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Measurement, STD2 M1 2021 HSC 16

The volume, `V`, of a sphere is given by the formula

`V = frac{4}{3} pi r^3,`

where `r` is the radius of the sphere.

A tank consists of the bottom half of a sphere of radius 2 metres, as shown.
 

Find the volume of the tank in cubic metres, correct to one decimal place.   (2 marks)

Show Answers Only

`16.8\ text{m}^3`

Show Worked Solution
 `V` `= frac{1}{2} times frac{4}{3} pi r^3`
  `= frac{1}{2} times frac{4}{3} times pi times 2^3`
  `= 16.755…`
  `= 16.8\ text{m}^3\ \ text{(1 d.p.)}`

Measurement, STD2 M1 2019 HSC 16

A bowl is in the shape of a hemisphere with a diameter of 16 cm.
 

What is the volume of the bowl, correct to the nearest cubic centimetre?  (2 marks)

Show Answers Only

`1072\ text(cm)^3`

Show Worked Solution
`V` `= 1/2 xx 4/3pir^3`
  `= 1/2 xx 4/3 xx pi xx 8^3`
  `= 1072.3…`
  `= 1072\ text{cm}^3\ text{(nearest cm}^3 text{)}`

Measurement, STD2 M1 2008 HSC 21 MC

A sphere and a closed cylinder have the same radius.

The height of the cylinder is four times the radius.

What is the ratio of the volume of the cylinder to the volume of the sphere?

  1. `2 : 1`
  2. `3 : 1`
  3. `4 : 1`
  4. `8 : 1`
Show Answers Only

`B`

Show Worked Solution

♦♦ Mean mark 33%.

`V_text(cylinder)` `: V_text(sphere)`
`pir^2h` `: 4/3pir^3`
`underbrace(pir^2 4r)_(h = 4r)` `: 4/3pir^3`
`4pir^3` `: 4/3pir^3`
`3` `: 1`

  
`=> B`

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)

Show Answers Only

`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 2017 HSC 30e

A solid is made up of a sphere sitting partially inside a cone.

The sphere, centre `O`, has a radius of 4 cm and sits 2 cm inside the cone. The solid has a total height of 15 cm. The solid and its cross-section are shown.
 


 

Using the formula  `V=1/3 pi r^2h`  where `r`  is the radius of the cone's circular base and `h` is the perpendicular height of the cone, find the volume of the cone, correct to the nearest cm³?  (3 marks)

Show Answers Only

`113\ text{cm}^3`

Show Worked Solution

`V = 1/3 xx text(base of cone × height)`

`text(Consider the circular base area of the cone,)`

`text(Find)\ x\ \ text{(using Pythagoras):}`

`x^2` `= 4^2-2^2 = 16-4 = 12`
`x` `= sqrt12\ text(cm)`

 

`:. V` `= 1/3 xx pi xx (sqrt12)^2 xx (15-6)`
  `= 1/3 xx pi xx 12 xx 9`
  `= 113.097…`
  `= 113\ text{cm}^3\ text{(nearest cm}^3 text{)}`

Measurement, STD2 M1 SM-Bank 8

A cannon ball is made out of steel and has a diameter of 23 cm.

  1. Find the volume of the sphere in cubic centimetres (correct to 1 decimal place).  (2 marks)

  2. It is known that the mass of the steel used is 8.2 tonnes/m³. Use this information to find the mass of the cannon ball to the nearest gram.  (2 marks)

Show Answers Only
  1. `6370.6\ text{cm³  (to 1 d.p.)}`
  2. `52\ 239\ text(grams)`
Show Worked Solution

i.   `text(Radius)= 23/2 = 11.5\ text(cm)`

`text(Volume)` `= 4/3pir^3`
  `= 4/3 xx pi xx 11.5^3`
  `= 6370.626…`
  `= 6370.6\ text{cm³  (to 1 d.p.)}`

 

ii.   `text(Convert m³ to cm³:)`

`text(1 m³)` `= 100\ text(cm × 100 cm × 100 cm)`
  `= 1\ 000\ 000\ text(cm³)`

 

`text(Convert 8.2 tonnes to grams:)`

`text(8.2 tonnes)` `= 8200\ text(kg)`
  `= 8\ 200\ 000\ text(g)`

 

`:.\ text(Weight of cannon ball)`

`= 6370.6 xx (8\ 200\ 000)/(1\ 000\ 000)`

`= 52\ 238.92`

`= 52\ 239\ text(grams)`

Measurement, STD2 M1 2017 HSC 22 MC

A concrete water pipe is manufactured in the shape of an annular cylinder. The dimensions are shown in the diagrams.
 


 

What is the approximate volume of concrete needed to make the water pipe?

  1. `text(0.06 m)³`
  2. `text(0.09 m)³`
  3. `text(0.70 m)³`
  4. `text(0.99 m)³`
Show Answers Only

`C`

Show Worked Solution
`text(Volume)` `= text(Area of annulus) xx h`
  `= (piR^2 – pir^2) xx 2.8`
  `= (pi xx 0.45^2 – pi xx 0.35^2) xx 2.8`
  `= 0.7037…`
  `= 0.70\ text(m)³`

  
`=>C`

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)

Show Answers Only
  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 28a

The Mitchell family has moved to a new house which has an empty swimming pool. The base of the pool is in the shape of a rectangle, with a semicircle on each end.

2005 28a1

  1. Explain why the expression for the area of the base of the pool is  `2xy + πy^2`.   (1 mark) 

      
            2005 28a2
      

The pool is 1.1 metres deep.

  1. The sides and base of the pool are covered in tiles. If  `x =6`  and  `y = 2.5`, find the total area covered by tiles. (Give your answer correct to the nearest square metre.)   (4 marks)

Before filling the pool, the Mitchells need to install a new shower head, which saves 6 litres of water per minute.

The shower is used 5 times every day, for 3 minutes each time.

  1. If the charge for water is $1.013 per kilolitre, how much money would be saved in one year by using this shower head? (Assume there are 365 days in a year.)   (2 marks)

Show Answers Only
  1. `text(See Worked Solutions)`
  2. `text(80 m)^2\ text{(nearest m}^2text{)}`
  3. `$33.28\ \ text{(nearest cent)}`
Show Worked Solution
i.    `text(Area of base)` `=\ text(Area of rectangle +)`
    `\ \ \ \ \ \2 xx text(Area of semi-circle)`
    `= (x xx 2y) + 2 xx (1/2 xx pi xx y^2)`
    `= 2xy + piy^2`

 

ii.   `text(Area of base)` `= (2 xx 6 xx 2.5) + (pi xx 2.5^2)`
    `= 49.634…\ text(m²)`

 

`text(Area of walls) = text(Length) xx text(Height)`

`text(Length)` `= 2x + 2 xx text(semi-circle perimeter)`
  `= (2 xx 6) + 2 xx (1/2 xx 2 xx pi xx 2.5)`
  `= 12 + 15.707…`
  `= 27.707…\ text(m)`

 

`:.\ text(Area of walls)` `= 27.707 xx 1.1`
  `= 30.478…\ text(m)^2`

 

`:.\ text(Total Area covered by tiles)`

`= 49.634… + 30.478…`

`= 80.11…`

`= 80\ text{m²  (nearest m²)}`

 

iii.  `text(Water saved)` `= 5 xx 3 xx 6`
    `= 90\ text(L per day.)`

 

`text(Water saved per year)`

`= 90 xx 365`

`= 32\ 850\ text(L)`

`= 32.85\ text(kL)`

`:.\ text(Money saved)` `= 32.85 xx $1.013`
  `= $33.277…`
  `= $33.28\ text{(nearest cent)}`

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)

Show Answers Only
  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 2007 HSC 23b

A cylindrical water tank, of height 2 m, is placed in the ground at a school.

The radius of the tank is 3.78 metres. The hole is 2 metres deep. When the tank is placed in the hole there is a gap of 1 metre all the way around the side of the tank.

 

  1. When digging the hole for the water tank, what volume of soil was removed? Give your answer to the nearest cubic metre.  (3 marks)

  2. Sprinklers are used to water the school oval at a rate of 7500 litres per hour.   

     

    The water tank holds 90 000 litres when full. 

     

    For how many hours can the sprinklers be used before a full tank is emptied?   (1 mark)

  3. Water is to be collected in the tank from the roof of the school hall, which has an area of 400 m².

     

    During a storm, 20 mm of rain falls on the roof and is collected in the tank. 

     

    How many litres of water were collected?   (2 marks)

Show Answers Only
  1. `144\ text(m³)\ \ text{(nearest m³)}`
  2. `text(12 hours)`
  3. `8000\ text(litres)`
Show Worked Solution

i.  `V = pi r^2 h\ \ \ \ text(where)`

`h = 2\ text(and)\ r = 4.78\ text(m)`

`:.\ V` `= pi xx 4.78^2 xx 2`
  `= 143.56…`
  `= 144\ text(m³)\ \ text{(nearest m³)}`

 

ii.  `text(Total water) = 90\ 000\ text(litres)`

`text(Usage) = 7500\ \ text(litres/hr)`

`:.\ text(Hours before it is empty)`

`= (90\ 000)/7500`

`= 12\ text(hours)`

 

iii. `text(Water collected)`

`= 400 xx 0.020`

`= 8\ text(m²)`

`= 8000\ text(litres)`

Measurement, STD2 M6 2008 HSC 25c

Pieces of cheese are cut from cylindrical blocks with dimensions as shown.

 

Twelve pieces are packed in a rectangular box. There are three rows with four pieces of cheese in each row. The curved surface is face down with the pieces touching as shown.
  

  1. What are the dimensions of the rectangular box?  (4 marks)

     

    To save packing space, the curved section is removed.
     
             
     

  2. What is the volume of the remaining triangular prism of cheese? Answer to the nearest cubic centimetre.    (2 marks)

Show Answers Only
  1. `41\ text(cm) xx 21\ text(cm) xx 15\ text(cm)`
  2. `506\ text(cm)³\ text{(nearest whole)}`
Show Worked Solution

i.  `text(Box height) = 15\ text(cm)`

♦ Mean mark 45%.

`text{(radius of the arc)}`

`text(Box width)` `= 3 xx 7`
  `= 21\ text(cm)`
`text(Box length)` `= 4x`

`text(Using cosine rule)`

`c^2` `= a^2 + b^2 – 2ab cos C`
`x^2` `= 15^2 + 15^2 – 2 xx 15 xx 15 xx cos 40^@`
  `= 450 – 344.7199…`
  `= 105.2800…`
`x` `= 10.2606…`

 

`text(Box length)` `= 4 xx 10.2606…`
  `= 41.04…`

 
`:.\ text(Dimensions are)\ \ 41\ text(cm) xx 21\ text(cm) xx 15\ text(cm)`

 

ii.  `text(Volume) = Ah`

♦♦♦ Mean mark 22%.

`h = 7\ text(cm)`

`text(Find)\ A:`

`A` `= 1/2 ab sin C`
  `= 1/2 xx 15 xx 15 xx sin 40^@`
  `= 72.3136…`

 

`:. V` `= 72.3136… xx 7`
  `= 506.195…`
  `= 506\ text(cm³)\ \ text{(nearest whole)}`

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) 

Show Answers Only

`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)}`

Measurement, STD2 M1 2010 HSC 28b

Moivre’s manufacturing company produces cans of Magic Beans. The can has a diameter of  10 cm and a height of  10 cm.

2010 28b1

  1. Cans are packed in boxes that are rectangular prisms with dimensions  30 cm × 40 cm × 60 cm.

     

    What is the maximum number of cans that can be packed into one of these boxes?   (1 mark)

  2. The shaded label on the can shown wraps all the way around the can with no overlap. What area of paper is needed to make the labels for all the cans in this box when the box is full?   (2 marks)

  3. The company is considering producing larger cans. Monica says if you double the diameter of the can this will double the volume.

     

    Is Monica correct? Justify your answer with suitable calculations.   (2 marks)

      
    The company wants to produce a can with a volume of 1570 cm³, using the least amount of metal. Monica is given the job of determining the dimensions of the can to be produced. She considers the following graphs.
     
    2010 28b2

  4. What radius and height should Monica recommend that the company use to minimise the amount of metal required to produce these cans? Justify your choice of dimensions with reference to the graphs and/or suitable calculations.   (2 marks)

Show Answers Only
  1. `72\ text(cans)`
  2. `20\ 358\ text{cm²  (nearest cm²)}`
  3. `text(Monica is incorrect because the volume)`

     

    `text(doesn’t double.  It increases by a factor of 4.)`

  4. `text(Radius 6.3 cm and height 12.6 cm.)`
Show Worked Solution
♦♦ Mean mark 27%
i.    `text(Maximum # Cans)` `= 4 xx 3 xx 6`
    `= 72\ text(cans)`

 

♦ Mean mark 38%
MARKER’S COMMENT: Many students didn’t account for the clearance of 0.5 cm at the top and bottom of each can.
ii.    `text(Label Area)\ text{(1 can)}` `= 2 pi rh`
    `= 2 xx pi xx 5 xx 9`
    `= 90 pi`
    `= 282.7433…\ text(cm²)`

 

`:.\ text(Label Area)\ text{(72 cans)}`

`= 72 xx 282.7433…`

`= 20\ 357.52…`

`= 20\ 358\ text(cm²)`  `text{(nearest cm²)}`

 

♦ Mean mark 44%
MARKER’S COMMENT: Many students performed calculations in this part without concluding if Monica is correct or not. Read the question carefully.
iii.   `text(Original volume)` `= pi r^2 h`
    `= pi xx 5^2 xx 10`
    `= 785.398…\ text(cm³)`
`text(If the diameter doubles, radius) = 10\ text(cm)`
`text(New volume)` `= pi xx 10^2 xx 10`
  `= 3141.592…\ text(cm³)`

 

`:.\ text(Monica is incorrect because the volume)`
`text(doesn’t double.  It increases by a factor of 4.)`

 

♦♦ Mean mark 26%
iv.
`text(Minimum metal used when surface area is a minimum.)`
`text(From graph, minimum surface area when)\ r = 6.3\  text(cm)`
`text(When)\ r = 6.3\  text(cm,)\ h = 12.6\  text(cm)\ \ \ text{(from graph)}`
`:.\ text(She should recommend radius 6.3 cm and height 12.6 cm)`

Measurement, STD2 M1 2009 HSC 19 MC

Two identical spheres fit exactly inside a cylindrical container, as shown.
 

The diameter of each sphere is 12 cm.

What is the volume of the cylindrical container, to the nearest cubic centimetre?

  1. `1357\ text(cm)^3`
  2. `2714\ text(cm)^3`
  3. `5429\ text(cm)^3`
  4. `10\ 857\ text(cm)^3`
Show Answers Only

`B`

Show Worked Solution

`text(S)text(ince diameter sphere = 12 cm) `

`=>\ text(Radius of cylinder = 6 cm)`

`text(Height of cylinder)` `= 2 xx text(diameter of sphere)`
  `= 2 xx 12`
  `= 24\ text(cm)`
   
`:.\ text(Volume cylinder)` `= pi r^2 h`
  `= pi xx 6^2 xx 24`
  `= 2714.336…\ text(cm³)`

 
`=>  B`