smc-1105-20-Maps and Scale Drawings

Measurement, STD1 M3 2025 HSC 20

A map of a park containing a duck pond is shown.

A fence is built passing through the points $A$, $B$ and $C$ around the duck pond.

Using the scale provided on the map, calculate the length of the fence $AB$. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---

The length of $AB$ is equal to the length of $BC$.
Use Pythagoras’ theorem to calculate the length of $AC$ in metres. Give your answer correct to 3 significant figures. (3 marks)
--- 7 WORK AREA LINES (style=lined) ---
What is the true bearing of point $A$ from point $C$ ? (2 marks)
--- 7 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $75\ \text{metres}$

b. $106\ \text{metres}$

c. $225^\circ$

Show Worked Solution

a. $\text{Scale: 1 grid width = 5 metres}$

$AB = 15 \times 5 = 75\ \text{metres}$

b. $\text{Using Pythagoras:}$

$AC^2=AB^2+BC^2$

$AC^2=75^2+75^2=11250$

	$\therefore\ AC$	$=\sqrt{11250}$
		$=106.066…$
		$\approx 106\ \text{m (3 sig fig)}$

c. $\text{Since }AB=BC:$

$\angle BAC=\angle BCA=45^\circ$

$\text{Bearing of \(A$ from $C$}\ =180+45=225^\circ\)

Measurement, STD1 M5 2025 HSC 26

A scale of $1 : 50$ is used to draw a rectangular area on a 2 mm grid as shown. The actual rectangular area is to be tiled.

The tiles cost $150 per square metre and the tiler orders 15% extra tiles to allow for cutting and breakage.

The tiler charges $90 per hour and will take 20 hours to complete the tiling.

Calculate the total cost of the tiles and tiling. Give your answer to the nearest dollar. (4 marks)

--- 10 WORK AREA LINES (style=lined) ---

Show Answers Only

$$6105.60$

Show Worked Solution

$\text{Using scale where each grid is 2mm × 2 mm:}$

$2\ \text{mm}\ =50 \times 2 = 100\ \text{mm in actual length}$

$\text{100 mm = 10 cm}$

$\Rightarrow \ \text{i.e. each grid represents 10 cm in actual length.}$

$\text{Width}\ = 52 \times 10 = 520\ \text{cm}\ = 5.2\ \text{m}$

$\text{Height}\ = 48 \times 10 = 480\ \text{cm}\ = 4.8\ \text{m}$

$\text{Area}=5.2 \times 4.8=24.96\ \text{m}^2$

$\text{Cost of tiles}=24.96\times $150\times 1.15=$4305.60$

$\text{Labour Cost}=90\times 20=$1800$

$\therefore\ \text{Total Cost}=$4305.60+$1800=$6105.60$

Measurement, STD1 M5 2025 HSC 10 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?

3.1
7.7
13.1
59.1

Show Answers Only

$B$

Show Worked Solution

$\text{Actual length}=4.9\ \text{m}\ =490\ \text{cm}$

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

$\Rightarrow B$

v1 Measurement, STD2 M7 2012 HSC 27c

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

Two lookouts are 3.6 cm apart on the map.

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

--- 3 WORK AREA LINES (style=lined) ---
Two towns are 42.5 km apart. How far apart are the two towns on the map, in centimetres? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(9 km)`
`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.}`

v1 Measurement, STD2 M7 2005 HSC 4 MC

The diagram is a scale drawing of a paper plane.

What is the actual wingspan of the paper plane?

4 cm
8 cm
12 cm
16 cm

Show Answers Only

`B`

Show Worked Solution

`text(The diagram shows a scale of 1)`

`text(The measured wingspan is 8 cm)`

`:.\ text(Wingspan)`	`= 8 × 1`
	`= 8 \ \text{cm}`

`=> B`

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)
--- 7 WORK AREA LINES (style=lined) ---
What is the distance from $B$ to $C$, in kilometres? (2 marks)
--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $1:20\,000$

b. $\text{1.4 km}$

Show Worked Solution

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\ $	$:\ 20\,000$

♦♦♦ Mean mark 15%.

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

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

♦♦♦ Mean mark 28%.

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)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$4\ \text{m}$

Show Worked Solution

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

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

♦ Mean mark 48%.

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)

Show Only

$10.113\ \text{m}^3$

Show Worked Solution

$\text{Scale }\longrightarrow 1\ \text{cm}=2\ \text{m}$

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

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

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

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

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

♦♦♦♦ 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)

--- 4 WORK AREA LINES (style=lined) ---
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)

--- 5 WORK AREA LINES (style=lined) ---
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)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $5.2\times 5.94$

b. $72\text{ tiles}$

c. $8\text{ boxes}$

Show Worked Solution

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?

$5\text{ km}$
$50\text{ km}$
$500\text{ km}$
$5000\text{ km}$

Show Answers Only

$A$

Show Worked Solution

$5\text{ cm : }$	$ 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)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`3.9 \ text{km/hr}`

Show Worked Solution

`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)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`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 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)

--- 1 WORK AREA LINES (style=lined) ---
How much fertiliser would be needed to fertilise the grassed area `ABFG` at the rate of 26.5 g /m²? (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

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

Show Worked Solution

♦♦ 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

Show Answers Only

`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.

Two mountain peaks are 2 cm apart on the map.

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

--- 3 WORK AREA LINES (style=lined) ---
Two cities are 75 km apart. How far apart are the two cities on the map, in centimetres? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(10 km)`
`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.)`

	\(\therefore\ AC\)	\(=\sqrt{11250}\)
		\(=106.066…\)
		\(\approx 106\ \text{m (3 sig fig)}\)

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

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

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

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

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

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

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

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