Aussie Maths & Science Teachers: Save your time with SmarterEd
The amount of paint (\(P\)) needed to cover a wall varies directly with the area (\(A\)) of the wall.
A painter uses 3.5 litres of paint to cover a wall with an area of 28 square metres.
How much paint is needed to cover a wall with an area of 42 square metres?
\(C\)
\(P \propto A\)
\(P=kA\)
\(\text{When } P = 3.5 \text{ and } A = 28:\)
| \(3.5\) | \(=k \times 28\) |
| \(k\) | \(=\dfrac{3.5}{28}\) |
| \(k\) | \(=0.125\) |
\(\text{When } A = 42:\)
| \(P\) | \(=0.125\times 42\) |
| \(=5.25\) |
\(\Rightarrow C\)
The energy (\(E\)) required to heat water varies directly with the mass (\(m\)) of the water.
It takes 2100 joules of energy to heat 500 grams of water by 1°C.
Which equation represents the relationship between \(E\) and \(m\)?
\(A\)
\(E \propto m\)
\(E=km\)
\(\text{When } E = 2100 \text{ and } m = 500:\)
| \(2100\) | \(=k \times 500\) |
| \(k\) | \(=\dfrac{2100}{500}\) |
| \(k\) | \(=4.2\) |
\(\therefore\ E=4.2m\)
\(\Rightarrow A\)
Jordan visits Italy on his holidays. He pays €180 (180 euros) for a pair of Italian leather boots.
How much is €180 in Australian dollars if AUD1 is worth €0.58? (2 marks)
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\($310.34\)
\(0.58\ \)€ \(\ =\ \text{AUD 1}\)
\(\rightarrow\ 1\ \)€\(\ =\)\(\ \text{AUD }\)\(\dfrac{1}{0.58}\)
\(\rightarrow\ 180\ \)€\(=180\times\dfrac{1}{0.58}=310.344…\)
\(\therefore\ \text{Jordan’s boots cost }$310.34\ \text{in Australian dollars.}\)
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.
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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:}`
| `1.2` | `= 0.005 xx N` |
| `therefore N` | `= frac{1.2}{0.005}` |
| `= 240 \ text{sheets}` |
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)`
`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)` |
John knows that
• one Australian dollar is worth 0.62 euros
• one Vistabella dollar `text{($V)}` is worth 1.44 euros.
John changes 25 Australian dollars to Vistabella dollars.
How many Vistabella dollars will he get?
`A`
`text(John has 25 Aust dollars.)`
`text(Converting to Euros)`
| `text(25 Aust)` | `= 25 xx 0.62` |
| `= 15.5\ text(Euros)` |
`text(Converting to Vistabella dollars)`
| `text(15.5 Euros)` | `= 15.5/1.44` |
| `=\ text($V10.76)` |
`=> A`
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)`
`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)`