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Volume, SMB-016

The truncated cone, pictured below, is made by cutting a right cone of height 60 centimetres.
 

Find the volume of the truncated cone, giving your answer to the nearest cubic centimetre.   (4 marks)

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`7916\ text(cm)^3`

Show Worked Solution

`text{Radius (original cone) = 12 cm}`

`text{Radius (small cone) = 6 cm}`

`text{Volume}` `=\ text{Original cone – Small cone}`  
  `=(1/3 xx pi xx 12^2 xx 60)-(1/3 xx pi xx 6^2 xx 30)`  
  `= 7916.81…`  
  `=7916\ text{cm}^3\ \ text{(nearest cm}^3 text{)}`  

Volume, SMB-015

A funnel is made in the shape of a square cone with radius 9.5 centimetres and height 19.5 centimetres.
 

Find the volume of the funnel in cubic centimetres, giving your answer correct to 2 decimal places.   (2 marks)

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`1842.94\ text(cm)^3`

Show Worked Solution
`text{Volume}` `= 1/3 xx A xx h`  
  `=1/3 xx pi xx 9.5^2 xx 19.5`  
  `= 1842.936…`  
  `=1842.94\ text{cm}^3`  

Volume, SMB-014

The storage building below is constructed by joining a square pyramid to a cube, with all measurements in metres.
 

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

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`150\ text(m)^3`

Show Worked Solution
`text{Volume (cube)}` `= 5 xx 5 xx 5`  
  `=125\ text{m}^3`  

 

`text{Volume (pyramid)}` `= 1/3 A h`
  `=  1/3 xx 5 xx 5 xx 3`
  `= 25\ text(cm)^3`

 
`text{Total volume}\ = 125 + 25 = 150\ text{m}^3`

Volume, SMB-013

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)

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`312\ 000\ text(m)^3`

Show Worked Solution
`text{Volume}` `= 1/3 xx A xx h`  
  `=1/3 xx 120 xx 120 xx 65`  
  `=312\ 000\ text{m}^3`  

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`
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`A`

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

 
`=>A`

Measurement, STD2 M1 2017 HSC 30e

A solid is made up of a sphere sitting partially inside a cone.

The sphere, centre `O`, has a radius of 4 cm and sits 2 cm inside the cone. The solid has a total height of 15 cm. The solid and its cross-section are shown.
 


 

Using the formula  `V=1/3 pi r^2h`  where `r`  is the radius of the cone's circular base and `h` is the perpendicular height of the cone, find the volume of the cone, correct to the nearest cm³?  (3 marks)

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`113\ text{cm}^3`

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`V = 1/3 xx text(base of cone × height)`

`text(Consider the circular base area of the cone,)`

`text(Find)\ x\ \ text{(using Pythagoras):}`

`x^2` `= 4^2-2^2 = 16-4 = 12`
`x` `= sqrt12\ text(cm)`

 

`:. V` `= 1/3 xx pi xx (sqrt12)^2 xx (15-6)`
  `= 1/3 xx pi xx 12 xx 9`
  `= 113.097…`
  `= 113\ text{cm}^3\ text{(nearest cm}^3 text{)}`

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`
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`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 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
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`B`

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

 
`=>\ B`