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v1 Measurement, STD2 M1 2023 HSC 24

The diagram shows the cross-section of a wall across a creek. 
 


 
  1. Use two applications of the trapezoidal rule to estimate the area of the cross-section of the wall.  (2 marks)

  2. The wall has a uniform thickness of 0.9 m. The weight of 1 m³ of concrete is 3.52 tonnes.  
  3. How many tonnes of concrete are in the wall? Give the answer to two significant figures.  (3 marks)

Show Answers Only
  1. `21.25\ text{m}^2`
  2. `text{67 tonnes}`
Show Worked Solution

a.    `h=10.0/2=5`

`A` `~~h/2[2+1.5+2(2.5)]`  
  `~~5/2(8.5)`  
  `~~21.25\ text{m}^2`  

 
b.   `V_text{wall}=21.25 xx 0.9=19.13\ text{m}^3`

`text{Mass of concrete}` `=19.13 xx 3.52`  
  `=67.33`  
  `=67\ text{tonnes (2 sig.fig.)}`  
♦ Mean mark (b) 46%.
 

v1 Measurement, STD2 M1 SM-Bank 8

A cannonball is made out of lead and has a diameter of 18 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 lead is 11.3 tonnes/m³. Use this information to find the mass of the cannonball to the nearest gram. (2 marks)
Show Answers Only
  1. `3053.6\ \text{cm³  (1 d.p.)}`
  2. `34\ 506\ \text{grams}`
Show Worked Solution

i.     `text(Radius) = 18 / 2 = 9\ \text{cm}`

`text(Volume)` `= \frac{4}{3} \pi r^3`
  `= \frac{4}{3} \pi (9)^3`
  `= \frac{4}{3} \pi (729) = 3053.628…`
  `= 3053.6\ \text{cm³ (1 d.p.)}`

 

ii.     `text(1 m³ = 1,000,000 cm³)`

`11.3\ \text{tonnes} = 11,300\ \text{kg} = 11,300,000\ \text{grams}`

`text(Mass of cannon ball)`

`= 3053.6 \times ({11,300,000} / {1,000,000})`

`= 3053.6 \times 11.3 = 34,505.68…`

`= 34,506\ \text{grams}`

Measurement, STD2 M1 2023 HSC 24

The diagram shows the cross-section of a wall across a creek. 
 


 
  1. Use two applications of the trapezoidal rule to estimate the area of the cross-section of the wall.  (2 marks)

  2. The wall has a uniform thickness of 0.80 m. The weight of 1 m³ of concrete is 3.52 tonnes.  
  3. How many tonnes of concrete are in the wall? Give the answer to two significant figures.  (3 marks)

Show Answers Only
  1. `18\ text{m}^2`
  2. `text{51 tonnes}`
Show Worked Solution

a.    `h=8.0/2=4`

`A` `~~h/2[1.9+1.7+2(2.7)]`  
  `~~4/2(9)`  
  `~~18\ text{m}^2`  

 
b.   `V_text{wall}=18 xx 0.8=14.4\ text{m}^3`

`text{Mass of concrete}` `=14.4 xx 3.52`  
  `=50.688`  
  `=51\ text{tonnes (2 sig.fig.)}`  
♦ Mean mark (b) 46%.
 

Measurement, STD2 M1 SM-Bank 4

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

i.   `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`
  `=3464.10…`
  `=3464.1\ text{cm³  (to 1 d.p.)}`

 

ii.  `text(Converting kg to g:)`

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

 

`text(Converting m³ to cm³:)`

`1\ text(m³)` `=100\ text(cm) xx100\ text(cm) xx100\ text(cm)` 
  `=1\ 000\ 000\ text(cm³)`

 

`:.\ 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 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 16 MC

The benchmark for annual greenhouse gas emissions from the residential sector is 3292 kg of carbon dioxide per person per year.

A new building, planned to house 6 people, has been designed to achieve a 25% reduction on this benchmark.

What is the maximum amount of carbon dioxide per year, to the nearest kilogram, that this building is designed to emit when fully occupied?

A.     823 kg

B.     2469 kg

C.     4938 kg

D.     14 814 kg

Show Answers Only

`D`

Show Worked Solution
`text(Benchmark emissions)` `= 6 xx 3292`
  `= 19\ 752\ text(kg)`

 
`:.\ text(Max emissions of new building)`

`= 75text(%) xx 19\ 752`

`= 14\ 814\ text(kg)`

`=> D`