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Measurement, STD2 EQ-Bank 21

Sam is installing a lap pool in the shape of a trapezoidal prism, with dimensions as shown.
 

  1. Calculate the volume of the lap pool in cubic metres.   (2 marks)
  2. What is the capacity of the pool in litres?   (1 mark)
Show Answers Only

a.    \(5.4\ \text{m}^3\)

b.    \(5400\ \text{L}\)

Show Worked Solution
a.     \(\text{Area of trapezoid}\) \(=\dfrac{h}{2}\big(a + b\big)\)
    \(=\dfrac{1.2}{2}\big(2.4+3.6\big)\)
    \(=3.6\ \text{m}^2\)

 
\(\text{Volume} = Ah=3.6\times 1.5=5.4\ \text{m}^3\)

  
b.    \(\text{Using 1 m}^3=1000\ \text{L}\)

\(\text{Capacity }=\ 5.4\times 1000=5400\ \text{Litres}\)

Measurement, STD2 M1 EQ-Bank 12 MC

During a flood, 12.5 hectares of land was covered by water to a depth of 30 cm.

How many kilolitres of water covered the land?

(1 hectare = 10 000 m² and 1 m³ = 1000 L)

  1. 3.75 kL
  2. 37.5 kL
  3. 37 500 kL
  4. 37 500 000 kL
Show Answers Only

`C`

Show Worked Solution

`text(Volume of water in m³:)`

`text(Vol)` `= Ah`
  `= 12.5 xx 10\ 000 xx 30\ text(cm)`
  `= 12.5 xx 10\ 000 xx 0.3`
  `= 37\ 500 \ text(m³)`
  `= 37\ 500\ text(kL)`

 
`=>C`

Measurement, STD2 M1 EQ-Bank 22

Steel rods are manufactured in the shape of equilateral triangular prisms.
 

  1. Find the volume of the prism (answer correct to 1 decimal place).   (2 marks)

  2. The mass of steel is 7850 kg/m³. Use this information to find the mass of the steel rod correct to the nearest gram.   (2 marks)

Show Answers Only
  1. `3464.1\ text(cm³)`
  2. `27\ 193\ text(g)`
Show Worked Solution

a.   `text{Area of triangular face (using sine rule)}`

`=1/2 xx 10 xx 10 xx sin60°`

`=43.301…`
 

`text(Volume)` `=Ah`
  `=43.301… xx 80`
  `=3464.1\ text{cm}^3\ \text{(to 1 d.p.)}`

 

b.  `text(Converting kg to g:)`

`7850\ text(kg) = 7\ 850\ 000\ text(g)`
 

`text(Converting m)^3\ \text(to cm)^3:`

`text(1 m)^3=100\ text(cm) xx100\ text(cm) xx100\ text(cm)=1\ 000\ 000\ text(cm)^3`

`:.\ text(Weight of steel rod)` `=3464.1 xx (7\ 850\ 000)/(1\ 000\ 000)` 
  `=27\ 193.185`
  `=27\ 193\ text{g  (nearest gram)}`

Measurement, STD2 M1 2017 HSC 18 MC

A skip bin is in the shape of a trapezoidal prism, with dimensions as shown.
 

What is the volume of the skip bin?

  1. `5.4\ text(m)^3`
  2. `7.776\ text(m)^3`
  3. `10.8\ text(m)^3`
  4. `15.552\ text(m)^3`
Show Answers Only

`A`

Show Worked Solution
`text(Area of trapezoid)` `= 1/2h (a + b)`
  `= 1/2 xx 1.2 xx (3.6 + 2.4)`
  `= 3.6\ text(m)^2`

 

`:.\ text(Volume)` `= Ah`
  `= 3.6 xx 1.5`
  `= 5.4\ text(m)^3`

 
`=>A`

Measurement, STD2 M1 2016 HSC 30a

The area of a roof is 30 m². Any rain that falls on the roof flows directly onto a garden.

Calculate how many litres of water flow onto the garden when 20 mm of rain falls on the roof.  (2 marks)

Show Answers Only

`600\ text(L)`

Show Worked Solution
♦♦ Mean Mark 27%.
STRATEGY: Converting 20 mm into metres as a 1st step is an efficient approach here.
`text(Volume)` `= Ah`
  `= 30 xx 20/1000`
  `= 0.6\ text(m³)`
  `= 600\ text(L)`

Measurement, STD2 M1 2016 HSC 12 MC

A container is in the shape of a triangular prism which has a capacity of 12 litres. The area of the base is 240 cm².
 

What is the distance, `h`, between the two triangular ends of the container?

  1. 5 cm
  2. 20 cm
  3. 25 cm
  4. 50 cm
Show Answers Only

`=> D`

Show Worked Solution
♦♦ Mean mark 35%.

`text{1 mL = 1 cm}^3\ \ =>\ \ text{1 L = 1000 cm}^3`

`text(Volume)` `= Ah`
`12\ 000` `= 240 xx h`
`h` `= (12\ 000)/240`
  `= 50\ text(cm)`

 
`=> D`

Measurement, STD2 M1 2004 HSC 23a

The diagram shows the shape of Carmel’s garden bed. All measurements are in
metres.

  1. Show that the area of the garden bed is 57 square metres.   (2 marks)

  2. Carmel decides to add a 5 cm layer of straw to the garden bed.

     

    Calculate the volume of straw required. Give your answer in cubic metres.   (2 marks)

  3. Each bag holds 0.25 cubic metres of straw.

     

    How many bags does she need to buy?   (2 marks)

  4. A straight fence is to be constructed joining point A to point B.

     

    Find the length of this fence to the nearest metre.   (2 marks)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `text(2.85 m³)`
  3. `text(She needs to buy 12 bags)`
  4. `8\ text{m  (nearest metre)}`
Show Worked Solution
a.    `text(Area of)\ Delta ABC` `= 1/2 xx b xx h`
  `= 1/2 xx 10 xx 5.1`
  `= 25.5\ text(m²)`
`text(Area of)\ Delta ACD` `= 1/2 xx 10 xx 6.3`
  `= 31.5\ text(m²)`

 

`:.\ text(Total Area)` `= 25.5 + 31.5`
  `= 57\ text(m² … as required)`

 

b.    `V` `= Ah`
  `= 57 xx 0.05`
  `= 2.85\ text(m³)`

 

c.    `text(Bags to buy)` `= 2.85/0.25`
  `= 11.4`

 
`:.\ text(She needs to buy 12 bags.)`

 

d.   `text(Using Pythagoras,)`

`AB^2` `= 6.0^2 + 5.1^2`
  `= 36 + 26.01`
  `= 62.01`
`AB` `= 7.874…`
  `=8\ text{m  (nearest metre)}`

Measurement, STD2 M1 2010 HSC 17 MC

During a flood 1.5 hectares of land was covered by water to a depth of  17 cm.

How many kilolitres of water covered the land?   (1 hectare = 10 000 m²)  

  1.    2.55 kL
  2.    2550 kL
  3.    255 000 kL
  4.    2 550 000 kL
Show Answers Only

`B`

Show Worked Solution
♦♦ Mean mark 34%
NOTE: The unit conversion 1 m³ = 1000 L  is contained in the Formulae and Data sheet given out in the exam.
`text(Area covered)` `=1.5xx10\ 000`
  `=15\ 000\ text(m²)`
`text(Volume)` `=Ah`
  `=15\ 000xx0.17`
  `=2550\ text(m³)`

 

`text{1 m³}` `=1000\ text(L)=1\ text(kL)`
`:.\ text(Volume)` `=2550\ text(kL)`

`=>B`

Measurement, STD2 M1 2012 HSC 6 MC

What is the volume of this rectangular prism in cubic centimetres? 
  

  1. 6 cm³
  2. 600 cm³
  3. 60 000 cm³
  4. 6 000 000 cm³
Show Answers Only

`C`

Show Worked Solution

`text{Convert all measurements to centimetres:}`

`V` `= l xx b xx h`
  `=30 xx 5 xx 400`
  `=60\ 000\ \ text(cm)^3`

 
`=>  C`