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v1 Measurement, STD2 M1 2014 HSC 27c

A swimming pool is in the shape of a rectangle with a semicircle at each end, as shown.

The pool is 7000 mm long, 4000 mm wide, and has a depth of 2100 mm.  
  

How much water is needed to fill the pool, to the nearest litre?   (4 marks) 

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`51 \ 589 \ text(L)`

Show Worked Solution

`V = Ah` 

♦ Mean mark 41%
STRATEGY: Adjusting measurements to metres makes the final conversion to litres simple.

`text(Finding Area of base)`

`text(Semi-circles have radius 2000 mm) = 2 \ text(m)`

`:.\ text(Area of 2 semicircles)`

`=2 xx 1/2 xx pi r^2`

`= pi xx 2^2`

`= 12.56 \ text(m)^2`
 

`text(Area of rectangle)`

`= l xx b`

`= (7-2 xx 2) xx 4`

`= 12\ text(m)^2`

 

`:.\ text(Volume)` `= Ah`
  `= (12.56… + 12) xx 2.1`
  `= 51.589…\ text(m)^3`
  `= 51 \ 589 \ text(L)\ \ text{(using 1m³} = 1000\ text{L)}`
   

v1 Measurement, STD2 M1 2018 HSC 30a

A cylindrical oil tank has a height of 7 metres and a capacity of 1.5 megalitres.
 

What is the diameter of the oil tank? Give your answer in metres, correct to two decimal places.  (3 marks)

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`16.50\ text{m}`

Show Worked Solution

`text{Converting megalitres to m³  (using 1 m³ = 1000 L):}`

♦ Mean mark 48%.

`1.5\ text(ML)` `= (1.5 xx 10^6)/(10^3)`
  `= 1.5 xx 10^3\ text(m)^3`
  `= 1500\ text(m)^3`

 

`V` `= pir^2h`
`1500` `= pi xx r^2 xx 7`
`r^2` `= 1500/(pi xx 7)`
`sqrt(r^2)` `= sqrt(68.21)`
  `= 8.26\ text{m}`

 

`text{Diameter}` `=2r`  
  `=2xx8.26 \ text{m}`  
  `=16.52\ text{m}`  

 

Volume, SMB-011

The cylinder shown below has a diameter of 18 centimetres and a length of 45 centimetres.
 

Find the volume of the cylinder, giving your answer to the nearest cubic centimetre.   (2 marks)

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`11\ 451\ text(cm)^3`

Show Worked Solution
`text(Volume)` `= pi r^2 h`
  `= pi xx 9^2 xx 45`
  `= 11\ 451.10…`
  `= 11\ 451\ text(cm)^3\ \ text{(nearest cm}^3 text{)}`

Measurement, STD2 M1 2018 HSC 30a

A cylindrical water tank has a radius of 9 metres and a capacity of 1.26 megalitres.
 

What is the height of the water tank? Give your answer in metres, correct to two decimal places.  (3 marks)

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`4.95\ text{m}`

Show Worked Solution

`text{Converting megalitres to m³  (using 1 m³ = 1000 L):}`

♦ Mean mark 48%.

`1.26\ text(ML)` `= (1.26 xx 10^6)/(10^3)`
  `= 1.26 xx 10^3\ text(m)^3`
  `= 1260\ text(m)^3`

 

`V` `= pir^2h`
`1260` `= pi xx 9^2 xx h`
`h` `= 1260/(pi xx 9^2)`
  `= 4.951…`
  `= 4.95\ text{m  (2 d.p.)}`

Measurement, STD2 M1 2016 HSC 28e

A company makes large marshmallows. They are in the shape of a cylinder with diameter 5 cm and height 3 cm, as shown in the diagram.

2ug-2016-hsc-q28_4

  1. Find the volume of one of these large marshmallows, correct to one decimal place.  (2 marks)

A cake is to be made by stacking 24 of these large marshmallows and filling the gaps between them with chocolate. The diagrams show the cake and its top view. The shading shows the gaps to be filled with chocolate.
 

2ug-2016-hsc-q28_5

  1. What volume of chocolate will be required? Give your answer correct to the nearest whole number.  (3 marks)

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  1. `58.9\ text{cm}^3`
  2. `193\ text{cm}^3`
Show Worked Solution
i.    `V` `= pir^2h`
    `= pi xx 2.5^2 xx 3`
    `= 58.904…`
    `= 58.9\ text{cm³  (1 d.p.)}`

 

ii.    2ug-2016-hsc-q28-answer1

`text(Volume of rectangle)`

♦ Mean mark part (ii) 35%.

`= 15 xx 10 xx 6`

`= 900\ text(cm)^3`
 

`text(Volume of marshmallows in rectangle)`

`= 6 xx 2 xx 58.9`

`= 706.8\ text(cm)^3`

 

`:.\ text(Volume of chocolate)`

`= 900-706.8`

`= 193.2`

`= 193\ text{cm}^3 \ text{(nearest cm}^3 text{)}`

Measurement, STD2 M1 2005 HSC 23b

A clay brick is made in the shape of a rectangular prism with dimensions as shown.
 

  1. Calculate the volume of the clay brick.  (1 mark)

Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.  
 

  1. What is the volume of clay remaining in the brick after the holes have been made? (Give your answer to the nearest cubic centimetre.)  (3 marks)

  2. What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.)  (1 mark)

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  1. `text(1512 cm)^3`
  2. `text{1364 cm}^3`
  3. `text{9.8%}`
Show Worked Solution
i.    `V` `= l × b × h`
    `= 21 × 8 × 9`
    `= 1512\ text(cm)^3`

 

ii.  `text(Volume of each hole)`

`= pir^2h`

`= pi × 1.4^2 × 8`

`= 49.260…\ text(cm)^3`

 

`:.\ text(Volume of clay still in brick)`

`= 1512 − (3 × 49.260…)`

`= 1364.219…`

`= 1364\ text{cm}^3\ text{(nearest whole)}`

 

iii. `text(Percentage of clay removed)`

`= ((3 × 49.260…))/1512 × 100`

`= 9.773…`

`= 9.8 text{%   (1 d.p.)}`

Measurement, STD2 M1 2014 HSC 27c

The base of a water tank is in the shape of a rectangle with a semicircle at each end, as shown.

The tank is 1400 mm long, 560 mm wide, and has a height of 810 mm.  
  

What is the capacity of the tank, to the nearest litre?   (4 marks) 

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`581\ text(L)`

Show Worked Solution

`V = Ah` 

♦ Mean mark 41%
STRATEGY: Adjusting measurements to metres makes the final conversion to litres simple.

`text(Finding Area of base)`

`text(Semi-circles have radius 280 mm) = 0.28\ text(m)`

`:.\ text(Area of 2 semicircles)`

`=2 xx 1/2 xx pi r^2`

`= pi xx (0.28)^2`

`= 0.2463…\ text(m)^2`
 

`text(Area of rectangle)`

`= l xx b`

`= (1.4-2 xx 0.28) xx 0.56`

`= 0.4704\ text(m)^2`

 

`:.\ text(Volume)` `= Ah`
  `= (0.2463… + 0.4704) xx 0.810`
  `= 0.580527…\ text(m)^3`
  `= 580.527…\ text(L)\ \ text{(using 1m³} = 1000\ text{L)}`
  `= 581\ text(L)\ text{(nearest L)}`