smc-1105-20-Maps and Scale Drawings

Measurement, STD1 M5 2024 HSC 32

A scale diagram is shown with locations \(A, B\) and \(C\) marked (assume grid squares are 1 cm × 1 cm).

Jo takes 24 minutes to walk from \(A\) to \(B\) (in a straight line) when walking at 3 km per hour.

What is the scale used in the diagram? (3 marks)
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What is the distance from \(B\) to \(C\), in kilometres? (2 marks)
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a. \(1:20\,000\)

b. \(\text{1.4 km}\)

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a. \(\text{Find distance }A\rightarrow B:\)

\(D=S \times T = 3 \times \dfrac{24}{60} = 1.2\ \text{km}\)

	\(\text{Scale }\rightarrow\ 6\ \text{cm }\)	\(:\ 1.2\ \text{km}\)
	\(6\ \text{cm }\)	\(:\ 1200\ \text{m}\)
	\(6\ \text{cm }\)	\(:\ 1200\times 100\ \text{cm}\)
	\(6\ \)	\(:\ 120\,000\)
	\(1\ \)	\(:\ 20\,000\)

b. \(\text{Distance }B\rightarrow C = 7\ \text{grid units}\)

\(\text{Real distance }B\rightarrow C\)	\(=7 \times 20\,000\)
	\(=140\,000\ \text{cm}\)
	\(=1400\ \text{m}\)
	\(=1.4\ \text{km}\)

Measurement, STD1 M5 2024 HSC 11

A scale drawing of a tree is shown. The scale is \(1 : 50\).

What is the actual height of the tree, in metres? (2 marks)

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\(4\ \text{m}\)

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\(8\times 1\ \text{cm}\,\)	\(:\,8\times 50\ \text{cm}\)
\(8\ \text{cm}\,\)	\(:\,400\ \text{cm}\)
\(8\ \text{cm}\,\)	\(:\,4\ \text{m}\)

\(\therefore\ \text{The tree is 4 metres tall.}\)

Measurement, STD1 M5 2023 HSC 31

A scale drawing of a garden plan, where 1 cm represents 2 m, is shown.

The shaded areas in the diagram represent the garden beds.

Woodchips will be laid as a mulch on the garden beds to a depth of \(10 cm\).

What is the volume of woodchips required? (5 marks)

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\(10.113\ \text{m}^3\)

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\(\text{Scale }\longrightarrow 1\ \text{cm}=2\ \text{m}\)

\(\text{Area of triangle}\)	\(=\dfrac{1}{2}\times 8\times 4\)
	\(=16\ \text{m}^2\)

\(\text{Area of L-shape}\)	\(=2\times 10+2\times 20\)
	\(=60\ \text{m}^2\)

\(\text{Area of semi-cirle}\)	\(=\dfrac{1}{2}\times \pi\times 4^2\)
	\(=25.13\dots\ \text{m}^2\)

\(\text{Total Area}\)	\(=16+60+25.13\)
	\(=101.13\ \text{m}^2\)

\(\text{Volume}\)	\(=101.13\times 0.1\)
	\(=10.113\ \text{m}^3\)

♦♦♦♦ Mean mark 15%.

Measurement, STD1 M5 2023 HSC 12

A floor plan is shown.

What are the dimensions, in metres, of the living room? (2 marks)

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The shaded area of the kitchen floor is to be tiled. Each tile is 40 cm by 40 cm. How many tiles are needed to cover the kitchen floor? (2 marks)

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The tiles are supplied in boxes of 10 . Only full boxes can be purchased. How many boxes need to be purchased to tile the kitchen floor? (1 mark)

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a. \(5.2\times 5.94\)

b. \(72\text{ tiles}\)

c. \(8\text{ boxes}\)

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a. \(\text{Dimensions}=5.2\times (2.15+0.16+3.6)=5.2\times5.94\)

♦ Mean mark (a) 51%.

b. \(\text{Method 1}\)

\(\text{Kitchen Area}\)	\(=3.2\times 3.6\)
	\(=11.52\text{ m}^2\)
\(\text{Tile area}\)	\(=0.4\times 0.4\)
	\(=0.16\text{ m}^2\)
\(\text{Tiles needed}\)	\(=\dfrac{11.52}{0.16}\)
	\(=72\text{ tiles}\)

\(\text{Method 2}\)

\(\text{Tiles to fit width}\)	\(=\dfrac{3.6}{0.4}\)
	\(=9\)
\(\text{Tiles to fit length}\)	\(=\dfrac{3.2}{0.4}\)
	\(=8\)
\(\text{Tiles needed}\)	\(=9\times 8\)
	\(=72\text{ tiles}\)

♦ Mean mark (b) 43%.

c. \(\text{Boxes}=\dfrac{72}{10}=7.2\)

\(\therefore\ \text{Boxes }=8\)

Measurement, STD1 M5 2023 HSC 3 MC

Two towns are \(5\) cm apart on a map that uses a scale of \(1 : 100 000\).

What is the actual distance between the two towns?

\(5\text{ km}\)
\(50\text{ km}\)
\(500\text{ km}\)
\(5000\text{ km}\)

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\(A\)

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\(5\text{ cm : }\)	\( 500\ 000\text{ cm}\)
\(5\text{ cm : }\)	\( 5000\text{ m}\)
\(5\text{ cm : }\)	\( 5\text{ km}\)

\(\Rightarrow A\)

♦ Mean mark 23%.

Measurement, STD1 M5 2021 HSC 21

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}`

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`text{Distance walked on map} \ = \ 2 times (8 + 5) = 26 \ text{cm}`

♦♦♦ Mean mark 24%.

`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 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)`

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`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 2018 HSC 26g

A field diagram of a block of land has been drawn to scale. The shaded region `ABFG` is covered in grass.

The actual length of `AG` is 24 m.

If the length of `AG` on the field diagram is 8 cm, what is the scale of the diagram? (1 mark)

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How much fertiliser would be needed to fertilise the grassed area `ABFG` at the rate of 26.5 g /m²? (3 marks)

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`text(1 : 300)`
`5008.5\ text(grams)`

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♦♦ Mean mark 32%.

i.	`text(Scale 8 cm)`	`\ :\ text(24 m)`
	`text(1 cm)`	`\ :\ text(3 m)`
	`1`	`\ :\ 300`

ii. `text(Area of rectangle)\ ABFE`

`= 6\ text(cm × 3 cm)`

`= 18\ text(m × 9 m)`

`= 162\ text(m²)`

`text(Area of)\ DeltaEFG`

`= 1/2 xx 3\ text(cm) xx 2\ text(cm)`

`= 1/2 xx 9 xx 6`

`= 27\ text(m²)`

`:.\ text(Fertiliser needed)`	`= (162 + 27) xx 26.5`
	`= 5008.5\ text(grams)`

Measurement, STD2 M7 2005 HSC 4 MC

The diagram is a scale drawing of a butterfly.

What is the actual wingspan of the butterfly?

2.5 cm
3 cm
15 cm
8.75 cm

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`B`

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`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.

Two mountain peaks are 2 cm apart on the map.

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

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

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`text(10 km)`
`text(15 cm)`

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♦ 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.)`