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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`

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.)`