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Algebra, STD2 A4 2024 HSC 14 MC

Year 11 is making pizzas to raise money for a charity.

The cost \((C)\) and revenue \((R)\) in dollars, when \(x\) pizzas are sold are represented by the equations

\begin{aligned}
& C=2.5 x+6 \\
& R=8 x .
\end{aligned}

Enough pizzas are sold so that a profit is made.

By how much does the profit increase for each additional pizza sold?

  1. $2.50
  2. $5.50
  3. $6.00
  4. $8.00
Show Answers Only

\(B\)

Show Worked Solution

\(\text {Cost increases by } \$ 2.50 \text { for each extra pizza.}\)

\(\text {Revenue increases by } \$ 8 \text { for each extra pizza.}\)

\(\text {Profit (additional) }=8-2.5=\$ 5.50 \text { per pizza}\)

\(\Rightarrow B\)

♦ Mean mark 49%.

Algebra, STD2 A4 2019 HSC 36

A small business makes and sells bird houses.

Technology was used to draw straight-line graphs to represent the cost of making the bird houses `(C)` and the revenue from selling bird houses `(R)`. The `x`-axis displays the number of bird houses and the `y`-axis displays the cost/revenue in dollars.
 


 

  1. How many bird houses need to sold to break even?  (1 mark)

  2. By first forming equations for cost `(C)` and revenue `(R)`, determine how many bird houses need to be sold to earn a profit of $1900.  (3 marks)

Show Answers Only
  1. `20`
  2. `96`
Show Worked Solution

a.   `20\ \ (xtext(-value at intersection))`

 

b.   `text(Find equations of both lines):`

♦♦ Mean mark 28%.

`(0, 500)\ text(and)\ (20, 800)\ text(lie on)\ \ C`

`m_C = (800-500)/(20-0) = 15`

`=> C = 500 + 15x`
 

`(0,0)\ text(and)\ (20, 800)\ text(lie on)\ \ R`

`m_R = (800-0)/(20-0) = 40`

`=> R = 40x`
 

`text(Profit) = R-C`

`text(Find)\ \ x\ \ text(when Profit = $1900:)`

`1900` `= 40x-(500 + 15x)`
`25x` `= 2400`
`x` `= 96`

Algebra, STD2 A4 2018 HSC 27d

The graph displays the cost (`$c`) charged by two companies for the hire of a minibus for `x` hours.
 


  

Both companies charge $360 for the hire of a minibus for 3 hours.

  1. What is the hourly rate charged by Company A?  (1 mark)

  2. Company B charges an initial booking fee of $75.

     

    Write a formula, in the form of  `c = mx + b`, for the cost of hiring a minibus from Company B for `x` hours.  (2 marks)

  3. A minibus is hired for 5 hours from Company B.

     

    Calculate how much cheaper this is than hiring from Company A.  (2 marks)

Show Answers Only
  1. `$120`
  2. `c = 95x + 75`
  3. `$50`
Show Worked Solution
i.    `text(Hourly rate)\ (A)` `= 360 ÷ 3`
    `= $120`

 

ii.   `m = text(hourly rate)`

`text(Find)\ m,\ text(given)\ c = 360,\ text(when)\ \ x = 3\ \ text(and)\ \ b = 75`

`360` `= m xx 3 + 75`
`3m` `= 285`
`m` `= 95`

 
`:. c = 95x + 75`
 

iii.    `text(C)text(ost)\ (A)` `= 120 xx 5 = $600`
  `text(C)text(ost)\ (B)` `= 95 xx 5 + 75 = $550`

 
`:.\ text(Company)\ B’text(s hiring cost is $50 cheaper.)`

Algebra, STD2 A4 2015 HSC 28f

A charity seeks to raise money by telephoning people at random from a call centre and asking them to donate.

Over the years, this charity has found that the amount of money raised `($A)` is related to the number of telephone calls made `(n)`. A graph of this relationship is shown.
 


 

It costs the charity $2100 per week to run the call centre. It also costs an average of 50 cents per telephone call.

  1. Write an equation to represent the total cost, `C`, of running the call centre for a week in which `n` phone calls are made.  (1 mark)

  2. By graphing this equation on the axes above, determine the number of phone calls the charity needs to make in order to break even.  (2 marks)

Show Answers Only
  1. `C = $2100 + $0.50n`
  2. `text(700 calls)`
Show Worked Solution

i.   `C = $2100 + $0.50n`

♦ Mean marks of 48% and 32% for parts (i) and (ii) respectively.

 

ii.  

2UG 2015 28f Answer

`text(From the above graph, the charity needs to)`

`text(make 700 calls to break even.)`

Algebra, STD2 A4 2005 HSC 28b

Sue and Mikey are planning a fund-raising dance. They can hire a hall for $400 and a band for $300. Refreshments will cost them $12 per person.

  1. Write a formula for the cost ($C) of running the dance for `x` people. (1 mark)

The graph shows planned income and costs when the ticket price is $20 

2005 28b

  1. Estimate the minimum number of people needed at the dance to cover the costs.  (1 mark)

  2. How much profit will be made if 150 people attend the dance? (1 mark)

Sue and Mikey plan to sell 200 tickets. They want to make a profit of $1500.

  1. What should be the price of a ticket, assuming all 200 tickets will be sold?  (3 marks)

Show Answers Only
  1. `700 + 12x`
  2. `text(Approximately 90)`
  3. `$500`
  4. `$23`
Show Worked Solution
i.    `$C` `= 400 + 300 + (12 xx x)`
    `= 700 + 12x`

 

ii.  `text(Using the graph intersection)`

`text(Approximately 90 people are needed)`

`text(to cover the costs.)`

 

iii.  `text(If 150 people attend)`

`text(Income)` `= 150 xx $20`
  `= $3000`
`text(C)text(osts)` `= 700 + (12 xx 150)`
  `= $2500`

 

`:.\ text(Profit)` `= 3000 − 2500`
  `= $500`

 

iv.  `text(C)text(osts when)\ x = 200:`

`C` `= 700 + (12 xx 200)`
  `= $3100`

 

`text(Income required to make $1500 profit)`

`= 3100 + 1500`

`= $4600`
 

`:.\ text(Price per ticket)` `= 4600/200`
  `= $23`

Algebra, STD2 A4 2011 HSC 20 MC

A function centre hosts events for up to 500 people. The cost `C`, in dollars, for the centre
to host an event, where `x` people attend, is given by:

`C = 10\ 000 + 50x`

The centre charges $100 per person. Its income `I`, in dollars, is given by:

`I = 100x`
 

2UG 2011 20

How much greater is the income of the function centre when 500 people attend an event, than its income at the breakeven point?

  1.  `$15\ 000`
  2. `$20\ 000`
  3. `$30\ 000` 
  4. `$40\ 000`
Show Answers Only

`C`

Show Worked Solution
♦ Mean mark 50%
COMMENT: Students can read the income levels directly off the graph to save time and then check with the equations given.

`text(When)\ x=500,\ I=100xx500=$50\ 000`

`text(Breakeven when)\ \ x=200\ \ \ text{(from graph)}`

`text(When)\ \ x=200,\ I=100xx200=$20\ 000`

`text(Difference)` `=50\ 000-20\ 000`
  `=$30\ 000`

 
`=> C`

Algebra, STD2 A4 2010 HSC 24b

Ashley makes picture frames as part of her business. To calculate the cost,  `C`, in dollars, of making  `x`  frames, she uses the equation  `C=40+10x`.

She sells the frames for $20 each and determines her income,  `I`, in dollars, using the equation  `I=20x`.
 

Use the graph to solve the two equations simultaneously for  `x`  and explain the significance of this solution for Ashley's business.   (2 marks)

Show Answers Only

`x=4`

Show Worked Solution

`text(From the graph, intersection occurs at)\ x=4`

♦ Mean mark 36%.
MARKER’S COMMENT: The intersection on the graph is the same point at which the two simultaneous equations are solved for the given value of `x`.

`=>\ text(Break-even point occurs at)\ x=4`

`text(i.e. when 4 frames sold)`

`text(Income)` `=20xx4=$80\ \ \ text(is equal to)`
`text(C)text(osts)` `=40+(10xx4)=$80`

 

`text(If)\ <4\ text(frames sold)=>\ text(LOSS for business)`

`text(If)\ >4\ text(frames sold)=>\ text(PROFIT)`