SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Linear Applications, SMB-007

Fiona and John are planning to hold a fund-raising event for cancer research.  They can hire a function room for $650 and a band for $850.  Drinks will cost them $25 per person.

  1. Write a formula for the cost ($C) of holding the charity event for  `x`  people.  (1 mark)

  2. The graph below shows the planned income and costs if they charge $50 per ticket.  Estimate the number of guests they need to break even.    (1 mark)


     

  3. How much profit will Fiona and John make if 80 people attend their event?   (1 mark)

Show Answers Only
  1. `$C = 1500 + 25x`
  2. `60`
  3. `$500`
Show Worked Solution
i.    `text(Fixed C) text(osts)` `= 650 + 850`
    `= $1500`

 
`text(Variable C) text(osts) = $25x`

`:.\ $C = 1500 + 25x`

 

ii.   `text(From the graph)`
  `text(C) text(osts = Income when)\ x = 60`
  `text{(i.e. where graphs intersect)}`

 

iii.  `text(When)\ \ x = 80:` 

`text(Income)` `= 80 xx 50`  
  `= $4000`  

 

`$C` `= 1500 + 25 xx 80`
  `= $3500`

 

`:.\ text(Profit)` `= 4000-3500`
  `= $500`

Linear Applications, SMB-006

The average height, `C`, in centimetres, of a girl between the ages of 6 years and 11 years can be represented by a line with equation

`C = 6A + 79`

where `A` is the age in years. For this line, the gradient is 6.

  1. What does this indicate about the heights of girls aged 6 to 11?   (1 mark)

  2. Give ONE reason why this equation is not suitable for predicting heights of girls older than 12.   (1 mark)

Show Answers Only
  1. `text(It indicates that 6-11 year old girls, on average, grow 6 cm per year.)`
  2. `text(Girls eventually stop growing, and the equation doesn’t factor this in.)`
Show Worked Solution

i.    `text(It indicates that 6-11 year old girls, on average, grow)`

`text(6 cm per year.)`
 

ii.    `text(Girls eventually stop growing, and the equation doesn’t)`

`text(factor this in.)`

Linear Applications, SMB-005 MC

Renee went bike riding on a holiday.

The hiring charges are  listed in the table below:

\begin{array} {|l|c|c|}
\hline \text{Hours hired} \ (h) & 1 & 2 & 3 & 4 & 5 \\
\hline \text{Cost} \ (C) & 18 & 24 & 30 & 36 & 42 \\
\hline \end{array}

Which linear equation shows the relationship between `C` and `h`?

  1. `C = 12 + 6h`
  2. `C = 6 + 12h`
  3. `C=18 + 12h`
  4. `C=12 + 18h`
Show Answers Only

`A`

Show Worked Solution

`text(Consider Option 1:)`

`12 + (6 xx 1) = 12+6=18`

`12 + (6 xx 2) = 12+12=24`

`12 + (6 xx 3) = 12+18=30\ \ \ \ text(etc …)`

`:.\ text(The linear equation is:)\ \ C = 12 + 6h`

`=>A`

Linear Applications, SMB-004

The prices at an ice cream shop can be seen below.
  

  

Each extra scoop of ice cream costs the same amount of money.

How much will an ice cream with 5 scoops cost?  (2 marks)

Show Answers Only

`$8.80`

Show Worked Solution

`text{Cost of 1 extra scoop}`

`= 6.25-5.40`

`= 0.85`
 

`:.\ text{Cost of an icecream with 5 scoops}`

`= $5.40 + 4\ text(extra scoops)`

`= 5.40 + (4 xx 0.85)`

`= 5.40 + 3.40`

`= $8.80`

Linear Applications, SMB-003 MC

This chart shows the longest run, in kilometres, that Deek ran each week over 5 weeks.

\begin{array} {|l|c|c|c|c|c|}
\hline \textbf{Week} & 1 & 2 & 3 & 4 & 5\\
\hline \textbf{Longest run (km)} & 8 & 11 & 14 & 17 & 20\\
\hline \end{array}

If the pattern continues, in which week is Deek's longest run 29 km?

  1. `7`
  2. `8`
  3. `9`
  4. `10`
Show Answers Only

`B`

Show Worked Solution

`text(Deek’s longest run increases by 3 km each week.)`

`text(In week 6:  Longest run = 23 km)`

`text(In week 7:  Longest run = 26 km)`

`text(In week 8:  Longest run = 29 km)`

`=>B`

Linear Applications, SMB-002 MC

At an apple orchard, apples are picked and put in a basket.

The table below shows the total number of apples in the basket after each minute.

\begin{array} {|c|c|c|}
\hline \textbf{Minutes} & \textbf{Total number of apples} \\
\hline 1 & 4 \\
\hline 2 & 8 \\
\hline 3 & 12 \\
\hline 4 & 16 \\
\hline \end{array}

How many apples are in the basket after 10 minutes? 

  1. `20`
  2. `30`
  3. `35`
  4. `40`
Show Answers Only

 `D`

Show Worked Solution

`text(4 apples are put into the basket each minute.)`

`:.\ text(Apples in basket after 10 minutes)`

`=4 xx 10`

`= 40\ text(apples)`

`=>D`

Linear Applications, SMB-001 MC

Jeremy sold ice creams out of his ice cream truck.

He drew the graph below to show how the number of ice creams he sells in a week is related to their price.

 
Which statement best describes the graph?

  1. As the ice cream price goes up, the number sold goes down.
  2. As the ice cream price goes up, the number sold goes up.
  3. As the ice cream price goes down, the number sold goes down.
  4. As the ice cream price goes down, the number sold stays the same.
Show Answers Only

`A`

Show Worked Solution

`text(As the ice cream price goes up, the number)`

`text(sold goes down.)`

`=>A`

Algebra, STD2 A4 2022 HSC 22

The formula  `C=100 n+b`  is used to calculate the cost of producing laptops, where `C` is the cost in dollars, `n` is the number of laptops produced and `b` is the fixed cost in dollars.

  1. Find the cost when 1943 laptops are produced and the fixed cost is $20 180.  (1 mark)

  2. Some laptops have some extra features added. The formula to calculate the production cost for these is
  3.      `C=100 n+a n+20\ 180`
  4. where `a` is the additional cost in dollars per laptop produced.
  5. Find the number of laptops produced if the additional cost is $26 per laptop and the total production cost is $97 040.  (2 marks)

Show Answers Only
  1. `$214\ 480`
  2. `610\ text{laptops}`
Show Worked Solution

a.   `text{Find}\ \ C\ \ text{given}\ \ n=1943 and b=20\ 180`

`C` `=100 xx 1943 + 20\ 180`  
  `=$214\ 480`  

 

b.   `text{Find}\ \ n\ \ text{given}\ \ C=97\ 040 and a=26`

`C` `=100 n+a n+20\ 180`  
`97\ 040` `=100n + 26n +20\ 180`  
`126n` `=76\ 860`  
`n` `=(76\ 860)/126`  
  `=610 \ text{laptops}`  

Algebra, STD2 A2 2020 HSC 10 MC

A plumber charges a call-out fee of $90 as well as $2 per minute while working.

Suppose the plumber works for `t` hours.

Which equation expresses the amount the plumber charges ($`C`) as a function of time (`t` hours)?

  1.  `C = 2 + 90t`
  2.  `C = 90 + 2t`
  3.  `C = 120 + 90t`
  4.  `C = 90 + 120t`
Show Answers Only

`D`

Show Worked Solution

♦ Mean mark 42%.

`text(Hourly rate)\ = 60 xx 2=$120`

`:. C = 90 + 120t`

`=>D`

Algebra, STD2 A2 2019 HSC 14 MC

Last Saturday, Luke had 165 followers on social media. Rhys had 537 followers. On average, Luke gains another 3 followers per day and Rhys loses 2 followers per day.

If  `x`  represents the number of days since last Saturday and  `y`  represents the number of followers, which pair of equations model this situation?

A.  `text(Luke:)\ \ y = 165x + 3`

 

`text(Rhys:)\ \ y = 537x - 2`
B. `text(Luke:)\ \ y = 165 + 3x`

 

`text(Rhys:)\ \ y = 537 - 2x`
C. `text(Luke:)\ \ y = 3x + 165`

 

`text(Rhys:)\ \ y = 2x - 537`
D. `text(Luke:)\ \ y = 3 + 165x`

 

`text(Rhys:)\ \ y = 2 - 537x`
Show Answers Only

`B`

Show Worked Solution

`text(Luke starts with 165 and adds 3 per day:)`

`y = 165 + 3x`

`text(Rhys starts with 537 and loses 2 per day:)`

`y = 537 – 2x`

`=> B`

Algebra, STD2 A2 2017 HSC 3 MC

The graph shows the relationship between infant mortality rate (deaths per 1000 live births) and life expectancy at birth (in years) for different countries.
 

What is the life expectancy at birth in a country which has an infant mortality rate of 60?

  1. 68 years
  2. 69 years
  3. 86 years
  4. 88 years
Show Answers Only

\(A\)

Show Worked Solution

\(\text{When infant mortality rate is 60, life expectancy}\)

\(\text{at birth is 68 years (see below).}\)
 

\(\Rightarrow A\)

Algebra, STD2 A2 2015 HSC 27c

Ariana’s parents have given her an interest‑free loan of $4800 to buy a car. She will pay them back by paying `$x` immediately and `$y` every month until she has repaid the loan in full.

After 18 months Ariana has paid back $1510, and after 36 months she has paid back $2770.

This information can be represented by the following equations.

`x + 18y = 1510`

`x + 36y = 2770`

  1. Graph these equations below and use to solve simultaneously for the values of `x` and `y`.   (2 marks)

         

  2. How many months will it take Ariana to repay the loan in full? (2 marks)

Show Answers Only
  1. `x = 250, \ y = 70`
  2. `text(65 months)`
Show Worked Solution

i.

 
`:.\ text(Solution is)\ \ x = 250, \ y = 70`
 

ii.  `text(Let)\ \ A = text(the amount paid back after)\ n\ text(months)`

`A = 250 + 70n`

♦ Mean mark 44%.

`text(Find)\ n\ text(when)\ A = 4800`

`250 + 70n` `= 4800`
`70n` `= 4550`
`n` `= 65`

 

`:.\ text(It will take Ariana 65 months to repay)`

`text(the loan in full.)`

Algebra, STD2 A2 2005 HSC 17 MC

The total cost, `$C`, of a school excursion is given by  `C = 2n + 5`, where `n` is the number of students.

If three extra students go on the excursion, by how much does the total cost increase?

  1. `$6`
  2. `$11`
  3. `$15`
  4. `$16`
Show Answers Only

`A`

Show Worked Solution

`C = 2n + 5`

`text(If)\ n\ text(increases to)\ n + 3`

`C` `= 2(n + 3) + 5`
  `= 2n + 6 + 5`
  `= 2n + 11`

 

`:.\ text(Total cost increases by $6)`

`=>  A`

Algebra, STD2 A2 2009 HSC 24d

A factory makes boots and sandals. In any week

• the total number of pairs of boots and sandals that are made is 200
• the maximum number of pairs of boots made is 120
• the maximum number of pairs of sandals made is 150.

The factory manager has drawn a graph to show the numbers of pairs of boots (`x`) and sandals (`y`) that can be made.
 

 

  1. Find the equation of the line `AD`.   (1 mark)

  2. Explain why this line is only relevant between `B` and `C` for this factory.     (1 mark)

  3. The profit per week, `$P`, can be found by using the equation  `P = 24x + 15y`.

     

    Compare the profits at `B` and `C`.     (2 marks)

Show Answers Only
  1. `x + y = 200`
  2. `text(S)text(ince the max amount of boots = 120)`

     

    `=> x\ text(cannot)\ >120`

     

    `text(S)text(ince the max amount of sandals = 150`

     

    `=> y\ text(cannot)\ >150`

     

    `:.\ text(The line)\ AD\ text(is only possible between)\ B\ text(and)\ C.`

  3. `text(The profits at)\ C\ text(are $630 more than at)\ B.`
Show Worked Solution

i.   `text{We are told the number of boots}\ (x),` 

♦♦♦ Mean mark part (i) 14%. 
Using `y=mx+b` is a less efficient but equally valid method, using  `m=–1`  and  `b=200` (`y`-intercept).

`text{and shoes}\  (y),\ text(made in any week = 200)`

`=>text(Equation of)\ AD\ text(is)\ \ x + y = 200`

 

ii.  `text(S)text(ince the max amount of boots = 120)`

♦ Mean mark 49%

`=> x\ text(cannot)\ >120`

`text(S)text(ince the max amount of sandals = 150`

`=> y\ text(cannot)\ >150`

`:.\ text(The line)\ AD\ text(is only possible between)\ B\ text(and)\ C.`

 

iii.  `text(At)\ B,\ \ x = 50,\ y = 150`

♦ Mean mark 40%.
`=>$P  (text(at)\ B)` `= 24 xx 50 + 15 xx 150`
  `= 1200 + 2250`
  ` = $3450`

`text(At)\ C,\ \  x = 120 text(,)\ y = 80`

`=> $P  (text(at)\ C)` `= 24 xx 120 + 15 xx 80`
  `= 2880 + 1200`
  `= $4080`

 

`:.\ text(The profits at)\ C\ text(are $630 more than at)\ B.`

Algebra, STD2 A2 2009 HSC 14 MC

If   `A = 6x + 10`, and  `x`  is increased by  2, what will be the corresponding increase in `A` ?

  1. `2x` 
  2. `6x` 
  3. `2` 
  4. `12` 
Show Answers Only

`D`

Show Worked Solution
♦ Mean mark 50%.
STRATEGY: Substituting real numbers into the equation can work well in these type of questions. eg. If `x=0,\ A=10` and when `x=2,\ A=22`.

`A = 6x + 10`

`text(If)\ x\ text(increases by 2)`

`A\ text(increases by)\ 6 xx 2 = 12`

`=>  D`