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Functions, 2ADV F1 2022 HSC 12

A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating `M` and `T`.    (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from `T=5^@C` to `T=30^@ text{C}`.   (2 marks)
     

\begin{array} {|c|c|c|c|}
\hline  \ \ T\ \  & \ \ 5\ \  & \ 15\  & \ 30\  \\
\hline M &  &  &  \\
\hline \end{array}

 
                   

Show Answers Only

a.    `M=180/T`

 b.    

\begin{array} {|c|c|c|c|}
\hline  \ \ T\ \  & \ \ 5\ \  & \ 15\  & \ 30\  \\
\hline M & 36 & 12 & 6 \\
\hline \end{array}       

 

Show Worked Solution
a.    `M` `prop 1/T`
  `M` `=k/T`
  `12` `=k/15`
  `k` `=15 xx 12`
    `=180`

 
`:.M=180/T`
 


♦ Mean mark (a) 49%.

b.   

\begin{array} {|c|c|c|c|}
\hline  \ \ T\ \  & \ \ 5\ \  & \ 15\  & \ 30\  \\
\hline M & 36 & 12 & 6 \\
\hline \end{array}

Functions, 2ADV F1 EQ-Bank 8

Jacques is a marine biologist and finds that the mass of a crab is directly proportional to the cube of the diameter of its shell.

If a crab with a shell diameter of 15 cm weighs 680 grams, what will be the diameter of a crab that weighs 1.1 kilograms? Give your answer to 1 decimal place.  (2 marks)

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`17.6\ text(cm)`

Show Worked Solution
`M` `prop d^3`  
`M` `= kd^3`  

 
`text(When)\ \ M=680, \ d=15`

`680` `=k xx 15^3`  
`k` `=0.201481…`  

 
`text(Find)\ \ d\ \ text(when)\ \ M=1100:`

`1100` `=0.20148… xx d^3`  
`d` `=root3(1100/(0.20148…))`  
  `=17.608…`  
  `=17.6\ text{cm  (to 1 d.p.)}`  

Functions, 2ADV F1 EQ-Bank 7

The current of an electrical circuit, measured in amps `(A)`, varies inversely with its resistance, measured in ohms `(R)`.

When the resistance of a circuit is 28 ohms, the current is 3 amps.

What is the current when the resistance is 8 ohms?   (2 marks)

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`10.5`

Show Worked Solution

`A \prop 1/R\ \ =>\ \ A=k/R`

`text(When)\ \ A=3, \ R=28:`

`3` `=k/28`  
`k` `=84`  

 
`text(Find)\ A\ \text{when}\ R=8:`

`A=84/8=10.5`

Functions, 2ADV F1 EQ-Bank 27

The stopping distance of a car on a certain road, once the brakes are applied, is directly proportional to the square of the speed of the car when the brakes are first applied.

A car travelling at 70 km/h takes 58.8 metres to stop.

How far does it take to stop if it is travelling at 105 km/h?  (3 marks)

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`132.3\ text(metres)`

Show Worked Solution

`text(Let)\ \ d\ text(= stopping distance)`

`d ∝ s^2`

`d = ks^2`
 

`text(Find)\ k,`

`58.8` `= k xx 70^2`
`k` `= 58.8/(70^2)`
  `= 0.012`

 
`text(Find)\ \ d\ \ text(when)\ \ s = 105:`

`d` `= 0.012 xx 105^2`
  `= 132.3\ text(metres)`

Functions, 2ADV F1 EQ-Bank 26

Fuifui finds that for Giant moray eels, the mass of an eel is directly proportional to the cube of its length.

An eel of this species has a length of 25 cm and a mass of 4350 grams.

What is the expected length of a Giant moray eel with a mass of 6.2 kg? Give your answer to one decimal place.  (3 marks)

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`28.1\ text{cm}`

Show Worked Solution

`text(Mass) prop text(length)^3`

`m = kl^3`
 

`text(Find)\ k:`

`4350` `= k xx 25^3`
`k` `= 4350/25^3`
  `= 0.2784`

 
`text(Find)\ \ l\ \ text(when)\ \ m = 6200:`

`6200` `= 0.2784 xx l^3`
`l^3` `= 6200/0.2784`
`:. l` `= 28.13…`
  `= 28.1\ text{cm  (to 1 d.p.)}`