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Measurement, STD2 M6 2023 HSC 12 MC

A cylindrical pipe with a radius of 12.5 cm is filled with water to a depth, `d` cm, as shown.

The surface of the water has a width of 20 cm.
 

What is the depth of water in the pipe?

  1. 2.5 cm
  2. 5.0 cm
  3. 7.5 cm
  4. 12.5 cm
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`B`

Show Worked Solution

`text{Triangle base}\ = 20/2=10\ text{cm}`

`text{Let}\ x =\ text{⊥ distance from centre to water}`

`text{Using Pythagoras:}`

`12.5^2` `=x^2+10^2`  
`x^2` `=12.5^2-10^2`  
  `=56.25`  
`x` `=7.5\ text{cm}`  

 
`d=12.5-7.5=5.0\ text{cm}`

`=>B`

♦ Mean mark 42%.

Measurement, STD2 M6 2019 HSC 22

Two right-angled triangles, `ABC` and `ADC`, are shown.
 

Calculate the size of angle `theta`, correct to the nearest minute.  (3 marks)

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`41°4^{′}\ \ text{(nearest minute)}`

Show Worked Solution

`text(Using Pythagoras in)\ DeltaACD:`

Mean mark 51%.

`AC^2` `= 2.5^2 + 6^2`
  `= 42.25`
`:.AC` `= 6.5\ text(cm)`

 
`text(In)\ DeltaABC:`

`costheta` `= 4.9/6.5`
`theta` `= cos^(−1)\ 4.9/6.5`
  `= 41.075…`
  `= 41°4^{′}31^{″}`
  `= 41°5^{′}\ \ text{(nearest minute)}`

Measurement, STD2 M6 2005 HSC 25b

2UG-2005-25b

  1. Use Pythagoras’ theorem to show that `ΔABC` is a right-angled triangle.  (1 mark)

  2. Calculate the size of `∠ABC` to the nearest minute.  (2 marks)

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  1. `text(Proof)`
  2. `67°23^{′}`
Show Worked Solution

i.   `ΔABC\ text(is right-angled if)\ \ a^2 + b^2 = c^2`

`a^2 + b^2` `= 5^2 + 12^2`
  `= 169`
  `= 13^2`
  `= c^2…\ text(as required.)`

MARKER’S COMMENT: Know your calculator process for producing an angle in minutes/seconds. Note >30 “seconds” rounds up to the higher “minute”.

 
ii.  `sin ∠ABC = 12/13`

`:.∠ABC` `= 67.38…°`
  `=67°22^{′}48^{″}`
  `= 67°23^{′}\ \ \ text{(nearest minute)}`

Measurement, STD2 M6 2011 HSC 9 MC

Two trees on level ground, 12 metres apart, are joined by a cable. It is attached 2 metres above the ground to one tree and 11 metres above the ground to the other.

What is the length of the cable between the two trees, correct to the nearest metre? 

  1.  `9\ text(m)`
  2. `12\ text(m)`
  3. `15\ text(m)`
  4. `16\ text(m)`
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`C`

Show Worked Solution

`text(Using Pythagoras)`

`c^2` `=12^2+9^2`
  `=144+81`
  `=225`
`:.c` `=15,\ \ c>0`

 
`=>C`

Measurement, STD2 M6 2010 HSC 3 MC

A field diagram has been drawn from an offset survey.

What is the distance from `G` to `H` correct to the nearest metre?

  1. 11
  2. 13
  3. 16
  4. 20
Show Answers Only

`B`

Show Worked Solution

`text(Using Pythagoras:)`

`GH^2` `=12^2+(16-11)^2`
  `=144+25`
  `=169`
   
`:.\ GH` `=sqrt169 `
  `=13\ text(m)`

 
` =>  B`