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Volume, SMB-012

The solid below is made from joining a rectangular prism to a triangular prism, with all measurements in centimetres.
 

Find the volume of the solid, giving your answer to the nearest cubic centimetre.   (3 marks)

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`11\ 451\ text(cm)^3`

Show Worked Solution
`text{Area (face)}` `= (9 xx 13) + (1/2 xx 13 xx 7)`  
  `=162.5\ text{cm}^2`  

 

`text(Volume)` `= A h`
  `= 162.5 xx 6`
  `= 975\ text(cm)^3`

Volume, SMB-010

The solid below is a triangular prism. All measurements are in centimetres.
 

Find the volume of the solid, in cubic centimetres.  (2 marks)

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`378\ \text{cm}^3`

Show Worked Solution
`text{Area (triangle)}` `=1/2 xx b xx h`  
  `=1/2 xx 9 xx 12`  
  `=54\ text{cm}^2`  

 

`text(Volume)` `= A xx h`
  `= 54 xx 7`
  `= 378\ \text{cm}^3`

Volume, SMB-009

An ancient temple, modelled below, was constructed as a trapezoidal prism.
 

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

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`432\ \text{m}^3`

Show Worked Solution
`text{Area (trapezium)}` `=1/2h(a+b)`  
  `=1/2 xx 4 (10+14)`  
  `=48\ text{m}^2`  

 

`text(Volume)` `= A xx h`
  `= 48 xx 9`
  `= 432\ \text{m}^3`

Volume, SMB-008

A jewellery box in the shape of a rectangular prism is pictured below with all measurements in centimetres.
 

Find the volume of the jewellery box, giving your answer correct to the nearest cubic centimetre.  (2 marks)

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`271\ text{cm}^3`

Show Worked Solution
`text(Volume)` `= l xx b xx h`
  `= 9.5 xx 4.2 xx 6.8`
  `= 271.32`
  `= 271\ text{cm}^3\ text{(nearest cm}^3 text{)}`

Volume, SMB-004

Calculate the volume of earth that must be excavated so that the pool shell below fits exactly into the hole. All measurements are in metres.  (3 marks)
 

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\(136.8\ \text{m}^3\)

Show Worked Solution

\(\text{The face of the pool is a rectangle and trapezium (composite).}\)

\(\text{Area (face)}\) \(= (2.3 \times 4) + \dfrac{1}{2} h (a+b)\)
  \(= 9.2 + \dfrac{1}{2} \times 8 (2.3+1.1) \)
  \(=22.8\ \text{m}^2\)

 

\(\text{Volume}\) \(=Ah\)
  \(=22.8 \times 6\)
  \(=136.8\ \text{m}^3\)

Volume, SMB-003

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

 

Find the volume of the prism (answer correct to 1 decimal place).  (3 marks)

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`3464.1\ text(cm)^3`

Show Worked Solution

`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  (1 d.p.)}`

Volume, SMB-002

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 ?   (3 marks)

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`576\ text(cm)^3`

Show Worked Solution

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

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(cm)^3`

Volume, SMB-001

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

Find the volume of the skip bin in cubic metres.   (3 marks)

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`5.4\ text(m)^3`

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`

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)

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`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 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
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`=> 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 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)

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

 

ii.  `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)}`

 

iii. `text(Percentage of clay removed)`

`= ((3 × 49.260…))/1512 × 100`

`= 9.773…`

`= 9.8 text{%   (1 d.p.)}`

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`