smc-6920-10-Cost/Revenue

Algebra, STD2 A4 EQ-Bank 27

SunPower Solutions is a business that installs solar panels. Fixed costs are $1200. Each panel costs $150.00 to install and generates revenue of $350.00.

The spreadsheet below models the business's costs and revenue for different numbers of panels installed.

Calculate the spreadsheet values for the installation of 4 panels (cells B9, C9, D9) and 6 panels (cells B10, C10, D10). (2 marks)
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Using the spreadsheet, identify the break-even point and explain what it means for the business. (2 marks)
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SunPower Solutions has received a large order that will see them make a profit of $4200. Calculate the number of solar panels $(x)$ they will be installing. (2 marks)
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a. $\text{4 panels: TC (B9) = \$1800.00}$

$\text{Revenue (C9) = \$1400.00, Profit/Loss (D9) = }-\$400.00$

$\text{6 panels: TC (B10) = \$2100.00}$

$\text{Revenue (C10) = \$2100.00, Profit/Loss (D10) = \$0.00}$

b. $\text{Break-even = 6 panels, See worked solution}$

c. $27 \text{ panels}$

Show Worked Solution

a. $\text{4 panels:}$

$\text{Total Cost (B9)} = \$1200+4\times\$150 = \$1800.00$

$\text{Revenue (C9)} = 4\times\$350 = \$1400.00$

$\text{Profit/Loss (D9)} = \$1400.00-\$1800.00 = -\$400.00$

$\text{6 panels:}$

$\text{Total Cost (B10)} = \$1200+6\times\$150 = \$2100.00$

$\text{Revenue (C10)} = 6\times\$350 = \$2100.00$

$\text{Profit/Loss (D10)} = \$2100.00-\$2100.00 = \$0.00$

b. $\text{When 6 panels are installed:}$

$\text{Revenue = Total costs = \$2100.00 (breakeven)}$

$\text{This is the point at which the business covers all of its costs and}$

$\text{begins to make a profit.}$

$\text{OR}$

$\text{If fewer than 6 panels are installed the business will make a loss.}$

c. $\text{Let } x = \text{the number of solar panels to be installed.}$

$\text{Profit}$	$ = \text{Revenue}-\text{Total costs}$
$4200$	$ = 350x-(1200+150x)$
$4200$	$= 200x-1200$
$5400$	$=200x$
$x$	$=27$

$\text{SunPower Solutions will install 27 solar panels.}$

Algebra, STD2 A4 EQ-Bank 16

Harmony Arts Festival is a cultural event with fixed costs of $510. Each ticket costs $8.00 to provide and sells for $25.00.

The spreadsheet below models the festival's costs and revenue for different numbers of tickets sold.

Calculate the values for cells C12 and D12 in the spreadsheet. (2 marks)
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Using the spreadsheet, identify the break-even point and explain what it means for the festival. (2 marks)
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a. $\text{C12} = \$1250.00 \quad \text{D12} = \$340.00$

b. $\text{See worked solution}$

Show Worked Solution

a. $\text{C12: Revenue} = 50 \times \$25.00 = \$1250.00$

$\text{D12: Profit/Loss} = \$1250.00-\$910.00 = \$340.00$

b. $\text{When 30 tickets sold:}$

$\text{Revenue = Total costs = \$750.00 (Breakeven)}$

$\text{This is the point at which the festival covers all of its costs}$

$\text{and begins to make a profit.}$

$\text{OR}$

$\text{If less than 30 tickets are sold the festival will make a loss.}$

Algebra, STD2 A4 EQ-Bank 3 MC

Beachside Brew is a cafe with fixed weekly costs of $200. Each cup of coffee costs $1.50 to make and sells for $4.00.

The spreadsheet below models the cafe's weekly costs and revenue for different numbers of cups sold.

How many cups of coffee must be sold each week to break even?

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$B$

Show Worked Solution

$\text{Break-even occurs when Revenue = Total costs}\ \rightarrow\ \text{(i.e. Profit = \$0)}$

$\text{From the spreadsheet, Revenue = Total costs (\$320.00)}$

$\text{when 80 cups are sold.}$

$\Rightarrow B$

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?

$2.50
$5.50
$6.00
$8.00

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$B$

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.

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

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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)

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`20`
`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.

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

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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)

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A minibus is hired for 5 hours from Company B.

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

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a. `$120`

b. `c = 95x + 75`

c. `$50`

Show Worked Solution

a. `text(Hourly rate)\ (A)= 360 ÷ 3= $120`

b. `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`

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

`:.\ text(The hiring cost for Company)\ B\ text(is $50 cheaper.)`

Algebra, STD2 A4 EQ-Bank 28

Penny is a baker and makes meat pies every day.

The cost of making `p` pies, `$C`, can be calculated using the equation

`C = 675 + 3.5 p`

Penny sells the pies for $5.75 each, and her income is calculated using the equation

`I = 5.75 p`

On the graph, draw the graphs of `C` and `I`. (2 marks)

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On the graph, label the breakeven point and the loss zone. (2 marks)

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a. & b.

Show Worked Solution

b. `text(Loss zone occurs when)\ C > I,\ text(which is shaded in the diagram above.)`

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.

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)

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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)

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a. `C = $2100 + $0.50n`

b. `text(700 calls)`

Show Worked Solution

a. `C = $2100 + $0.50n`

♦ Mean marks of 48% and 32% for parts (a) and (b) respectively.

`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.

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

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

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

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

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

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

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a. `700 + 12x`

b. `text(Approximately 90)`

c. `$500`

d. `$23`

Show Worked Solution

a. `$C= 400 + 300 + (12 xx x)= 700 + 12x`

b. `text(Using the graph intersection:)`

`text(Approximately 90 people are needed to cover the costs.)`

c. `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`

d. `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 EQ-Bank 20

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.

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

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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)

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

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a. `$C = 1500 + 25x`

b. `60`

c. `$500`

Show Worked Solution

a. `text{Fixed Costs} = 650 + 850= $1500`

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

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

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

c. `text(When)\ \ x = 80:`

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

`$C= 1500 + 25 xx 80= $3500`

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

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`

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

`$15\ 000`
`$20\ 000`
`$30\ 000`
`$40\ 000`

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`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)

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`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)`

\(\text{Profit}\)	\( = \text{Revenue}-\text{Total costs}\)
\(4200\)	\( = 350x-(1200+150x)\)
\(4200\)	\(= 200x-1200\)
\(5400\)	\(=200x\)
\(x\)	\(=27\)