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v1 Measurement, STD2 M7 2012 HSC 27c

A topographic map has a scale of 1 : 250 000.

  1. Two lookouts are 3.6 cm apart on the map.

     

    What is the actual distance between the two lookouts, in kilometres?   (1 mark)

  2. Two towns are 42.5 km apart. How far apart are the two towns on the map, in centimetres?   (1 mark)

Show Answers Only
  1. `text(9 km)`
  2. `text(17 cm)`
Show Worked Solution
♦ Mean mark 41%
MARKER’S COMMENT: Better responses converted between centimetres, metres and kilometres systematically using the map scale (1 unit on the map represents 250 000 of the same unit in reality).
i.    `text{Actual distance (3.6 cm)}` `= 3.6 xx 250\ 000`
  `= 900\ 000\ text(cm)`
  `= 9\ 000\ text(m)`
  `= 9\ text(km)`

 

`:.\ text(The 2 lookouts are 9 km apart.)`

♦ Mean mark 45%

ii.   `text(Towns are 42.5 km apart.)`

`text{From the scale, } 1\ \text{cm} = 250\ 000\ \text{cm} = 2\ 500\ \text{m} = 2.5\ \text{km}`

`=>\ \text{On the map, } 42.5\ \text{km} = 42.5/2.5 = 17\ \text{cm}`

`:.\ \text{Distance on the map is 17 cm.}`

Similarity, SMB-028

The floor plan of a home unit has been drawn to scale.
 


 

What is the total floor area of the home unit in square metres?  (2 marks)

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`text{55.5 m}^2`

Show Worked Solution

`text{Conversion: 1000 mm = 1 metre}`

`text{Calculate area by splitting into 2 rectangles}`

`text{Area}`  `=  5.5 × 5 + (4.5 +2.5) xx 4`  
   `=  55.5\ text{m}^2`  

Similarity, SMB-015

Towns `A, B` and `C` are marked on the scale diagram below.
  

The distance from Town `A` to Town `B` is 9 km.

What is the distance, in kilometres, from Town `B` to Town `C`?   (2 marks)

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`3.6\ text(km)`

Show Worked Solution

`text(Let)\ \ d = text(distance from Town)\ B\ text(to Town)\ \ C`

`text(S)text(ince the diagram is to scale,)`

`d/9` `= 2/5`
`:. d` `= (9 xx 2)/5`
  `= 3.6\ text(km)`

Similarity, SMB-014 MC

Spiro is making a scale drawing of his house.

  • The height of the garage in the scale model is 6 centimetres.
  • The height of Spiro's actual garage is 3 metres.

What does 1 centimetre in Spiro's scale drawing represent in his real house?

  1. 1800 cm
  2. 200 cm
  3. 100 cm
  4. 50 cm
Show Answers Only

`D`

Show Worked Solution
`text(6 cm)` `= 3\ text(m)`
  `= 300\ text(cm)`
`:. 1\ text(cm)` `= 300 -: 6`
  `= 50\ text(cm)`

 
`=>D`

Similarity, SMB-013

Libby plays on a hockey field that is 120 metres long.

She makes a scale diagram of the field using a ratio of `1: 400`.
 

How long, in centimetres, should Libby make the scale diagram of the field?   (2 marks)

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`text(30 cm)`

Show Worked Solution

`text(Ratio is)\ \ 1:400`

`text{Converting to centimetres:}`

`text{120 m = 120 × 100 = 12 000 cm}`

`:.\ text(Length of the scale diagram)`

`=12\ 000 ÷ 400`

`= 30\ text(cm)`

Similarity, SMB-010 MC

The actual body length of a beetle Brad has caught is 24 mm.

A scale drawing of the beetle is shown below.

What scale is used in the drawing?

  1. \(1\ \text{cm represents } 5\ \text{mm}\)
  2. \(1\ \text{cm represents } 2\ \text{mm}\)
  3. \(2\ \text{cm represents } 1\ \text{mm}\)
  4. \(5\ \text{cm represents } 1\ \text{mm}\)
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\(B\)

Show Worked Solution
\(12\ \text{cm}\) \(: 24\ \text{mm}\)
\(120\ \text{mm}\) \(: 24\ \text{mm}\)
\(5\) \(: 1\)
\(1\ \text{cm}\) \(: 2\ \text{mm  (possible options)}\)

 
\(\Rightarrow B\)

Similarity, SMB-008 MC

The picture below shows a flower.
 

 
The picture is 2 cm wide. The actual flower is 40 cm wide.

What scale is used in the picture?

  1. 1 cm represents 80 cm
  2. 4 cm represents 20 cm
  3. 1 cm represents 20 cm
  4. 1 cm represents 2 cm
Show Answers Only

`C`

Show Worked Solution

`2:40\ \ =>\ \ 1:20`

`text(1 cm represents 20 cm)`

`=>C`

Similarity, SMB-003

On a map, the distance between two towns is measured at 54 millimetres.

  1. The actual distance between the two towns is 16.2 kilometres. What is the scale of the map?  (1 mark)

  2. If two other towns on the same map are 9.2 centimetres apart, what is the actual distance between the two towns in kilometres?  (1 mark)

Show Answers Only
  1. `1: 300\ 000`
  2. `27.6\ text(km)`
Show Worked Solution

i.   `text{Convert both distance to the same unit (cm).}`

`text(54 mm)` `= 5.4\ text(cm)`
`text(16.2 km)` `= 16\ 200\ text(m)`
  `= 1\ 620\ 000\ text(cm)`

 

`:.\ text(Scale) \ \ 5.4\ ` `: 1\ 620\ 000`
`1\ ` `: 300\ 000`

 

ii.    `text(Actual distance)` `= 9.2 xx 300\ 000`
    `= 2\ 760\ 000\ text(cm)`
    `= 27\ 600\ text(m)`
    `= 27.6\ text(km)`

Similarity, SMB-002

A map is drawn to scale, on 1-cm paper, showing the position of a supermarket and a cinema. A reservoir is also shown.
 


 

It takes 10 minutes to walk in a straight line from the cinema to the supermarket at a constant speed of 3 km/h. Show that the scale of the map is 1 cm = 100 m.  (3 marks)

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`text(See Worked Solutions)`

Show Worked Solution

`text(3 km/h = 3000 metres per 60 minutes)`

♦ Mean mark part 49%.

`text(In 10 minutes:)`

`text(Actual distance) = 3000 xx 10/60 = 500\ text(metres)`

`text(Distance on map = 5 cm)`

`:.\ text(Scale   5 cm)` `: 500\ text(metres)`
`text(1 cm)` `: 100\ text(metres)`

Similarity, SMB-001

The scale on a given map is `1:80\ 000`.

If the actual distance between two points is 3.4 kilometres, how far apart on the map would be the two points be, in centimetres?  (2 marks)

Show Answers Only

`4.25\ text(cm)`

Show Worked Solution

`text(Distance on map)`

`= text(real distance)/text(scale)`

`= text(3.4 km)/(80\ 000)`

`= text(3400 m)/(80\ 000)`

`= 0.0425\ text(m)`

`= 4.25\ text(cm)`

Measurement, STD2 M7 2012 HSC 27c

A map has a scale of  1 : 500 000.

  1. Two mountain peaks are 2 cm apart on the map.

     

    What is the actual distance between the two mountain peaks, in kilometres?  (1 mark)

  2. Two cities are 75 km apart. How far apart are the two cities on the map, in centimetres?   (1 mark)

Show Answers Only
  1. `text(10 km)`
  2. `text(15 cm)`
Show Worked Solution
♦ Mean mark 37%
MARKER’S COMMENT: Better responses realised that 1 unit on the map represented 1 unit x 500,000 in real life.
i.  `text{Actual distance (2 cm)}` `= 2 xx 500\ 000`
  `= 1\ 000\ 000\ text(cm)`
  `= 10\ 000\ text(m)`
  `=10\ text(km)`

 

`:.\ text(The 2 mountain peaks are 10 km apart.)`

 

♦ Mean mark 44%

ii.   `text(Cities are 75 km apart.)`

`text{From part (i), we know 2 cm = 10 km}`

`=>\ text(1 cm = 5 km)`

`=>\ text(On the map,  75 km)= 75/5=15\ text(cm)` 

`:.\ text(Distance on the map is 15 cm.)`