per square kilometre)

  1. are both categorical.
  2. are both numerical. 
  3. are categorical and numerical respectively.
  4. are numerical and categorical respectively.
Show Answers Only

\(C\)

Show Worked Solution

\(\text{Region is a categorical variable and population}\)

\(\text{density is a numerical variable (i.e. it can be}\)

\(\text{represented by countable numbers).}\)

\(\Rightarrow C\)

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)
Show Answers Only

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)
Show Answers Only

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 164

A half-cylinder has a height of 44 millimetres and a diameter of 20 millimetres. Calculate the volume of the half-cylinder in cubic centimetres, giving your answer as an exact value in terms of \(\pi\).  (2 marks)

Show Answers Only

\(2200\pi\ \text{cm}^3\)

Show Worked Solution

\(\text{NOTE: change measurements to centimetres before calculations}\)

\(V\) \(=\dfrac{1}{2}\pi r^2h\)
  \(=\dfrac{1}{2}\times\pi\times 10^2\times 44\)
  \(=2200\pi\)

 
\(\therefore\ \text{The exact volume of the half-cylinder is }2200\pi\ \text{cm}^3\)

Volume, SM-Bank 163

A quarter-cylinder has a height of 160 centimetres and a radius of 800 centimetres . Calculate the volume of the quarter-cylinder in cubic metres, giving your answer as an exact value in terms of \(\pi\).  (2 marks)

Show Answers Only

\(25.6\pi\ \text{m}^3\)

Show Worked Solution

\(\text{NOTE: change measurements to metres before calculations}\)

\(V\) \(=\dfrac{1}{4}\pi r^2h\)
  \(=\dfrac{1}{4}\times\pi\times 8^2\times 1.6\)
  \(=25.6\pi\)

 
\(\therefore\ \text{The exact volume of the quarter-cylinder is }25.6\pi\ \text{m}^3\)

Volume, SM-Bank 162

A half-cylinder has a height of 12 centimetres and a radius of 9 centimetres. Calculate the volume of the half-cylinder, giving your answer as an exact value in terms of \(\pi\).  (2 marks)

Show Answers Only

\(486\pi\ \text{cm}^3\)

Show Worked Solution
\(V\) \(=\dfrac{1}{2}\pi r^2h\)
  \(=\dfrac{1}{2}\times\pi\times 9^2\times 12\)
  \(=486\pi\)

 
\(\therefore\ \text{The exact volume of the half-cylinder is }486\pi\ \text{cm}^3\)

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)

Show Answers Only

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

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)

Show Answers Only

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

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)

Show Answers Only

\(9\ \text{m}\)

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)

Show Answers Only

\(2\ \text{m}\)

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)

Show Answers Only

\(30\ \text{mm}\)

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)

Show Answers Only

\(18\ \text{cm}\)

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)

Show Answers Only

\(4\ \text{mm}\)

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)

Show Answers Only

\(20\ \text{m}\)

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)

Show Answers Only

\(9\ \text{cm}\)

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 152

The 3D shape below is a composite prism consisting of a half-cylinder and a rectangular prism.
 

Calculate the volume of the of the prism in cubic centimetres, giving your answer correct to one decimal place.  (2 marks)

Show Answers Only

\( 4881.7\ \text{cm}^3\ (\text{1 d.p.})\)

Show Worked Solution

\(\text{Diameter of cylinders = 10 cm  }\longrightarrow\  r=5\ \text{cm}\)

\(\text{Volume = rectangular prism + ½ cylinder}\)

\(\therefore\ V\) \(=l\times b\times h+\dfrac{1}{2}\times\pi r^2h\)
  \(=24\times 6.5\times 25+\dfrac{1}{2}\times\pi\times 5^2\times 25\)
  \(=3900+981.747\dots\)
  \(=4881.747\dots\)
  \(\approx 4881.7\ \text{cm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 151

A piece of metal in the shape of a rectangular prism has had two cylindrical holes, each with a diameter of 8 millimetres, drilled through it.
 

Calculate the volume of the remaining metal in cubic millimetres, giving your answer correct to one decimal place.  (2 marks)

Show Answers Only

\( 4937.6\ \text{mm}^3\ (\text{1 d.p.})\)

Show Worked Solution

\(\text{Diameter of cylinders = 8 mm  }\longrightarrow\  r=4\ \text{mm}\)

\(\text{Volume = rectangular prism – 2 × cylinders}\)

\(\therefore\ V\) \(=l\times b\times h-2\times\pi r^2h\)
  \(=32\times 16\times 12-2\times\pi\times 4^2\times 12\)
  \(=6144-1206.371\dots\)
  \(=4937.628\dots\)
  \(\approx 4937.6\ \text{mm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 150

A a skateboard ramp has been constructed using a rectangular prism that has had a quarter-cylinder removed to create the curved surface.
 

Calculate the volume of the skateboard ramp, giving your answer correct to one decimal place.  (2 marks)

Show Answers Only

\( 92.2\ \text{m}^3\ (\text{1 d.p.})\)

Show Worked Solution

\( V\) \(=l\times b\times h-\dfrac{1}{4}\times\pi r^2h\)
  \(=7\times 5\times 6-\dfrac{1}{4}\times\pi\times 5^2\times 6\)
  \(=210-117.809\dots\)
  \(=92.190\dots\)
  \(\approx 92.2\ \text{m}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 149

A a chicken feeder has been constructed using a rectangular prism and a quarter-cylinder.
 

Calculate the volume of the chicken feeder, giving your answer correct to one decimal place.  (2 marks)

Show Answers Only

\( 61\ 703.4\ \text{cm}^3\ (\text{1 d.p.})\)

Show Worked Solution
\(V\) \(=\dfrac{1}{4}\times\pi r^2h+l\times b\times h\)
  \(=\dfrac{1}{4}\times\pi\times 24^2\times 60+24\times 24\times 60\)
  \(=27\ 143.360\dots+34\ 560\)
  \(=61\ 703.360\dots\)
  \(\approx 61\ 703.4\ \text{cm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 148

A large machinery storage shed has been constructed on a property. The shed is made up of a rectangular prism and a half cylinder.
 

Calculate the volume of the machinery shed, giving your answer to the nearest cubic metre.  (2 marks)

Show Answers Only

\(3126\ \text{m}^3\)

Show Worked Solution

\(\text{Diameter semi-circle = 12 m }\rightarrow\ \text{r = 6 m}\)

\(\therefore\ V\) \(=\dfrac{1}{2}\times\pi r^2h+l\times b\times h\)
  \(=\dfrac{1}{2}\times\pi\times 6^2\times 19+12\times 9\times 19\)
  \(=1074.424\dots+2052\)
  \(=3126.424\dots\)
  \(\approx 3126\ \text{m}^3\ (\text{nearest m}^3)\)

Volume, SM-Bank 147

A concrete half-pipe was constructed in a park. The pipe has a constant thickness 0.5 metres.
 

  1. Calculate the volume of the concrete used to create the half-pipe, giving your answer to the nearest cubic metre.  (2 marks)
  2. Calculate the capacity of the concrete used in kilolitres.  (1 mark)
Show Answers Only

a.    \(19\ \text{m}^3\)

b.    \(9\ \text{kL}\)

Show Worked Solution

a.    \(\text{diameter large semi-circle = 4 m }\rightarrow\ \text{R = 2 m}\)

\(\text{diameter small semi-circle = 3 m }\rightarrow\ \text{r = 1.5 m}\)

\(\therefore\ V\) \(=\dfrac{1}{2}\times\pi R^2h-\dfrac{1}{2}\times\pi r^2h\)
  \(=\dfrac{1}{2}\times\pi\times 2^2\times 7-\dfrac{1}{2}\times\pi\times 1.5^2\times 7\)
  \(=19.242\dots\)
  \(\approx 19\ \text{m}^3\ (\text{nearest m}^3)\)

 

b.    \(\text{1 kL = 1 m}^3\)

\(\therefore\ 19\ \text{m}^3\) \(=19\ \text{kL}\)

Volume, SM-Bank 146

Geraldine created a large chocolate mould in the shape of a half cylinder using her 3D printer.
 

  1. Calculate the volume of the mould, giving your answer to the nearest cubic centimetre.  (2 marks)
  2. Calculate the capacity of the mould in litres.  (1 mark)
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a.    \(8310\ \text{cm}^3\)

b.    \(8.31\ \text{L}\)

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a.    \(\text{diameter = 23 cm }\rightarrow\ \text{radius = 11.5 cm}\)

\(\therefore\ V\) \(=\dfrac{1}{2}\times \pi r^2h\)
  \(=\dfrac{1}{2}\times \pi\times 11.5^2\times 41\)
  \(=8309.512\dots\)
  \(\approx 8310\ \text{cm}^3\ \text{nearest cm}^3)\)

 

b.    \(\text{1 litre = 1000 cm}^3\)

\(\therefore\ 8310\ \text{cm}^3\) \(=\Bigg(\dfrac{8310}{1000}\Bigg)\ \text{L}\)
  \(=8.31\ \text{L}\)

Volume, SM-Bank 145

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

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\(942.5\ \text{mm}^3\)

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\(\text{Figure is }\dfrac{60}{360}=\dfrac{1}{6}\ \text{of a cylinder}\)

\(\therefore\ V\) \(=\dfrac{1}{6}\times \pi r^2h\)
  \(=\dfrac{1}{6}\times \pi\times 10^2\times 18\)
  \(=942.477\dots\)
  \(\approx 942.5\ \text{mm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 144

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

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\(17.9\ \text{m}^3\)

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\(\text{Figure is }\dfrac{3}{4}\ \text{of a cylinder}\)

\(\therefore\ V\) \(=\dfrac{3}{4}\times \pi r^2h\)
  \(=\dfrac{3}{4}\times \pi\times 1.3^2\times 4.5\)
  \(=17.918\dots\)
  \(\approx 17.9\ \text{m}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 143

Calculate the volume of the quarter cylinder below in cubic centimetres.  Give your answer correct to the nearest cubic centimetre.  (2 marks)
 

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\(113\ \text{cm}^3\)

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\(V\) \(=\dfrac{1}{4}\times \pi r^2h\)
  \(=\dfrac{1}{4}\times \pi\times 4^2\times 9\)
  \(=113.097\dots\)
  \(\approx 113\ \text{cm}^3\ (\text{nearest cubic cm.})\)

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

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

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

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

Volume, SM-Bank 136

  1. What shape best describes the uniform cross-section of this prism?  (1 mark)

  2. Calculate the volume of the prism in cubic centimetres.  (2 marks)
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a.    \(\text{A rhombus}\)

b.    \(497.664\ \text{cm}^3\)

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a.    \(\text{The uniform cross-section is a rhombus.}\)

b.    \(\text{Area of rhombus cross-section }\)

\(A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times 9.6\times 7.2\)
  \(=34.56\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=34.56\times 14.4\)
  \(=497.664\ \text{cm}^3\)

Volume, SM-Bank 135

The figure below is a prism with a rhombus as the uniform cross-section.

Calculate the value of \(x\), the length of the diagonal in the cross-section, given the volume of the prism is 1950 cubic metres.  (2 marks)

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\(20\ \text{m}\)

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\(\text{The uniform cross-section is a rhombus.}\)

\(\therefore\ A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times x\times 15\)
  \(=7.5x\ \text{m}^2\)

 

\(V\) \(=A\times h\)
\(1950\) \(=7.5x\times 13\)
\(1950\) \(=97.5x\)
\(\therefore\ x\) \(=\dfrac{1950}{97.5}\)
  \(=20\ \text{m}\)

Properties of Geometric Figures, SM-Bank 039

Determine the value of \(x^{\circ}\) in the quadrilateral above, giving reasons for your answer.     (2 marks)

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\(30^{\circ}\)

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\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\) \(=2x + 3x + 4x + 2x \)  
\(12x^{\circ}\) \(=360\)  
\(x^{\circ}\) \(=\dfrac{360}{12}\)  
  \(=30^{\circ}\)  

Volume, SM-Bank 134

The figure below is a prism with a kite as the uniform cross-section.

Calculate the value of \(x\), the length of the diagonal in the cross-section, given the volume of the prism is 1485 cubic centimetres.  (2 marks)

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\(11\ \text{cm}\)

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\(\text{The uniform cross-section is a kite.}\)

\(\therefore\ A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times x\times 18\)
  \(=9x\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
\(1485\) \(=9x\times 15\)
\(1485\) \(=135x\)
\(\therefore\ x\) \(=\dfrac{1485}{135}\)
  \(=11\ \text{cm}\)

Properties of Geometric Figures, SM-Bank 041

A pentagon is pictured below.
 

  1. By drawing triangles from one vertex, or otherwise, calculate the sum of the internal angles of a pentagon.   (1 mark)

  2. Determine the value of \(x^{\circ}\) in the pentagon.     (2 marks)
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i.    \(540^{\circ}\)

ii.   \(110^{\circ}\)

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i.    \(\text{Pentagon can be divided into 3 triangles (from one chosen vertex).}\)

\(\text{Sum of internal angles}\ = 3 \times 180 = 540^{\circ}\)
 

ii.    \(540\) \(=x + 2 \times 90 + 135+115 \)  
\(540\) \(=x+430\)  
\(x^{\circ}\) \(=540-430\)  
  \(=110^{\circ}\)  

Properties of Geometric Figures, SM-Bank 040

 

Determine the value of the two unknown angles in the quadrilateral above, giving reasons for your answer.     (3 marks)

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\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\) \(=5x+3x+79+105 \)  
\(8x\) \(=360-184\)  
\(x^{\circ}\) \(=\dfrac{176}{8}\)  
  \(=22^{\circ}\)  

 
\(\text{Unknown angle 1}\ = 3 \times 22 = 66^{\circ}\)

\(\text{Unknown angle 2}\ = 5 \times 22 = 110^{\circ}\)

Show Worked Solution

\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\) \(=5x+3x+79+105 \)  
\(8x\) \(=360-184\)  
\(x^{\circ}\) \(=\dfrac{176}{8}\)  
  \(=22^{\circ}\)  

 
\(\text{Unknown angle 1}\ = 3 \times 22 = 66^{\circ}\)

\(\text{Unknown angle 2}\ = 5 \times 22 = 110^{\circ}\)

Volume, SM-Bank 133

The figure below is a prism with a parallelogram as the uniform cross-section.

Calculate the value of \(x\), the perpendicular height of the cross-section, given the volume of the prism is 2880 cubic millimetres.  (2 marks)

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\(12\ \text{mm}\)

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\(\text{The uniform cross-section is a parallelogram.}\)

\(\therefore\ A\) \(=b\times h\)
  \(=10\times x\)
  \(=10x\ \text{mm}^2\)

 

\(V\) \(=A\times h\)
\(2880\) \(=10x\times 24\)
\(2880\) \(=240x\)
\(\therefore\ x\) \(=\dfrac{2880}{240}\)
  \(=12\ \text{mm}\)