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Algebra, STD1 A2 2025 HSC 16

The mass \((M )\) of a box with a square base, in grams, is directly proportional to the area of its base, in cm².
 

A box with a square base of side length 5 cm has a mass of 500 g.

What is the mass of a similar box with a square base of side length 3 cm?    (3 marks)

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\(M=180\ \text{grams}\)

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\(\text{Area of square base }= s^2\)

\(M \propto s^2\ \ \Rightarrow \ \ M=k\times s^2\)

\(\text{Find \(k\) given \(\ s=5\ \) when \(\ M=500\):}\)

\(500\) \(=k\times 5^2\)
\(25k\) \(=500\)
\(k\) \(=\dfrac{500}{25}=20\)

 
\(\text{Find \(M\) when  \(s=3\):}\)

\(M=20\times 3^2=180\ \text{grams}\)

Algebra, STD1 A2 2020 HSC 20

The weight of a bundle of A4 paper (`W` kg) varies directly with the number of sheets (`N`) of A4 paper that the bundle contains.

This relationship is modelled by the formula  `W = kN`, where  `k`  is a constant.

The weight of a bundle containing 500 sheets of A4 paper is 2.5 kilograms.

  1. Show that the value of  `k`  is 0.005.   (1 mark)

  2. A bundle of A4 paper has a weight of 1.2 kilograms. Calculate the number of sheets of A4 paper in the bundle.   (2 marks)

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  1. `text{Show Worked Solutions}`
  2. `240 \ text{sheets}`
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a.     `W = 2.5\ text{kg when} \  N = 500:`

`2.5` `= k xx 500`
`therefore \ k` `= frac{2.5}{500}`
  `= 0.005`

 

b.     `text{Find}\ \ N \ \ text{when} \ \ W = 1.2\ text{kg:}`

♦ Mean mark 50%.
`1.2` `= 0.005 xx N`
`therefore N` `= frac{1.2}{0.005}`
  `= 240 \ text{sheets}`

Algebra, STD2 A2 SM-Bank 2

The weight of a steel beam, `w`, varies directly with its length, `ℓ`.

A 1200 mm steel beam weighs 144 kg.

Calculate the weight of a 750 mm steel beam.  (2 marks)

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`90\ text(kg)`

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`w propto ℓ`

`w = kℓ`

`text(When)\ \ w = 144\ text(kg),\ \ ℓ = 1200\ text(mm)`

`144` `= k xx 1200`
`k` `= 144/1200`
  `= 3/25`

 

`text(When)\ \ ℓ = 750\ text(mm),`

`w` `= 3/25 xx 750`
  `= 90\ text(kg)`

Algebra, STD2 A2 2014 HSC 26f

The weight of an object on the moon varies directly with its weight on Earth.  An astronaut who weighs 84 kg on Earth weighs only 14 kg on the moon.

A lunar landing craft weighs 2449 kg when on the moon. Calculate the weight of this landing craft when on Earth.   (2 marks)

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 `14\ 694\ text(kg)`

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`W_text(moon) prop W_text(earth)`

`=> W_text(m) = k xx W_text(e)`

`text(Find)\ k\ text{given}\  W_text(e) = 84\ text{when}\ W_text(m) = 14`

`14` `= k xx 84`
`k` `= 14/84 = 1/6`

 

`text(If)\ W_text(m) = 2449\ text(kg),\ text(find)\ W_text(e):`

`2449` `= 1/6  xx W_text(e)`
`W_text(e)` `= 14\ 694\ text(kg)`

 

`:.\ text(Landing craft weighs)\ 14\ 694\ text(kg on earth)`