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Measurement, STD2 EQ-Bank 21

Sam is installing a lap pool in the shape of a trapezoidal prism, with dimensions as shown.
 

  1. Calculate the volume of the lap pool in cubic metres.   (2 marks)
  2. What is the capacity of the pool in litres?   (1 mark)
Show Answers Only

a.    \(5.4\ \text{m}^3\)

b.    \(5400\ \text{L}\)

Show Worked Solution
a.     \(\text{Area of trapezoid}\) \(=\dfrac{h}{2}\big(a + b\big)\)
    \(=\dfrac{1.2}{2}\big(2.4+3.6\big)\)
    \(=3.6\ \text{m}^2\)

 
\(\text{Volume} = Ah=3.6\times 1.5=5.4\ \text{m}^3\)

  
b.    \(\text{Using 1 m}^3=1000\ \text{L}\)

\(\text{Capacity }=\ 5.4\times 1000=5400\ \text{Litres}\)

Measurement, STD2 EQ-Bank 14 MC

A square pyramid has a volume of 384 cm³ and a perpendicular height of 8 cm.

 Calculate the side length of the square base. Use the formula \(V = \frac{1}{3}Ah\) where \(A\) is the area of the base.

  1. 8 cm
  2. 12 cm
  3. 16 cm
  4. 24 cm
Show Answers Only

\(B\)

Show Worked Solution
\(V\) \(=\frac{1}{3}Ah\)
\(384\) \(=\frac{1}{3}\times A\times 8\)
\(384\) \(=\dfrac{8A}{3}\)
\(8A\) \(=384\times 3=1152\)
\(A\) \(=\dfrac{1152}{8}\)
  \(=144\ \text{cm}^2\)
\(\text{Since the base is a square: }x^2\) \(=144\)
\(x\) \(=\sqrt{144}\)
  \(=12\ \text{cm}\)

 

\(\Rightarrow B\)

Measurement, STD2 EQ-Bank 13 MC

A square pyramid has a base with side length 9 cm and a perpendicular height of 8 cm.

Calculate the volume of the pyramid. 

  1. 72 cm³
  2. 144 cm³
  3. 192 cm³
  4. 216 cm³
Show Answers Only

\(D\)

Show Worked Solution

\(A=9^2=81\)

\(V\) \(=\frac{1}{3}Ah\)
  \(=\frac{1}{3}\times 81\times 8\)
  \(=216\ \text{cm}^3\)

  
\(\Rightarrow D\)

Measurement, STD2 EQ-Bank 4 MC

A cone has a radius of 6 cm and a height of 8 cm.

Calculate the volume of the cone, giving your answer in cubic centimetres correct to 1 decimal place.

  1. 100.5 cm³
  2. 150.8 cm³
  3. 301.6 cm³
  4. 904.8 cm³
Show Answers Only

\(C\)

Show Worked Solution
\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\times \pi \times (6)^2\times 8\)
  \(=301.592\dots\)
  \(=301.6\ \text{cm}^3 \text{(1 d.p.)}\)

  
\(\Rightarrow C\)

Measurement, STD2 EQ-Bank 19

The cone below has a diameter of 13.5 centimetres and a height of 0.4 metres.
 

Calculate the volume of the cone, giving your answer in cubic centimetres correct to 1 decimal place.   (2 marks)

Show Answers Only

\(1908.5\ \text{cm}^3 \text{(1 d.p.)}\)

Show Worked Solution

\(\text{Radius}\ =\dfrac{13.5}{2}=6.75\ \text{cm}\)

\(0.4\ \text{m}\ =\ 40\ \text{cm}\)

\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\times \pi \times (6.75)^2\times 40\)
  \(=1908.517\dots\)
  \(=1908.5\ \text{cm}^3 \text{(1 d.p.)}\)

Measurement, STD2 EQ-Bank 18

The cone below has a radius of 4.2 metres and a height of 6.1 metres.
 

Calculate the volume of the cone, giving your answer in cubic metres correct to 1 decimal point.   (2 marks)

Show Answers Only

\(112.7\ \text{m}^3 \text{(1 d.p.)}\)

Show Worked Solution
\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\times \pi \times (4.2)^2\times 6.1\)
  \(=112.682\dots\)
  \(=112.7\ \text{m}^3 \text{(1 d.p.)}\)

Measurement, STD2 EQ-Bank 16

The square pyramid below, has a side measurement of 120 metres and a perpendicular height \((h)\) of 65 metres.
 

Find the volume of the pyramid in cubic metres.   (2 marks)

Show Answers Only

\(312\, 000\ \text{m}^3\)

Show Worked Solution
\(V\) \(=\frac{1}{3}Ah\)
  \(=\frac{1}{3}\times 120\times 120\times 65\)
  \(=312\, 000\ \text{m}^3\)

Measurement, STD2 EQ-Bank 23

A cone with a radius 9.5 cm and height 19.5 cm is shown.
 

The cone is filled with liquid. 

Find the amount of liquid in the cone. Give your answer in litres correct to 3 decimal places.   (3 marks)

Show Answers Only

\(1.843\ \text{L}\)

Show Worked Solution
\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\pi (9.5)^2\times 19.5\)
  \(=1842.93679\ \text{cm}^3\)

 
\(\text{Since 1 cm\(^3\) = 1 mL  and  1 L = 1000 mL:}\)

\(\text{Liquid in cone}=\dfrac{1842.9…}{1000}=1.8429…=1.843\ \text{litres (3 d.p.)}\)

Measurement, STD2 M1 2021 HSC 16

The volume, `V`, of a sphere is given by the formula

`V = frac{4}{3} pi r^3,`

where `r` is the radius of the sphere.

A tank consists of the bottom half of a sphere of radius 2 metres, as shown.
 

Find the volume of the tank in cubic metres, correct to one decimal place.   (2 marks)

Show Answers Only

`16.8\ text{m}^3`

Show Worked Solution
 `V` `= frac{1}{2} times frac{4}{3} pi r^3`
  `= frac{1}{2} times frac{4}{3} times pi times 2^3`
  `= 16.755…`
  `= 16.8\ text{m}^3\ \ text{(1 d.p.)}`

Measurement, STD2 M1 2019 HSC 16

A bowl is in the shape of a hemisphere with a diameter of 16 cm.
 

What is the volume of the bowl, correct to the nearest cubic centimetre?  (2 marks)

Show Answers Only

`1072\ text(cm)^3`

Show Worked Solution
`V` `= 1/2 xx 4/3pir^3`
  `= 1/2 xx 4/3 xx pi xx 8^3`
  `= 1072.3…`
  `= 1072\ text{cm}^3\ text{(nearest cm}^3 text{)}`

Measurement, STD2 M1 EQ-Bank 12 MC

During a flood, 12.5 hectares of land was covered by water to a depth of 30 cm.

How many kilolitres of water covered the land?

(1 hectare = 10 000 m² and 1 m³ = 1000 L)

  1. 3.75 kL
  2. 37.5 kL
  3. 37 500 kL
  4. 37 500 000 kL
Show Answers Only

`C`

Show Worked Solution

`text(Volume of water in m³:)`

`text(Vol)` `= Ah`
  `= 12.5 xx 10\ 000 xx 30\ text(cm)`
  `= 12.5 xx 10\ 000 xx 0.3`
  `= 37\ 500 \ text(m³)`
  `= 37\ 500\ text(kL)`

 
`=>C`

Measurement, STD2 M1 EQ-Bank 22

Steel rods are manufactured in the shape of equilateral triangular prisms.
 

  1. Find the volume of the prism (answer correct to 1 decimal place).   (2 marks)

  2. The mass of steel is 7850 kg/m³. Use this information to find the mass of the steel rod correct to the nearest gram.   (2 marks)

Show Answers Only
  1. `3464.1\ text(cm³)`
  2. `27\ 193\ text(g)`
Show Worked Solution

a.   `text{Area of triangular face (using sine rule)}`

`=1/2 xx 10 xx 10 xx sin60°`

`=43.301…`
 

`text(Volume)` `=Ah`
  `=43.301… xx 80`
  `=3464.1\ text{cm}^3\ \text{(to 1 d.p.)}`

 

b.  `text(Converting kg to g:)`

`7850\ text(kg) = 7\ 850\ 000\ text(g)`
 

`text(Converting m)^3\ \text(to cm)^3:`

`text(1 m)^3=100\ text(cm) xx100\ text(cm) xx100\ text(cm)=1\ 000\ 000\ text(cm)^3`

`:.\ text(Weight of steel rod)` `=3464.1 xx (7\ 850\ 000)/(1\ 000\ 000)` 
  `=27\ 193.185`
  `=27\ 193\ text{g  (nearest gram)}`

Measurement, STD2 M1 EQ-Bank 17

A cannon ball is made out of steel and has a diameter of 23 cm.

  1. Find the volume of the sphere in cubic centimetres (correct to 1 decimal place).   (2 marks)

  2. It is known that the mass of the steel used is 8.2 tonnes/m³. Use this information to find the mass of the cannon ball to the nearest gram.   (2 marks)

Show Answers Only
  1. `6370.6\ text{cm}^3\ \text{(to 1 d.p.)}`
  2. `52\ 239\ text(grams)`
Show Worked Solution

a.   `text(Radius)= 23/2 = 11.5\ text(cm)`

`text(Volume)` `= 4/3pir^3`
  `= 4/3 xx pi xx 11.5^3`
  `= 6370.626…`
  `= 6370.6\ text{cm}^3\ \text{(to 1 d.p.)}`

 

b.   `text(Convert m)^3\ \text{to cm}^3:`

`text(1 m)^3= 100\ text(cm × 100 cm × 100 cm)= 1\ 000\ 000\ text(cm)^3`
 

`text(Convert 8.2 tonnes to grams:)`

`text(8.2 tonnes)= 8200\ text(kg)= 8\ 200\ 000\ text(g)`
 

`:.\ text(Weight of cannon ball)`

`= 6370.6 xx (8\ 200\ 000)/(1\ 000\ 000)`

`= 52\ 238.92`

`= 52\ 239\ text(grams)`

Measurement, STD2 M1 2017 HSC 22 MC

A concrete water pipe is manufactured in the shape of an annular cylinder. The dimensions are shown in the diagrams.
 


 

What is the approximate volume of concrete needed to make the water pipe?

  1. `text(0.06 m)³`
  2. `text(0.09 m)³`
  3. `text(0.70 m)³`
  4. `text(0.99 m)³`
Show Answers Only

`C`

Show Worked Solution
`text(Volume)` `= text(Area of annulus) xx h`
  `= (piR^2 – pir^2) xx 2.8`
  `= (pi xx 0.45^2 – pi xx 0.35^2) xx 2.8`
  `= 0.7037…`
  `= 0.70\ text(m)³`

  
`=>C`

Measurement, STD2 M1 2017 HSC 18 MC

A skip bin is in the shape of a trapezoidal prism, with dimensions as shown.
 

What is the volume of the skip bin?

  1. `5.4\ text(m)^3`
  2. `7.776\ text(m)^3`
  3. `10.8\ text(m)^3`
  4. `15.552\ text(m)^3`
Show Answers Only

`A`

Show Worked Solution
`text(Area of trapezoid)` `= 1/2h (a + b)`
  `= 1/2 xx 1.2 xx (3.6 + 2.4)`
  `= 3.6\ text(m)^2`

 

`:.\ text(Volume)` `= Ah`
  `= 3.6 xx 1.5`
  `= 5.4\ text(m)^3`

 
`=>A`

Measurement, STD2 M1 2016 HSC 30a

The area of a roof is 30 m². Any rain that falls on the roof flows directly onto a garden.

Calculate how many litres of water flow onto the garden when 20 mm of rain falls on the roof.  (2 marks)

Show Answers Only

`600\ text(L)`

Show Worked Solution
♦♦ Mean Mark 27%.
STRATEGY: Converting 20 mm into metres as a 1st step is an efficient approach here.
`text(Volume)` `= Ah`
  `= 30 xx 20/1000`
  `= 0.6\ text(m³)`
  `= 600\ text(L)`

Measurement, STD2 M1 2016 HSC 12 MC

A container is in the shape of a triangular prism which has a capacity of 12 litres. The area of the base is 240 cm².
 

What is the distance, `h`, between the two triangular ends of the container?

  1. 5 cm
  2. 20 cm
  3. 25 cm
  4. 50 cm
Show Answers Only

`=> D`

Show Worked Solution
♦♦ Mean mark 35%.

`text{1 mL = 1 cm}^3\ \ =>\ \ text{1 L = 1000 cm}^3`

`text(Volume)` `= Ah`
`12\ 000` `= 240 xx h`
`h` `= (12\ 000)/240`
  `= 50\ text(cm)`

 
`=> D`

Measurement, STD2 M1 2015 HSC 8 MC

The Louvre Pyramid in Paris has a square base with side length 35 m and a perpendicular height of 22 m.
 

What is the volume of this pyramid, to the nearest m³?

  1. `257\ text(m)^3`
  2. `1027\ text(m)^3`
  3. `8983\ text(m)^3`
  4. `26\ 950\ text(m)^3`
Show Answers Only

`C`

Show Worked Solution
`V` `= 1/3Ah`
`A` `= 35 xx 35`
  `= 1225\ text(m)^2`

 

`:.V` `= 1/3 xx 1225 xx 22`
  `= 8983.33…\ text(m)^3`

 
`=>C`

Measurement, STD2 M1 2005 HSC 23b

A clay brick is made in the shape of a rectangular prism with dimensions as shown.
 

  1. Calculate the volume of the clay brick.  (1 mark)

Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.  
 

  1. What is the volume of clay remaining in the brick after the holes have been made? (Give your answer to the nearest cubic centimetre.)  (3 marks)

  2. What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.)  (1 mark)

Show Answers Only
  1. `text(1512 cm)^3`
  2. `text{1364 cm}^3`
  3. `text{9.8%}`
Show Worked Solution
a.    `V` `= l × b × h`
    `= 21 × 8 × 9`
    `= 1512\ text(cm)^3`

 

b.  `text(Volume of each hole)`

`= pir^2h`

`= pi × 1.4^2 × 8`

`= 49.260…\ text(cm)^3`

 

`:.\ text(Volume of clay still in brick)`

`= 1512 − (3 × 49.260…)`

`= 1364.219…`

`= 1364\ text{cm}^3\ text{(nearest whole)}`

 

c.  `text(Percentage of clay removed)`

`= ((3 × 49.260…))/1512 × 100`

`= 9.773…`

`= 9.8 text{%   (1 d.p.)}`

Measurement, STD2 M1 2004 HSC 23a

The diagram shows the shape of Carmel’s garden bed. All measurements are in
metres.

  1. Show that the area of the garden bed is 57 square metres.   (2 marks)

  2. Carmel decides to add a 5 cm layer of straw to the garden bed.

     

    Calculate the volume of straw required. Give your answer in cubic metres.   (2 marks)

  3. Each bag holds 0.25 cubic metres of straw.

     

    How many bags does she need to buy?   (2 marks)

  4. A straight fence is to be constructed joining point A to point B.

     

    Find the length of this fence to the nearest metre.   (2 marks)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `text(2.85 m³)`
  3. `text(She needs to buy 12 bags)`
  4. `8\ text{m  (nearest metre)}`
Show Worked Solution
a.    `text(Area of)\ Delta ABC` `= 1/2 xx b xx h`
  `= 1/2 xx 10 xx 5.1`
  `= 25.5\ text(m²)`
`text(Area of)\ Delta ACD` `= 1/2 xx 10 xx 6.3`
  `= 31.5\ text(m²)`

 

`:.\ text(Total Area)` `= 25.5 + 31.5`
  `= 57\ text(m² … as required)`

 

b.    `V` `= Ah`
  `= 57 xx 0.05`
  `= 2.85\ text(m³)`

 

c.    `text(Bags to buy)` `= 2.85/0.25`
  `= 11.4`

 
`:.\ text(She needs to buy 12 bags.)`

 

d.   `text(Using Pythagoras,)`

`AB^2` `= 6.0^2 + 5.1^2`
  `= 36 + 26.01`
  `= 62.01`
`AB` `= 7.874…`
  `=8\ text{m  (nearest metre)}`

Measurement, STD2 M1 2007 HSC 23b

A cylindrical water tank, of height 2 m, is placed in the ground at a school.

The radius of the tank is 3.78 metres. The hole is 2 metres deep. When the tank is placed in the hole there is a gap of 1 metre all the way around the side of the tank.

 

  1. When digging the hole for the water tank, what volume of soil was removed? Give your answer to the nearest cubic metre.  (3 marks)

  2. Sprinklers are used to water the school oval at a rate of 7500 litres per hour.   

     

    The water tank holds 90 000 litres when full. 

     

    For how many hours can the sprinklers be used before a full tank is emptied?   (1 mark)

  3. Water is to be collected in the tank from the roof of the school hall, which has an area of 400 m².

     

    During a storm, 20 mm of rain falls on the roof and is collected in the tank. 

     

    How many litres of water were collected?   (2 marks)

Show Answers Only
  1. `144\ text(m³)\ \ text{(nearest m³)}`
  2. `text(12 hours)`
  3. `8000\ text(litres)`
Show Worked Solution

a.  `V = pi r^2 h\ \ \ \ text(where)`

`h = 2\ text(and)\ r = 4.78\ text(m)`

`:.\ V` `= pi xx 4.78^2 xx 2`
  `= 143.56…`
  `= 144\ text(m³)\ \ text{(nearest m³)}`

 

b.  `text(Total water) = 90\ 000\ text(litres)`

`text(Usage) = 7500\ \ text(litres/hr)`

`:.\ text(Hours before it is empty)`

`= (90\ 000)/7500`

`= 12\ text(hours)`

 

c.  `text(Water collected)`

`= 400 xx 0.020`

`= 8\ text(m²)`

`= 8000\ text(litres)`

Measurement, STD2 M1 2009 HSC 19 MC

Two identical spheres fit exactly inside a cylindrical container, as shown.
 

The diameter of each sphere is 12 cm.

What is the volume of the cylindrical container, to the nearest cubic centimetre?

  1. `1357\ text(cm)^3`
  2. `2714\ text(cm)^3`
  3. `5429\ text(cm)^3`
  4. `10\ 857\ text(cm)^3`
Show Answers Only

`B`

Show Worked Solution

`text(S)text(ince diameter sphere = 12 cm) `

`=>\ text(Radius of cylinder = 6 cm)`

`text(Height of cylinder)` `= 2 xx text(diameter of sphere)`
  `= 2 xx 12`
  `= 24\ text(cm)`
   
`:.\ text(Volume cylinder)` `= pi r^2 h`
  `= pi xx 6^2 xx 24`
  `= 2714.336…\ text(cm³)`

 
`=>  B`

Measurement, STD2 M1 2010 HSC 17 MC

During a flood 1.5 hectares of land was covered by water to a depth of  17 cm.

How many kilolitres of water covered the land?   (1 hectare = 10 000 m²)  

  1.    2.55 kL
  2.    2550 kL
  3.    255 000 kL
  4.    2 550 000 kL
Show Answers Only

`B`

Show Worked Solution
♦♦ Mean mark 34%
NOTE: The unit conversion 1 m³ = 1000 L  is contained in the Formulae and Data sheet given out in the exam.
`text(Area covered)` `=1.5xx10\ 000`
  `=15\ 000\ text(m²)`
`text(Volume)` `=Ah`
  `=15\ 000xx0.17`
  `=2550\ text(m³)`

 

`text{1 m³}` `=1000\ text(L)=1\ text(kL)`
`:.\ text(Volume)` `=2550\ text(kL)`

`=>B`

Measurement, STD2 M1 2012 HSC 6 MC

What is the volume of this rectangular prism in cubic centimetres? 
  

  1. 6 cm³
  2. 600 cm³
  3. 60 000 cm³
  4. 6 000 000 cm³
Show Answers Only

`C`

Show Worked Solution

`text{Convert all measurements to centimetres:}`

`V` `= l xx b xx h`
  `=30 xx 5 xx 400`
  `=60\ 000\ \ text(cm)^3`

 
`=>  C`

Measurement, STD2 M1 2013 HSC 12 MC

A square pyramid fits exactly on top of a cube to form a solid.

2013 12 mc

What is the volume of the solid?

  1. 513 cm³
  2. 999 cm³
  3. 1242 cm³
  4. 1539 cm³
Show Answers Only

`B`

Show Worked Solution
`text(Volume )` `=text{Vol (cube)} +text{Vol (pyramid)}`
  `=l^3+1/3Ah`
  `=(9xx9xx9)+(1/3xx9xx9xx10)`
  `=999\ text(cm)^3`

 
`=>\ B`