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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`

Show Worked Solution

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

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

`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 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)

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  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 M6 2008 HSC 25c

Pieces of cheese are cut from cylindrical blocks with dimensions as shown.

 

Twelve pieces are packed in a rectangular box. There are three rows with four pieces of cheese in each row. The curved surface is face down with the pieces touching as shown.
  

  1. What are the dimensions of the rectangular box?   (4 marks)

     

    To save packing space, the curved section is removed.
     
             
     

  2. What is the volume of the remaining triangular prism of cheese? Answer to the nearest cubic centimetre.   (2 marks)

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a.    `41\ text(cm) xx 21\ text(cm) xx 15\ text(cm)`

b.    `506\ text(cm)^3`

Show Worked Solution

a.  `text(Box height) = 15\ text(cm)`

♦ Mean mark (a) 45%.

`text{(radius of the arc)}`

`text(Box width)` `= 3 xx 7= 21\ text(cm)`
`text(Box length)` `= 4x`

`text(Using cosine rule:)`

`c^2` `= a^2 + b^2-2ab\ cos C`
`x^2` `= 15^2 + 15^2-2 xx 15 xx 15 xx cos 40^@`
  `= 450-344.7199…`
  `= 105.2800…`
`x` `= 10.2606…`

 
`text{Box length} = 4 xx 10.2606…= 41.04…`

`:.\ text(Dimensions are)\ \ 41\ text(cm) xx 21\ text(cm) xx 15\ text(cm)`
 

b.  `text(Volume) = Ah`

♦♦♦ Mean mark (b) 22%.

`h = 7\ text(cm)`

`A= 1/2 ab\ sin C= 1/2 xx 15 xx 15 xx sin 40^@= 72.3136…`

`:. V` `= 72.3136… xx 7`
  `= 506.195…`
  `= 506\ text(cm)^3\ \ text{(nearest whole)}`