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Measurement, STD2 M1 2022 HSC 28

A dam is in the shape of a triangular prism which is 50 m long, as shown.

Both ends of the dam, `A B C` and `D E F`, are isosceles triangles with equal sides of length 25 metres. The included angles `B A C` and `E D F` are each `150^@`.

 
     

Calculate the number of litres of water the dam will hold when full.  (4 marks)

Show Answers Only

`7\ 812\ 500\ text{L}`

Show Worked Solution

`V=Ah`

`text{Use sine rule to find}\ A:`

`A` `=1/2 ab\ sinC`  
  `=1/2 xx 25 xx 25 xx sin150^@`  
  `=156.25\ text{m}^2`  

 

`:.V` `=156.25 xx 50`  
  `=7812.5\ text{m}^3`  

 

`text{S}text{ince 1 m³ = 1000 litres:}`

`text{Dam capacity}` `=7812.5 xx 1000`  
  `=7\ 812\ 500\ text{L}`  

♦♦ Mean mark 33%.

Measurement, STD2 M1 SM-Bank 26 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

`=>\ text(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)`

 
`=>\ text(C)`

Measurement, STD2 M1 2018 HSC 13 MC

A rectangular pyramid has base side lengths `3x` and `4x`. The perpendicular height of the pyramid is `2x`. All measurements are in metres.
 

What is the volume of the pyramid in cubic metres?

  1. `8x^3`
  2. `9x^3`
  3. `12x^3`
  4. `24x^3`
Show Answers Only

`A`

Show Worked Solution
`text(Volume)` `= 1/3Ah`
  `= 1/3(4x xx 3x xx 2x)`
  `= 8x^3`

 
`=>A`

Measurement, STD2 M1 SM-Bank 4

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

i.   `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`
  `=3464.10…`
  `=3464.1\ text{cm³  (to 1 d.p.)}`

 

ii.  `text(Converting kg to g:)`

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

 

`text(Converting m³ to cm³:)`

`1\ text(m³)` `=100\ text(cm) xx100\ text(cm) xx100\ text(cm)` 
  `=1\ 000\ 000\ text(cm³)`

 

`:.\ 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 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)

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`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 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
i.  `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)`

 

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

 

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

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

 

iv.  `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 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 25 MC

A net is made using four rectangles and two trapeziums. It is folded to form a solid.
  

 What is the volume of the solid, in cm3 ?

  1. `360\ text(cm)^3`
  2. `434\ text(cm)^3`
  3. `440\ text(cm)^3`
  4. `576\ text(cm)^3`
Show Answers Only

`D`

Show Worked Solution

`text(Volume)=Ah\ \ \ text(where)\ \ A\ text(is the area of a trapezium)`

♦ Mean mark 35%
COMMENT: Note that `h` in the “area” formula is different to the `h` used in the “volume” formula.
`A` `=1/2 h(a+b)`
  `=1/2xx8(11+5)`
  `=64\ text(cm²)`

 

`:.V=Ah=64xx9=576\ text(cm³)`

`=>  D`

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`