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Area, SMB-035

The surface area of a brick is given by the rule:

total surface area = 2 × [(width × height) + (width × length) + (height × length)]

The brick shown has a total surface area of 982 square centimetres.
 

What is the width of the brick?   (2 marks)

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`10.5\ text(cm)`

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`text(Let)\ \ w = text(width of the brick)`

`text(Surface Area)` `= 2 xx [8w + 22w + (8 xx 22)]`
`982` `= 2(30w + 176)`
`982` `= 60w + 352`
`60w` `= 630`
`:. w` `= 10.5\ text(cm)`

Area, SMB-034

The total surface area of a cube is 150 cm².
 

 How long is an edge of the cube?   (2 marks)

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`text(5 cm)`

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`text(Surface area of 1 face of the cube)`

`=150/6`

`=25\ text(cm)^2`
 

`text(Let)\ \ x=\ text(length of 1 side)`

`x^2` `=25`
`:.\ x` `=5\ text(cm)`

Area, SMB-033

The rectangular prism, shown below, is cut in half to form 2 equal cubes.
 

What is the ratio of the surface area of the rectangular prism to one of the cubes?   (3 marks)

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`5 : 3`

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`text(S.A. of rectangular prism)`

`= 2 xx (h xx h) + 4 xx (2h xx h)`

`= 2h^2 + 8h^2`

`= 10 h^2`
 

`text(S.A. of cube)\ = 6 xx (h xx h) = 6h^2`
 

`:.\ text(Ratio of rectangular prism to cube)`

`= 10h^2 : 6h^2`

`= 5 : 3`

Area, SMB-032

A net of a cube is pictured below.
  

The cube has a volume of 512 cubic centimetres.

What is the height of the net in centimetres?   (3 marks)

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`24\ text(cm)`

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`text(Let)\ \ s` `=\ text(length of 1 side of the cube)`
`s^3` `= 512`
`s` `= 8\ text(cm)`

 

`:.\ text(Height of net)` `= 3 xx 8`
  `= 24\ text(cm)`

Area, SMB-026

Clancy needs to paint all sides of a triangular prism.

The area of each triangular face is 12 cm².

What is the total area Clancy needs to paint?   (2 marks)

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`78\ text(cm²)`

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`text(Total Area)` `= (2 xx 12) + 2 xx (3 xx 5) + (8 xx 3)`
  `= 78\ text(cm²)`

Area, SMB-022

A cube has a side length of 6 cm.

Two smaller cubes of side length 3 cm are attached to the larger cube as shown in the diagram below.
  

What is the surface area of the new object?   (3 marks)

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`270\ text(cm²)`

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`text(One strategy:)`

`text{Calculate the surface area (S.A.) of the 2 objects joined}`

`text(separately and deduct the area where they join.)`
 

`text{S.A. (large cube)} = 6 xx 6xx 6 = 216\ text(cm²)`

`text{S.A. (2 smaller cubes joined together)}`

`=2 xx (3 xx 3) + 4 xx (6 xx 3)`

`=90\ text(cm²)`

`text{S.A. to be deducted (where cubes are attached)}`

`= 2 xx (6 xx 3)`

`= 36\ text(cm²)`

 

`:.\ text{S.A. (new object)}` `= 216 + 90-36`
  `=270\ text(cm²)`

Area, SMB-012

Find the surface area of the solid below where all measurements are in metres.  (2 marks)
 

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\(104\ \text{m}^2\)

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\(\text{Area (front side)} = (4 \times 2) + (2 \times 2) = 12\ \text{m}^2 \)

\(\text{S.A.}\) \(=2 \times 12 + 2 \times (5 \times 4) + 4 \times (5 \times 2) \)  
  \(= 24 + 2(20) + 4(10) \)  
  \(=104\ \text{m}^2\)  

Area, SMB-011

A triangular prism has the following dimensions (all measurements are in metres).
 

  1. Draw a net of the solid   (1 mark)
  1. Find the surface area of the prism.  (2 marks)
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i. 

ii.    \(348\ \text{m}^2\)

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i.    

ii.    \(\text{S.A.}\) \(=2 \times (5 \times 18) + (8 \times 18) + 2 \times (\dfrac{1}{2} \times 8 \times 3) \)  
  \(= 2(90) + 144 + 2(12) \)  
  \(=348\ \text{m}^2\)  

Area, SMB-010

A triangular prism has the following dimensions (all measurements are in centimetres).
 

  1. Draw a net for this solid.   (1 mark)
  1. Find the surface area of the prism.  (2 marks)
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i.  

 

ii.    \(408\ \text{cm}^2\)

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i.   

ii.    \(\text{S.A.}\) \(=(15 \times 6) + (15 \times 10) + (15 \times 8) + 2 \times (\dfrac{1}{2} \times 8 \times 6) \)  
  \(=90+150+120+2(24) \)  
  \(=408\ \text{cm}^2\)  

Area, SMB-009

Find the surface area of the rectangular prism pictured below, correct to 1 decimal place. All measurements are in metres.  (2 marks)
 

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\(68.0\ \text{m}^2\)

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\(\text{S.A.}\) \(=2 \times (2.8 \times 1.7) + 2 \times (2.8 \times 6.5) + 2 \times (1.7 \times 6.5) \)  
  \(= 2(4.76 + 18.2 + 11.05) \)  
  \(=68.02\)  
  \(=68.0\ \text{m}^2\ \ \text{(1 d.p.)}\)  

Area, SMB-008

Find the surface area of the rectangular prism pictured below. All measurements are in metres.  (2 marks)
 

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\(368\ \text{m}^2\)

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\(\text{S.A.}\) \(=2 \times (6 \times 14) + 2 \times (6 \times 5) + 2 \times (5 \times 14) \)  
  \(=2(84+30+70)\)  
  \(=368\ \text{m}^2\)  

Measurement, STD2 M1 2020 HSC 25

A composite solid consists of a triangular prism which fits exactly on top of a cube, as shown.
 

Find the surface area of the composite solid.   (3 marks)

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`424 \ text{cm}^2`

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`text{S.A. of 1 face of cube} = 8 xx 8 = 64 \ text{cm}^2`

`text{Height of triangle} = 11 – 8 = 3 \ text{cm}`

`therefore \ text{S.A. (triangular prism)}` `= 2 xx ( frac{1}{2} xx 8 xx 3 ) + 2 xx (5 xx 8)`
  `= 24 + 80`
  `= 104 \ text{cm}^2`

 

`therefore \ text{Total S.A.}` `= 5 xx 64 + 104`
  `= 424 \ text{cm}^2`