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Functions, 2ADV EQ-Bank 5 MC

The time taken to clean a warehouse varies inversely with the number of cleaners employed.

It takes 8 cleaners 60 hours to clean a warehouse.

Working at the same rate, how many hours would it take 10 cleaners to clean the same warehouse.

  1. 45
  2. 48
  3. 62
  4. 75
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`B`

Show Worked Solution

`text{Time to clean}\ (T) prop 1/text{Number of cleaners (C)}`

`T=k/C`

`text(When)\ \ T=60, C=8:`

`60` `=k/8`
`k` `=480`

 
`text{Find}\ \ T\ \ text(when)\ \ C=10:`

`T=480/10=48\ text(hours)`

`=>  B`

Functions, 2ADV EQ-Bank 10

The number of trees that can be planted along the fence line of a paddock varies inversely with the distance between each tree.

There will be 108 trees if the distance between them is 5 metres.

  1. How many trees can be planted if the distance between them is 6 metres?   (2 marks)

  2. What is the distance between the trees if 120 trees are planted.   (1 mark)

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a.   `90`

b.   `4.5\ text(metres)`

Show Worked Solution

a.   `t \prop 1/d \ \ =>\ \ t=k/d`

`108` `= k/5`
`k` `= 540`

 
`text(Find)\ t\ \text(when)\ \ d = 6:`

`t= 540/6= 90`
 

b.   `text(Find)\ d\ text(when)\ \ t = 120:`

`120` `= 540/d`
`d` `= 540/120= 4.5\ text(metres)`

Functions, 2ADV EQ-Bank 9

It is known that the volume of a hailstone \((V)\) is directly proportional to the cube of its radius \((r)\).

A hailstone with a radius of 1.25 cm has a volume of 8.2 cm\(^3\).

  1. Find the equation relating \(V\) and \(r\).   (2 marks)

  2. What is the expected radius of a hailstone with a volume of 51.1 cm\(^3\) ? Give your answer to 1 decimal place.   (1 mark)

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a.    \(V=4.1984 \times r^{3}\)

b.    \(2.3\ \text{cm (2 d.p.)}\)

Show Worked Solution

a.    \(V \propto r^3 \ \Rightarrow \ V=kr^3\)

\(\text{Find} \ k \ \text{given} \ \ V=8.2 \ \ \text{when}\ \  r=1.25:\)

\(8.2\) \(=k \times 1.25^3\)
\(k\) \(=\dfrac{8.2}{1.25^3}=4.1984\)

 
\(\therefore V=4.1984 \times r^{3}\)

  

b.    \(\text{Find \(r\) when \(\ V=51.1\):}\)

\(51.1\) \(=4.1984 \times r^3\)
\(r\) \(=\sqrt[3]{\dfrac{51.1}{4.1984}}=2.3\ \text{cm (1 d.p.)}\)

Functions, 2ADV EQ-Bank 8

The intensity of light \((I)\), measured in lux, from a lamp varies inversely with the square of the distance from the lamp \((d)\), measured in metres.

At a distance of 2.2 metres from the lamp, the light intensity is 400 lux.

What is the light intensity at a distance of 5.0 metres from the lamp?   (2 marks)

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\(77.44\ \text{lux}\)

Show Worked Solution

\(I \propto \dfrac{k}{d^2} \ \Rightarrow \ I=\dfrac{k}{d^2}\)

\(\text{Find k given} \ \ I=400 \ \ \text {when} \ \ d=2.2:\)

\(400=\dfrac{k}{2.2^2} \ \Rightarrow \ k=1936\)
 

\(\text{Find \(I\) when \(\ d=5\):}\)

\(I=\dfrac{1936}{5.0^2}=77.44\ \text{lux}\)

Functions, 2ADV EQ-Bank 7

Energy \((E)\) stored in a spring, measured in joules, varies directly with the square of its compression distance \(d\), measured in centimetres.

When a spring is compressed by 4 cm, it stores 48 joules of energy.

How much energy is stored when the spring is compressed by 7 cm?   (3 marks)

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\(\text{147 joules}\)

Show Worked Solution

\(E \propto d^2 \ \Rightarrow \ E=k d^2\)

\(\text{Find \(k\) given  \(\ E=48 \ \)  when  \(\ d=4\):}\)

\(48\) \(=k \times 4^2\)
\(k\) \(=3\)

 
\(\text{Find \(E\) when  \(\ d=7\):}\)

\(E=3 \times 7^2=147 \ \text{joules}\)