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v1 Algebra, STD2 A4 2019 HSC 33

The time taken for a student to type an assignment varies inversely with their typing speed.

It takes the student 180 minutes to finish the assignment when typing at 40 words per minute.

  1. Calculate the length of the assignment in words.   (1 mark)
  2. By first plotting four points, draw the curve that shows the time taken to complete the assignment at different constant typing speeds.   (3 marks)
     

 

Show Answers Only

a.   `7200\ text(words)`

b.   
     

Show Worked Solution

a.   `text{Assignment length}\ = 180 xx 40=7200\ \text{words}`

b.   `text{Time} (T) prop 1/{\text{Typing speed}\ (S)} \ \ =>\ \ T = k/S`
 

\begin{array} {|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & 720 & 360 & 180 & 90 \\
\hline
\rule{0pt}{2.5ex} S \rule[-1ex]{0pt}{0pt} & 10 & 20 & 40 & 80 \\
\hline
\end{array}

v1 Algebra, STD2 A4 2004 HSC 28a

A fitness index, `F`, is calculated by dividing a person’s weight, `w`, in kilograms by the square of the person’s height, `h`, in metres.

  1. Albert is 160 cm and weighs 76.8 kg. Calculate Albert’s health rating.   (1 mark)

  2. In the next few years, Albert expects to grow 20 cm taller. By then he wants her fitness index to be 23. How much weight should he gain or lose to achieve this? Justify your answer with mathematical calculations.   (2 marks)

Show Answers Only

a.   `30`

b.   `text(Albert needs to gain 4.2 kg)`

Show Worked Solution

a.   `\text{160 cm = 1.60 m.}`

`text(When)\ w = 76.8 and h = 1.60:`

`F=w/h^2 = 76.8/1.60^2 = 30`
 

b.   `text(Find)\ w\ text(given)\ F= 25 and h = 1.80:`

`25` `= w/1.8^2`
`w` `= 25 xx 1.8^2`
  `= 81\ text(kg)`

 

`:.\ text(Weight Albert should gain)`

`= 81-76.8`

`= 4.2\ text(kg)`

v1 Algebra, STD2 A4 2014 HSC 29a

A golf club hires an entire course for a charity event at a total cost of `$40\ 000`. The cost will be shared equally among the players, so that `C` (in dollars) is the cost per player when `n` players attend.

  1. Complete the table below by filling in the three missing values.   (1 mark)
    \begin{array} {|l|c|c|c|c|c|c|}
    \hline
    \rule{0pt}{2.5ex}\text{Number of players} (n) \rule[-1ex]{0pt}{0pt} & \ 50\ & \ 100 \ & 200 \ & 250 \ & 400\ & 500 \ \\
    \hline
    \rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} &  &  &  & 160 & 100\ & 80 \ \\
    \hline
    \end{array}
  2. Using the values from the table, draw the graph showing the relationship between `n` and `C`.   (2 marks)
     
  3. What equation represents the relationship between `n` and `C`?   (1 mark)

  4. Give ONE limitation of this equation in relation to this context.   (1 mark)

  5. Is it possible for the cost per person to be $94? Support your answer with appropriate calculations.   (1 mark)

Show Answers Only

a.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of players} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
 

b. 

c.   `C = (40\ 000)/n`

`n\ text(must be a whole number)`
 

d.    `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`
 

e.   `text(If)\ C = 94:`

`94` `= (120\ 000)/n`
`94n` `= 120\ 000`
`n` `= (120\ 000)/94`
  `= 1276.595…`

 
`:.\ text(C)text(ost cannot be $94 per person, because)\ n\ text(isn’t a whole number.)`

Show Worked Solution

a.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of players} (n) \rule[-1ex]{0pt}{0pt} & \ 50\ & \ 100 \ & 200 \ & 250 \ & 400\ & 500 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 800 & 400 & 200 & 160 & 100\ & 80 \ \\
\hline
\end{array}

b.   
       
      

c.   `C = (40\ 000)/n`

 

TIP: Limitations require looking at possible restrictions of both the domain and range.

d.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`
 

e.   `text(If)\ C = 120`

`120` `= (40\ 000)/n`
`120n` `= 40\ 000`
`n` `= (40\ 000)/120`
  `= 333.33..`

  
`:.\ text{Cost cannot be $120 per person because the required}\ n\ \text{is not a whole number.}`

v1 Algebra, STD2 A4 EO-Bank 2

The number of chairs that can be placed around a circular stage varies inversely with the space left between each chair.

There will be 72 chairs if the distance between them is 0.5 metres.

  1. How many chairs can be placed if the distance between them is 0.9 metres?   (2 marks)

  2. What is the spacing between chairs if 90 chairs are placed around the stage?   (1 mark)

Show Answers Only

a.   40 chairs

b.   0.4 metres

Show Worked Solution

a.   `c \prop 1/d\ \ =>\ \ c=k/d`

`72` `= k/0.5`
`k` `= 0.5 xx 72 = 36`

 
`text(Find)\ c\ text(when)\ \ d = 0.9:`

`c=36/0.9=40\ \text{chairs}`
 

b.   `text(Find)\ d\ text(when)\ \ c = 90:`

`90` `= 36/d`
`d` `= 36/90`
  `= 0.4\ text(metres)`

v1 Algebra, STD2 A4 2022 HSC 24

A chef believes that the time it takes to defrost a turkey (`D` hours) varies inversely with the room temperature (`T^\circ \text{C}`). The chef observes that at a room temperature of `20^\circ \text{C}`, it takes 15 hours for the turkey to fully defrost.

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

  2. By first completing this table of values, graph the relationship between temperature and time.   (2 marks)

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 10\ \ \  & \ \ 15\ \ \  & \ \ \ 20\ \ \  & \ \ \ 25\ \ \ & \ \ \ 30\ \ \ \\
\hline
\rule{0pt}{2.5ex} \ \ D\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ \  & \ \ \  & \ \ \ \  & \ \ \ \ & \ \ \ \\
\hline
\end{array}

 
           

Show Answers Only

a.   `D \prop 1/T\ \ =>\ \ D=k/T`

  `15` `=k/20`
  `k` `=15 xx 20=300`

 
`:.D=300/T`

b.   

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 10\ \ \  & \ \ 15\ \ \  & \ \ \ 20\ \ \  & \ \ \ 25\ \ \ & \ \ \ 30\ \ \ \\
\hline
\rule{0pt}{2.5ex} \ \ D\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 30\ \ \  & \ \ 20\ \ \  & \ \ \ 15\ \ \  & \ \ \ 12\ \ \ & \ \ \ 10\ \ \ \\
\hline
\end{array}

 

     

Show Worked Solution

a.   `D \prop 1/T\ \ =>\ \ D=k/T`

  `15` `=k/20`
  `k` `=15 xx 20=300`

 
`:.D=300/T`

b.   

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ D\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 10\ \ \  & \ \ 15\ \ \  & \ \ \ 20\ \ \  & \ \ \ 25\ \ \ & \ \ \ 30\ \ \ \\
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & 30 & 20 & 15 & 12 & 10  \\
\hline
\end{array}

 

     

v1 Algebra, STD2 A4 2018 HSC 29c

Snowhound makes snow shoes of various sizes. In its design phase, Snowhound collect data on the different footprint depths of snow shoes of different sizes, all worn by the same person.

 The footprint depth (`d` cm) is then graphed against the area of the sole of the snow shoe (`A` cm).
 


 

  1. The graph shape shows that `d` is inversely proportional to `A`. The point `X` lies on the graph.

     

    Find the equation relating `d` and `A`.  (2 marks)

  2. A man from this group walks in snow and the depth of his footprint is 5 cm.

     

    Use your equation from part (a) to calculate the area of his shoe sole.  (1 mark)

Show Answers Only

a.   `D = 4500/A`

b.   `480\ text(cm)^2`

Show Worked Solution

a.   `d prop 1/A \ =>\ d = k/A`

`text(When)\ D = 12, A = 200:`

`12` `= k/200`
`k` `= 12 xx 200 = 2400`
`:. d` `=2400/A`

 

b.    `5` `= 2400/A`
  `:. A` `= 2400/5= 480\ text(cm)^2`

v1 Algebra, STD2 A4 2011 HSC 28a

The intensity of light, `I`, from a lamp varies inversely with the square of the distance, `d`, from the lamp.

  1. Write an equation relating `I`, `d` and `k`, where `k` is a constant.    (1 mark)

  2. It is known that `I = 20` when `d = 2`.

     

    By finding the value of the constant, `k`, find the value of `I` when `d = 5`.    (2 marks)

  3. Sketch a graph to show how `I` varies for different values of `d`.

     

    Use the horizontal axis to represent distance and the vertical axis to represent light intensity.   (2 marks)


Show Answers Only

a.   `I = k/d^2`

b.   `P = 1 1/2`

c.   
         

Show Worked Solution

a.   `I prop 1/d\ \ =>\ \ I=k/d^2`
 

b.  `text(When)\ I=20, d=2:`

`20` `= k/2^2`
`k` `=4 xx 20=80`

 
`text(Find)\ I\ text(when)\ d = 5:`

`I=80/5^2=16/5`
 

c.

v1 Algebra, STD2 A4 2021 HSC 13 MC

The time taken to harvest a field varies inversely with the number of workers employed.

It takes 12 workers 36 hours to harvest the field.

Working at the same rate, how many hours would it take 27 workers to harvest the same field?

  1. 12
  2. 16
  3. 24
  4. 54
Show Answers Only

`B`

Show Worked Solution

`text{Time to harvest}\ (T) prop 1/text{Number of workers (W)}`

`T=k/W`

`text(When)\ \ T=36, W=12:`

`36=k/12\ \ =>\ \ k=36 xx 12 = 432`  
 

`text{Find}\ T\ text(when)\ \ W=27:`

`T=432/27=16\ \text{hours}`

 `=>  B`

v1 Algebra, STD2 A4 2007 HSC 15 MC

If the speed `(s)` of a journey varies inversely with the time `(t)` taken, which formula correctly expresses `s` in terms of `t` and `k`, where `k` is a constant?

  1. `s = k/t`
  2. `s = kt`
  3. `s = k + t`
  4. `s = t/k`
Show Answers Only

`A`

Show Worked Solution

`s prop 1/t \ \ => \ s = k/t`

`=>  A`

v1 Algebra, STD2 A4 2010 HSC 13 MC

The time taken to charge a battery varies inversely with the charging voltage. At 24 volts \((V)\) it takes 15 hours to fully charge a battery.

How long will it take the same battery to fully charge at 40 volts?

  1. 8 hours
  2. 9 hours
  3. 10.5 hours
  4. 12 hours
Show Answers Only

`B`

Show Worked Solution
 
♦ Mean mark 50% 

`text{Time to charge}\ (T) prop 1/text(Voltage) \ => \ T=k/V`

`text(When) \ T=15, V = 24:`

`15=k/24\ \ => \ k=15 xx 24=360` 
   

`text{Find}\ T\ text{when}\ \ V= 40:}`

`T=360/40=9\ \text{hours}`

 `=>  B`

v1 Algebra, STD2 A4 2024 HSC 9 MC

The time taken to fill a swimming pool varies inversely with the number of hoses being used.

Using 4 hoses, it takes 18 hours to fill the pool.

How many hours would it take 9 hoses to fill the same pool?

  1. 6
  2. 7.5
  3. 8
  4. 12
Show Answers Only

\(C\)

Show Worked Solution

\(T \propto \dfrac{1}{H} \ \Rightarrow \ \ T=\dfrac{k}{H}\)

\(\text {Find}\ k\ \text{given}\ \  T=18\ \ \text {when}\ \ H=4 \text {:}\)

\(18=\dfrac{k}{4} \ \Rightarrow\ \ k=72\)
 

\(\text {Find}\ T \ \text {if}\ \ H=9:\)

\(T=\dfrac{72}{9}=8 \text { hours}\)

\(\Rightarrow C\)

Algebra, STD2 A4 2024 HSC 9 MC

The time taken to paint a school varies inversely with the number of painters completing the task.

It takes 6 painters a total of 20 days to paint a school.

How many days would it take 15 painters to paint the same school?

  1. 4.5
  2. 8
  3. 15.5
  4. 50
Show Answers Only

\(B\)

Show Worked Solution

\(T \propto \dfrac{1}{N} \ \Rightarrow \ \ T=\dfrac{k}{N}\)

\(\text {Find}\ k\ \text{given}\ \  T=20\ \ \text {when}\ \ N=6 \text {:}\)

\(20=\dfrac{k}{6} \ \Rightarrow\ \ k=120\)

\(\therefore\ T=\dfrac{120}{N}\)

\(\text {Find}\ T \ \text {if}\ \ N=15:\)

\(T=\dfrac{120}{15}=8 \text { days }\)

\(\Rightarrow B\)

Algebra, STD2 A4 2022 HSC 24

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} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \  & \ \ 15\ \ \  & \ \ \ 30\ \ \  \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}

 
           

Show Answers Only

a.  `M prop 1/T \ \ =>\ \ M=k/T`

  `12` `=k/15`
  `k` `=15 xx 12=180`

 
`:.M=180/T`

b.   

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \  & \ \ 15\ \ \  & \ \ 30\ \  \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & 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 part (a) 29%.

b.   

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \  \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \  & \ \ 15\ \ \  & \ \ 30\ \  \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & 36 & 12 & 6 \\
\hline
\end{array}

 

     


♦ Mean mark 44%.

Algebra, STD2 A4 2021 HSC 13 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
Show Answers Only

`B`

Show Worked Solution

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

♦ Mean mark 41%.

`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`

Algebra, STD2 A4 2019 HSC 33

The time taken for a car to travel between two towns at a constant speed varies inversely with its speed.

It takes 1.5 hours for the car to travel between the two towns at a constant speed of 80 km/h.

  1. Calculate the distance between the two towns.  (1 mark)

  2. By first plotting four points, draw the curve that shows the time taken to travel between the two towns at different constant speeds.  (3 marks)

Show Answers Only
  1. `120\ text(km)`
  2.  
Show Worked Solution
a.    `D` `= S xx T`
    `= 80 xx 1.5`
    `= 120\ text(km)`

 
b.   

\begin{array} {|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ s\ \  \rule[-1ex]{0pt}{0pt} & 20 & 40 & 60 & 80 \\
\hline
\rule{0pt}{2.5ex} t \rule[-1ex]{0pt}{0pt} & 6 & 3 & 2 & 1.5 \\
\hline
\end{array}

Algebra, STD2 A4 SM-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)

Show Answers Only

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`
 

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

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

Algebra, STD2 A4 2018 HSC 29c

When people walk in snow, the depth (`D` cm) of each footprint depends on both the area (`A` cm²) of the shoe sole and the weight of the person. The graph shows the relationship between the area of the shoe sole and the depth of the footprint in snow, for a group of people of the same weight.
 


 

  1. The graph is a hyperbola because `D` is inversely proportional to `A`. The point `P` lies on the hyperbola.

     

    Find the equation relating `D` and `A`.  (2 marks)

  2. A man from this group walks in snow and the depth of his footprint is 4 cm.

     

    Use your equation from part (i) to calculate the area of his shoe sole.  (1 mark)

Show Answers Only
  1. `D = 4500/A`
  2. `1125\ text(cm²)`
Show Worked Solution

i.     `D prop 1/A \ =>\ D = k/A`

♦♦ Mean mark part (i) 22%.

 

`text(When)\ D = 15, A = 300`

`15` `= k/300`
`k` `= 4500`
`:. D` `=4500/A`

 

♦♦ Mean mark part (ii) 33%.

ii.    `4` `= 4500/A`
  `:. A` `= 4500/4`
    `= 1125\ text(cm²)`

Algebra, STD2 A4 2004 HSC 28a

A health rating, `R`, is calculated by dividing a person’s weight, `w`, in kilograms by the square of the person’s height, `h`, in metres.

  1. Fred is 150 cm and weighs 72 kg. Calculate Fred’s health rating.  (1 mark)

  2. Over several years, Fred expects to grow 10 cm taller. By this time he wants his health rating to be 25. How much weight should he gain or lose to achieve his aim? Justify your answer with mathematical calculations.  (2 marks)

Show Answers Only

i.   `32`

ii.  `text(8 kg)`

Show Worked Solution

i.    `R = w/h^2`

`text(When)\ w = 72 and h = 1.5:`

`R=72/1.5^2= 32`
 

ii.  `text(Find)\ w\ text(if)\ R = 25 and h = 1.6:`

`25` `= w/1.6^2`
`w` `= 25 xx 1.6^2`
  `= 64\ text(kg)`

 
`:.\ text(Weight Fred should lose)`

`= 72-64`

`= 8\ text(kg)`

Algebra, STD2 A4 2004 HSC 21 MC

The time `(t)` taken to clean a house varies inversely with the number `(n)` of people
cleaning the house.

Which graph represents this relationship?

Show Answers Only

`D`

Show Worked Solution

`t ∝ 1/n \ => \ t = k/n`

`text(The graph is a hyperbola that sees)\ t\ text(decrease as)\ n\ text{increases (eliminate A and B).}`

`text(Also,)\ t\ text{cannot be zero (eliminate C).}`

`=>  D`

Algebra, STD2 A4 2007 HSC 15 MC

If pressure (`p`) varies inversely with volume (`V`), which formula correctly expresses  `p`  in terms of  `V`  and  `k`, where  `k`  is a constant?

  1. `p = k/V`
  2. `p = V/k`
  3. `p = kV`
  4. `p = k + V`
Show Answers Only

`A`

Show Worked Solution

`p prop 1/V`

`p = k/V`

`=>  A`

Algebra, STD2 A4 2014 HSC 29a

The cost of hiring an open space for a music festival is  $120 000. The cost will be shared equally by the people attending the festival, so that  `C`  (in dollars) is the cost per person when  `n`  people attend the festival.

  1. Complete the table below by filling in the THREE missing values.   (1 mark)
    \begin{array} {|l|c|c|c|c|c|c|}
    \hline
    \rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
    \hline
    \rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} &  &  &  & 60 & 48\ & 40 \ \\
    \hline
    \end{array}
  2. Using the values from the table, draw the graph showing the relationship between  `n`  and  `C`.   (2 marks)
     
  3. What equation represents the relationship between `n` and `C`?   (1 mark)

  4. Give ONE limitation of this equation in relation to this context.   (1 mark)

  5. Is it possible for the cost per person to be $94? Support your answer with appropriate calculations.   (1 mark)

Show Answers Only

i.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
 

ii. 

iii.   `C = (120\ 000)/n`

`n\ text(must be a whole number)`
 

iv.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`
 

v.   `text(If)\ C = 94:`

`94` `= (120\ 000)/n`
`94n` `= 120\ 000`
`n` `= (120\ 000)/94`
  `= 1276.595…`

 
`:.\ text(C)text(ost cannot be $94 per person,)`

`text(because)\ n\ text(isn’t a whole number.)`

Show Worked Solution

i.   

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
 

ii. 

 

♦ Mean mark (iii) 48%

iii.   `C = (120\ 000)/n`

 

♦♦♦ Mean mark (iv) 7%
COMMENT: When asked for limitations of an equation, look carefully at potential restrictions with respect to both the domain and range.

iv.   `text(Limitations can include:)`

  `•\ n\ text(must be a whole number)`

  `•\ C > 0`

 

v.   `text(If)\ C = 94`

`=> 94` `= (120\ 000)/n`
`94n` `= 120\ 000`
`n` `= (120\ 000)/94`
  `= 1276.595…`
♦ Mean mark (v) 38%

 

`:.\ text(C)text(ost cannot be $94 per person,)`

`text(because)\ n\ text(isn’t a whole number.)`

Algebra, STD2 A4 2011 HSC 28a

The air pressure, `P`, in a bubble varies inversely with the volume, `V`, of the bubble. 

  1. Write an equation relating `P`, `V` and `a`, where `a` is a constant.    (1 mark)

  2. It is known that `P = 3` when `V = 2`.

     

    By finding the value of the constant, `a`, find the value of `P` when `V = 4`.    (2 marks)

  3. Sketch a graph to show how `P` varies for different values of `V`.

     

    Use the horizontal axis to represent volume and the vertical axis to represent air pressure.   (2 marks)


Show Answers Only
  1. `P = a/V`
  2. `P = 1 1/2`
  3.  
     
Show Worked Solution
♦ Mean mark (i) 39%
COMMENT: Expressing the proportional relationship `P prop 1/V` as the equation `P=k/V` is a core skill here.
i. `P` `prop 1/V`
    `= a/V`

 

ii. `text(When)\ P=3,\ V = 2`
`3` `= a/2`
`a` `=6`

 

`text(Need to find)\ P\ text(when)\ V = 4`  

♦ Mean mark (ii) 47%
`P` `=6/4`
  `= 1 1/2`

  

♦♦ Mean mark (iii) 26%
COMMENT: An inverse relationship is reflected by a hyperbola on the graph.
iii.

Algebra, STD2 A1 2009 HSC 16 MC

The time for a car to travel a certain distance varies inversely with its speed.

Which of the following graphs shows this relationship?
 

Show Answers Only

`A`

Show Worked Solution
`T` `prop 1/S`
`T` `= k/S`

 
`text{By elimination:}`

`text(As   S) uarr text(, T) darr => text(cannot be B or D)`

♦ Mean mark 38%

`text(C  is incorrect because it graphs a linear relationship)`

`=>  A`

Algebra, STD2 A4 2010 HSC 13 MC

The number of hours that it takes for a block of ice to melt varies inversely with the temperature. At 30°C it takes 8 hours for a block of ice to melt.

How long will it take the same size block of ice to melt at 12°C?  

  1. 3.2 hours
  2. 20 hours
  3. 26  hours
  4. 45 hours
Show Answers Only

`B`

Show Worked Solution
 
♦ Mean mark 50% 

`text{Time to melt}\ (T) prop1/text(Temp) \ => \ T=k/text(Temp)`

`text(When) \ T=8, text(Temp = 30)`

`8` `=k/30`
`k` `=240`

  

`text{Find}\ T\ text{when  Temp = 12:}`

`T` `=240/12`
  `=20\ text(hours)`

 
`=>  B`