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Measurement, STD2 M7 2025 HSC 9 MC

The ratio of the dimensions of a model car to the dimensions of an actual car is \(1:64\). The actual car has a length of 4.9 m.

What is the length of the model car in cm, correct to 1 decimal place?

  1. 3.1
  2. 7.7
  3. 13.1
  4. 59.1
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\(B\)

Show Worked Solution

\(\text{Actual length}=4.9\ \text{m}\ =490\ \text{cm}\)

\(\therefore\ \text{Model car length}\ =490\times\dfrac{1}{64}=7.65625\approx 7.7\ \text{cm}\)

 \(\Rightarrow B\)

v1Measurement, STD2 M7 2021 HSC 25

A rectangular sportsground has been drawn to scale on a 1-cm grid as shown. The scale used is `1:2000`.
 

Johnny took 12 minutes to walk around the perimeter of this sportsground.

What was Johnny's average speed in kilometres per hour?  (4 marks)

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`3.4 \ text{km/hr}`

Show Worked Solution
`text{Distance walked on map} \ = \ 2 times  (10 + 7) = 34 \ text{cm}`
Mean mark 54%.

 

`text{Actual distance}` ` =34 times 2000`  
  `= 68\ 000 \ text{cm}`  
  `=680 \ text{m}`  
  `= 0.68 \ text{km}`  

 
`12 \ text{minutes}\ = 12/60 = 0.2 \ text{hours}`
 

`text{Speed}` `= text{distance}/text{time}`  
  `= 0.68/0.2`  
  `= 3.4 \ text{km/hr}`  

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)

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  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.}`

Measurement, STD2 M7 2021 HSC 25

A rectangular sportsground has been drawn to scale on a 1-cm grid as shown. The scale used is `1:3000`.
 

Kerry took 12 minutes to walk around the perimeter of this sportsground.

What was Kerry's average speed in kilometres per hour?  (4 marks)

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`3.9 \ text{km/hr}`

Show Worked Solution
`text{Distance walked on map} \ = \ 2 times  (8 + 5) = 26 \ text{cm}`
Mean mark 54%.

 

`text{Actual distance}` ` =26 times 3000`  
  `= 78\ 000 \ text{cm}`  
  `=780 \ text{m}`  
  `= 0.78 \ text{km}`  

 
`12 \ text{minutes}\ = 12/60 = 0.2 \ text{hours}`
 

`text{Speed}` `= text{distance}/text{time}`  
  `= 0.78/0.2`  
  `= 3.9 \ text{km/hr}`  

Measurement, STD2 M7 2019 HSC 41

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


 

  1. 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)

  2. The reservoir is initially empty. During a storm 20 mm of rain falls on the reservoir.

     

    With the aid of one application of the trapezoidal rule, estimate the amount of water in the reservoir immediately after the storm. Assume that all rain which falls over the reservoir is stored. Give your answer in cubic metres.  (3 marks)

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  1. `text(See Worked Solutions)`
  2. `1600\ text(m)^3`
Show Worked Solution

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

♦ Mean mark part (a) 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)`

 

b.   

 

`A` `~~ h/2(a + b)`
  `~~ 400/2(100 + 300)`
  `~~ 80\ 000\ text(m²)`

 
`text(Converting mm to metres:)`

♦♦ Mean mark part (b) 28%.

`text(20 mm) = 20/1000 text(m = 0.02 metres)`

 
`:.\ text(Volume of water)`

`= A xx h`

`= 80\ 000 xx 0.02`

`= 1600\ text(m)^3`

Measurement, STD2 M7 SM-Bank 11

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)

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  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)`

Measurement, STD2 M7 SM-Bank 7

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)

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`4.25\ text(cm)`

Show Worked Solution

`text(Actual 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 2005 HSC 4 MC

The diagram is a scale drawing of a butterfly.
 

2UG-2005-4MC 
 

What is the actual wingspan of the butterfly?

  1.    2.5 cm
  2.    3 cm
  3.    15 cm
  4.    8.75 cm
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`B`

Show Worked Solution

`text(Wingspan is 3 times the scale distance)`

`text(that equals 1 cm.)`

`:.\ text(Wingspan)` `= 3 × 1`
  `= 3\ text(cm)`

`=> B`

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)

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  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.)`