Applications: Currency, Fuel and Other Problems (Std 2-X)

v1 Algebra, STD2 A2 2012 HSC 13 MC

Conversion graphs can be used to convert from one currency to another.

Abbie converted 70 New Zealand dollars into Euros. She then converted all of these Euros into Australian dollars.

How much money, in Australian dollars, should Abbie have?

Show Answers Only

$C$

Show Worked Solution

$\text{Using the graphs:}$

$$70\ \text{New Zealand}$	$=40\ \text{Euro}$
$40\ \text{Euro}$	$=$55\ \text{Australian}$

$\Rightarrow C$

v1 Algebra, STD2 A2 2009 HSC 13 MC

The volume of water in a tank changes over six months, as shown in the graph.

Consider the overall decrease in the volume of water.

What is the average percentage decrease in the volume of water per month over this time, to the nearest percent?

Show Answers Only

$B$

Show Worked Solution

$\text{Initial Volume}$	$=50\ 000\ \text{L}$
$\text{Final volume}$	$=15\ 000\ \text{L}$
$\text{Decrease}$	$=50\ 000-15\ 000$
	$=35\ 000\ \text{L (over 6 months)}$

$\text{Loss per month}$	$=\dfrac{35\ 000}{6}$
	$=5833.33\dots\ \text{L per month}$
$\text{% loss per month}$	$=\dfrac{5833.33\dots}{50\ 000}\times 100\%$
	$=11.666\dots \%$

$\Rightarrow B$

♦ Mean mark 48%.
COMMENT: Remember that % decrease requires the decrease in volume to be divided by the original volume (50,000L)

v1 Algebra, STD2 A2 2009 HSC 24d

A factory makes both cloth and leather lounges. In any week

• the total number of cloth lounges and leather lounges that are made is 400
• the maximum number of leather lounges made is 270
• the maximum number of cloth lounges made is 325.

The factory manager has drawn a graph to show the numbers of leather lounges ($x$) and cloth lounges ($y$) that can be made.

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

--- 2 WORK AREA LINES (style=lined) ---
Explain why this line is only relevant between $B$ and $C$ for this factory. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
The profit per week, $$P$, can be found by using the equation $P = 2520x + 1570y$.

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

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

$x+y=400$
$\text{Since the max amount of leather lounges}=270$

$\rightarrow\ x\ \text{cannot be}\ >270$

$\text{Since the max amount of cloth lounges}=325$

$\rightarrow\ y\ \text{cannot be}\ >325$

$\therefore\ \text{The line}\ AD\ \text{is only possible between}\ B\ \text{and}\ C.$
$\text{The profits at}\ C\ \text{are }$185\ 250\ \text{more than at}\ B.$

Show Worked Solution

i. $\text{We are told the number of leather lounges}\ (x),$

$\text{and cloth lounges}\ (y),\ \text{made in any week} = 400$

$\rightarrow\ \text{Equation of}\ AD\ \text{is}\ x+y=400$

♦♦♦ Mean mark part (i) 14%.
Using $y=mx+c$ is a less efficient but equally valid method, using $m=–1$ and $b=400$ ($y$-intercept).

ii. $\text{Since the max amount of leather lounges}=270$

$\rightarrow\ x\ \text{cannot}\ >270$

$\text{Since the max amount of cloth lounges}=325$

$\rightarrow\ y\ \text{cannot}\ >325$

$\therefore\ \text{The line}\ AD\ \text{is only possible between}\ B\ \text{and}\ C.$

♦ Mean mark part (ii) 49%.

iii. $\text{At}\ B,\ x=75,\ y=325$

$\rightarrow\ $P (\text{at}\ B)$	$=2520\times 75+1570\times 325$
	$=189\ 000+510\ 250$
	$=$699\ 250$

$\text{At}\ C,\ x=270,\ y=130$

$\rightarrow\ $P (\text{at}\ C)$	$=2520\times 270+1570\times 130$
	$=680\ 400+204\ 100$
	$=$884\ 500$

$\text{Difference in profits}=$884\ 500-$699\ 250=$185\ 250$

$\text{The profits at}\ C\ \text{are } $185\ 250\ \text{more than at}\ B.$

♦ Mean mark (iii)40%.

v1 Algebra, STD2 A2 2010 HSC 27c

The graph shows tax payable against taxable income, in thousands of dollars.

Use the graph to find the tax payable on a taxable income of $$18\ 000$. (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
Use suitable points from the graph to show that the gradient of the section of the graph marked $A$ is $\dfrac{7}{15}$. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
How much of each dollar earned between $$18\ 000$ and $$33\ 000$ is payable in tax? Give your answer correct to the nearest whole number. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Write an equation that could be used to calculate the tax payable, $T$, in terms of the taxable income, $I$, for taxable incomes between $$18\ 000$ and $$33\ 000$. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$$3000\ \ \text{(from graph)}$
$\text{See worked solution}$
$46\frac{2}{3}\approx 47\ \text{cents per dollar earned}$
$\text{Tax payable →}\ T=\dfrac{7}{15}I-5400$

Show Worked Solution

$\text{Income on}\ $18\ 000=$3000\ \ \text{(from graph)}$

ii. $\text{Using the points}\ (18, 3)\ \text{and}\ (33, 10)$

$\text{Gradient at}\ A$	$=\dfrac{y_2-y_1}{x_2-x_1}$
	$=\dfrac{10\ 000-3000}{33\ 000-18\ 000}$
	$=\dfrac{7000}{15\ 000}$
	$=\dfrac{7}{15}\ \ \ \ \text{… as required}$

♦♦ Mean mark (ii) 25%.

iii. $\text{The gradient represents the tax applicable on each dollar}$

$\text{Tax}$	$=\dfrac{7}{15}\ \text{of each dollar earned}$
	$=46\frac{2}{3}\approx 47\ \text{cents per dollar earned (nearest whole number)}$

♦♦♦ Mean mark (iii) 12%!
MARKER’S COMMENT: Interpreting gradients is an examiner favourite, so make sure you are confident in this area.

iv. $\text{Tax payable up to }$18\ 000 = $3000$

$\text{Tax payable on income between }$18\ 000\ \text{and }$33\ 000$

$=\dfrac{7}{15}(I-18\ 000)$

$\therefore\ \text{Tax payable →}\ \ T$	$=3000+\dfrac{7}{15}(I-18\ 000)$
	$=3000+\dfrac{7}{15} I-8400$
	$=\dfrac{7}{15}I-5400$

♦♦♦ Mean mark (iv) 15%.
STRATEGY: The earlier parts of this question direct students to the most efficient way to solve this question. Make sure earlier parts of a question are front and centre of your mind when devising strategy.

v1 Algebra, STD2 A2 2014 HSC 27b

Clara is comparing the costs of two different ways of travelling to work.

Clara’s motor scooter uses one litre of fuel for every 22 km travelled. The cost of fuel is $2.24/L and the distance from her home to the work car park is 33 km. The cost of travelling by bus and light rail is $35.80 for 10 single trips.

Which way of travelling is cheaper and by how much? Support your answer with calculations. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$\text{Motor scooter is }$0.22 \text{ cheaper per one-way trip.}$

Show Worked Solution

$\text{Compare cost of a one-way trip}$

$\text{Motor scooter}$

$\text{Fuel used}=\dfrac{33}{22}=1.5\ \text{L}$

$\text{Cost}=1.5\times 2.24=$3.36$

$\text{Bus and light rail}$

$\text{Cost}=\dfrac{35.80}{10}=$3.58$

$\text{Difference}=$3.58-3.36=$0.22$

$\therefore\ \text{Motor scooter is }$0.22 \text{ cheaper per one-way trip.}$

v1 Algebra, STD2 A2 2007 HSC 24c

Blythe travels to France via the USA. She uses this graph to calculate her currency conversions.

After leaving the USA she has US$750 to add to the A$2150 that she plans to spend in France.

She converts all of her money to euros. How many euros does she have to spend in France? (3 marks)

--- 6 WORK AREA LINES (style=lined) ---
If the value of the US dollar rises in comparison to the Australian dollar, what will be the effect on the gradient of the line used to convert US dollars to Australian dollars? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

$1890\ \text{€}$
$\text{See worked solutions}$

Show Worked Solution

i. $\text{From graph:}$

$75\ \text{US}$$	$=100\ \text{A}$$
$\rightarrow\ 750\ \text{US}$$	$=1000\ \text{A}$$

$\therefore\ \text{Blythe has a total of }$1000+$2150=\text{A}$3150 $

$\text{Converting A}$\ \text{to €}:$

$100\ \text{A}$$	$=60\ \text{€}$
$\therefore\ 3150\ \text{A}$$	$=\dfrac{3150}{100}\times 60$
	$=1890\ \text{€}$

ii. $\text{If the value of the US}\ $\ \text{rises against the}$

$\text{Australian }$\ \text{then 1 A}\ $\ \text{will buy less US}\ $$

$\text{than before and the gradient used to convert}$

$\text{the currencies will steepen (increase).}$

v1 Algebra, STD2 A2 2014 HSC 22 MC

Lisa’s motorbike uses fuel at the rate of 1.8 L per 100 km for long-distance driving and 2.3 L per 100 km for short-distance driving.

She used the motorbike to make a journey of 840 km, which included 108 km of short-distance driving.

Approximately how much fuel did Lisa’s motorbike use on the journey?

9 L
16 L
18 L
34 L

Show Answers Only

$B$

Show Worked Solution

$\text{Fuel used in short distance}$

$=\dfrac{108}{100}\times 2.3\ \text{L}=2.484\ \text{L}$

$\text{Fuel used in long distance}$

$=\dfrac{840-108}{100}\times 1.8\ \text{L}=13.176\ \text{L}$

$\therefore\ \text{Total Fuel}$	$=2.484+13.176$
	$=15.66\ \text{L}$

$\Rightarrow B$

v1 Algebra, STD2 A2 2014 HSC 26f

The weight of an object on the moon varies directly with its weight on Earth. An astronaut who weighs 63 kg on Earth weighs only 9 kg on the moon.

A lunar landing craft weighs 2449 kg when on the moon. Calculate the weight of this landing craft when on Earth. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$17\ 143\ \text{kg}$

Show Worked Solution

$W_{\text{moon}}\propto W_{\text{earth}}$

$\rightarrow\ W_{\text{m}}=k\times W_{\text{e}}$

$\text{Find}\ k,\ \text{given}\ W_{\text{e}}=63\ \text{when}\ W_{\text{m}}=9$

$9$	$=k\times 63$
$k$	$=\dfrac{9}{63}=\dfrac{1}{7}$

$\text{If}\ W_{\text{m}}=2449\ \text{kg, find}\ W_{\text{e}}:$

$2449$	$=\dfrac{1}{7}\times W_{\text{e}}$
$W_{\text{e}}$	$=7\times 2449=17\ 143$

$\text{Landing craft weighs}\ 17\ 143\ \text{kg on earth}$

v1 Algebra, STD2 A2 2007 HSC 27b

A cafe uses eight long-life light globes for 7 hours every day of the year. The purchase price of each light globe is $11.00 and they each cost $$f$ per hour to run.

Write an equation for the total cost ($$c$) of purchasing and running these eight light globes for one year in terms of $f$. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
Find the value of $f$ (correct to three decimal places) if the total cost of running these eight light globes for one year is $850. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
If the use of the light globes increases to ten and a half hours per night every night of the year, does the total cost increase by one-and-a-half times? Justify your answer with appropriate calculations. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

$$c=88+20\ 440f$
$0.037\ $/\text{hr}\ \text{(3 d.p.)}$
$\text{Proof: (See Worked Solutions)}$

Show Worked Solution

i. $\text{Purchase price}=8\times 11=$88$

$\text{Running cost}$	$=\text{No. of hours}\times \text{Cost per hour}$
	$=8\times 7\times 365\times f$
	$=20\ 440f$

$\therefore\ $c=88+20\ 440f$

ii. $\text{Given}\ \ $c=$850$

$850$	$=88+20\ 440f$
$20\ 440f$	$=850-88$
$f$	$=\dfrac{762}{20440}$
	$= 0.03727\dots$
	$=0.037\ $/\text{hr}\ \text{(3 d.p.)}$

iii. $\text{If}\ f\ \text{is multiplied by }1.5 =\dfrac{10.5}{7}$

$f=1.5\times0.037=0.0555\ \ $/\text{hr}$

$\therefore\ $c$	$=88+20\ 440\times 0.0555$
	$=$1222.42$

$\text{Since }$1222.42\ \text{is less than}\ 1.5\times $850 = $1275,$

$\text{the total cost increases to less than 1.5 times the}$

$\text{the original cost.}$

v1 Algebra, STD2 A2 2004 HSC 22 MC

Mary-Anne knows that

• one Australian dollar (AUD) is worth 0.64 euros, and
• one Canadian dollar (CAD) is worth 0.97 euros.

Mary-Anne changes 75 AUD to Canadian dollars.

How many Canadian dollars will she get?

46.56 CAD
49.48 CAD
113.67 CAD
120.75 CAD

Show Answers Only

$B$

Show Worked Solution

$\text{Mary-Anne has 75 AUD.}$

$\text{Converting to Euros}$

$25\ \text{AUD}$	$=75\times 0.64$
	$=48\ \text{Euros}$

$\text{Converting to CAD}$

$48\ \text{euros}$	$=\dfrac{48}{0.97}$
	$=49.484\dots$
	$=49.48\ \text{CAD}$

$\Rightarrow B$

v1 Algebra, STD2 A2 2016 HSC 26c

Bonn’s car uses fuel at the rate of 6.1 L /100 km for country driving and 8.3 L /100 km for city driving. On a trip, he drives 350 km in the country and 40 km in the city.

Calculate the amount of fuel he used on this trip. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$24.67\ \text{L}$

Show Worked Solution

$\text{Fuel used in country}$

$=350\times \dfrac{6.1}{100}$

$=21.35\ \text{L}$

$\text{Fuel used in city}$

$=40\times \dfrac{8.3}{100}$

$=3.32\ \text{L}$

$\therefore\ \text{Total fuel used}$

$=21.35+3.32$

$=24.67\ \text{L}$

v1 Algebra, STD2 A2 2016 HSC 29e

The graph shows the life expectancy of people born between 1900 and 2010.

According to the graph, what is the life expectancy of a person born in 1968? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
With reference to the value of the gradient, explain the meaning of the gradient in this context. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$\text{76 years}$
$\text{After 1900, life expectancy increases by 0.38 years for}$
$\text{each year later that someone is born.}$

Show Worked Solution

i. $\text{76 years}$

ii. $\text{Using (2000, 88) and (1900, 50):}$

$\text{Gradient}$	$= \dfrac{y_2-y_1}{x_2-x_1}$
	$= \dfrac{88-50}{2000-1900}$
	$= 0.38$

$\text{After 1900, life expectancy increases by 0.38 years for}$

$\text{each year later that someone is born.}$

♦♦ Mean mark (ii) 33%.

v1 Algebra, STD2 A2 SM-Bank 2

The cost of apples per kilogram, $C$, varies directly with the weight of apples purchased, $w$.

If 12 kilograms costs $56.64, calculate the cost of 4.5 kilograms of apples. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$$21.24$

Show Worked Solution

$C\ \propto \ w$

$C=kw$

$\text{When}\ C=$56.64\ \text{kg},\ w=12\ \text{kg}$

$56.64$	$=k\times 12$
$k$	$=\dfrac{56.64}{12}$
	$=$4.72$

$\text{When}\ \ w=4.5\ \text{kg,}$

$C$	$=4.72\times 4.5$
	$=$21.24$

v1 Algebra, STD2 A2 2018 HSC 5 MC

The driving distance from Burt's home to his work is 15 km. He drives to and from work five times each week. His car uses fuel at the rate of 12 L/100 km.

How much fuel does he use driving to and from work each week?

15 L
18 L
27 L
36 L

Show Answers Only

$B$

Show Worked Solution

$\text{Total distance travelled each week}$

$=5\times 2\times 15$

$=150\ \text{km}$

$\therefore\ \text{Total fuel used}$

$=\dfrac{150}{100}\times 12\ \text{L}$

$=18\ \text{L}$

$\Rightarrow B$

v1 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 80?

20 years
21 years
61 years
62 years

Show Answers Only

$D$

Show Worked Solution

$\text{When infant mortality rate is 80, life expectancy}$

$\text{at birth is 62 years (see below).}$

$\Rightarrow D$

v1 Algebra, STD2 A2 2017 HSC 14 MC

Christopher is comparing two different models of 4WD cars. Car A uses fuel at the rate of 11.4 L/100 km. Car B uses 9.6 L/100 km.

Suppose Christopher plans on driving $11\ 000$ km in the next year.

How much less fuel will he use driving car B instead of car A?

198 L
440 L
720 L
1000 L

Show Answers Only

$A$

Show Worked Solution

$\text{Difference in fuel usage}$

$=(11.4-9.6)\ \text{L/100 km}$

$=(1.8\ \text{L/100 km}$

$\therefore\ \text{Fuel saved using car}\ B$

$=\dfrac{11\ 000}{100}\times 1.8$

$=198\ \text{L}$

$\Rightarrow A$

v1 Algebra, STD2 A2 SM-Bank 3

The average height, $L$, in centimetres, of a boy between the ages of 7 years and 10 years can be represented by a line with equation

$L=7A+85$

where $A$ is the age in years. For this line, the gradient is 7.

What does this indicate about the heights of boys aged 7 to 10? (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Give ONE reason why this equation is not suitable for predicting heights of boys older than 10. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

$\text{It indicates that 7-10 year old boys, on average, grow 7 cm per year.}$
$\text{Boys eventually stop growing, and the equation doesn’t factor this in.}$

Show Worked Solution

i. $\text{It indicates that 7-10 year old boys, on average, grow}$

$\text{7 cm per year.}$

ii. $\text{Boys eventually stop growing, and the equation doesn’t}$

$\text{factor this in.}$

v1 Algebra, STD2 A4 EQ-Bank 8 MC

Water was poured into a container at a constant rate. The graph shows the depth of water in the container as it was being filled.

Which of the following containers could have been used to produce this result?

A.		B.
C.		D.

Show Answers Only

$C$

Show Worked Solution

$\text{Since the graph is a straight line, the container must have}$
$\text{vertical sides so the container will fill up at a constant rate.}$

$\Rightarrow C$

v1 Algebra, STD2 A2 SM-Bank 4 MC

A car travels 480 km on 37 L of petrol.

What is its fuel consumption, correct to 1 decimal place?

5.2 L/100 km
7.7 L/100 km
13.0 L/100 km
15.4 L/100 km

Show Answers Only

$B$

Show Worked Solution

$37\text{ litres are used to travel}\ 4.8\times 100\ \text{km}$

$\text{Fuel consumption (L/100 km)}$

$=\dfrac{37}{4.8}$

$=7.7083\dots$

$=7.7\ \text{L/100 km}$

$\Rightarrow B$

v1 Algebra, STD2 A2 2019 HSC 14 MC

Last Friday, Jake had 98 marbles and Jack had 79 Marbles. On average, Jake wins 5 marbles per day and Jack loses 4 marbles per day.

If $x$ represents the number of days since last Friday and $y$ represents the number of marbles, which pair of equations model this situation?

A.	$\text{Jake:}\ \ y=98x+5$ $\text{Jack:}\ \ y=79x-4$
B.	$\text{Jake:}\ \ y=5+98x$ $\text{Jack:}\ \ y=4-79x$
C.	$\text{Jake:}\ \ y=5x+98$ $\text{Jack:}\ \ y=4x-79$
D.	$\text{Jake:}\ \ y=98+5x$ $\text{Jack:}\ \ y=79-4x$

Show Answers Only

$D$

Show Worked Solution

$\text{Jake starts with 98 and adds 5 per day:}$

$y=98+5x$

$\text{Jack starts with 79 and loses 4 per day:}$

$y=79-4x$

$\Rightarrow D$

v1 Algebra, STD2 A2 2019 HSC 34

The relationship between British pounds $(p)$ and Australian dollars $(d)$ on a particular day is shown in the graph.

Write the direct variation equation relating British pounds to Australian dollars in the form $p=md$. Leave $m$ as a fraction. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
The relationship between Japanese yen $(y)$ and Australian dollars $(d)$ on the same day is given by the equation $y=84d$.

Convert $107\ 520$ Japanese yen to British pounds. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$p=\dfrac{5}{8}d$
$107\ 520\ \text{yen = 800 pounds}$

Show Worked Solution

a. $m=\dfrac{\text{rise}}{\text{run}}=\dfrac{5}{8}$

$p=\dfrac{5}{8}d$

♦ Mean mark 42%.

b. $\text{Yen to Australian dollars:}$

$y$	$=84d$
$107\ 520$	$=84d$
$d$	$=\dfrac{107\ 520}{84}$
	$= 1280\ $\text{A}$

$\text{Australian dollars to pounds:}$

$p$	$=\dfrac{5}{8}\times 1280$
	$=800\ \text{pounds}$

$\therefore\ 107\ 520\ \text{yen = 800 pounds}$

v1 Algebra, STD1 A3 2021 HSC 25

The diagram shows a container which consists of a large hexagonal prism on top of a smaller hexagonal prism.

The container is filled with water at a constant rate into the top of the larger hexagonal prism.

The smaller prism is totally filled before the larger prism begins to fill.

It takes 5 minutes to fill the smaller cylinder.

Draw a possible graph of the water level in the container against time. (2 marks)

--- 0 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

♦♦ Mean mark 38%.

v1 Algebra, STD2 A2 2020 HSC 10 MC

An electrician charges a call-out fee of $75 as well as $1.50 per minute while working.

Suppose the electrician works for $t$ hours.

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

$C=75+1.50t$
$C=150+75t$
$C=75+90t$
$C=90+75t$

Show Answers Only

$C$

Show Worked Solution

$\text{Hourly rate}=60\times 1.50=$90$

$\therefore\ C=75+90t$

$\Rightarrow C$

♦ Mean mark 42%.

v1 Algebra, STD1 A2 2020 HSC 20

The height of a bundle of photographic paper ($H$ mm) varies directly with the number of sheets ($N$) of photographic paper that the bundle contains.

This relationship is modelled by the formula $H=kN$, where $k$ is a constant.

The height of a bundle containing 150 sheets of photographic paper is 2.7 centimetres.

Show that the value of $k$ is 0.18. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
A bundle of photographic paper has a height of 36 centimetres. Calculate the number of sheets of photographic paper in the bundle. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$\text{See Worked Solutions}$
$2000\ \text{sheets}$

Show Worked Solution

a. $H=2.7\ \text{cm }=27\ \text{mm, when}\ N=150:$

$H$	$=kN$
$2.7$	$=k\times 150$
$\therefore\ k$	$=\dfrac{2.7}{150}$
	$=0.18$

b. $\text{Find}\ \ N \ \text{when} \ \ H=36\ \text{cm}=360\ \text{mm:}$

$360$	$=0.18\times N$
$\therefore\ N$	$=\dfrac{360}{0.18}$
	$=2000\ \text{sheets}$

♦ Mean mark 50%.

v1 Algebra, STD2 A2 2021 HSC 18

The fuel consumption for a medium SUV vehicle is 7.2 litres/100 km. On a road trip, the SUV travels a distance of 1325 km and the fuel cost is $2.15 per litre.

What is the total fuel cost for the trip? (2 marks)

Show Answers Only

$$205.11$

Show Worked Solution

$\text{Total fuel used}$	$=7.2\times \dfrac{1325}{100}$
	$=95.4\ \text{litres}$

$\text{Total fuel cost}$	$=95.4\times 2.15$
	$=$205.11$

v1 Algebra, STD2 A2 2022 HSC 16

Rhonda is 38 years old, and likes to keep fit by doing cross-fit classes.

Use this formula to find her maximum heart rate (bpm). (1 mark)
Maximum heart rate = 220 – age in years

--- 2 WORK AREA LINES (style=lined) ---
Rhonda will get the most benefit from this exercise if her heart rate is between 65% and 85% of her maximum heart rate.
Between what two heart rates should Rhonda be aiming for to get the most benefit from her exercise? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

a. $182\ \text{bpm}$

b. $118-155\ \text{bpm}$

Show Worked Solution

a.	$\text{Max heart rate}$	$=220-38$
		$=182\ \text{bpm}$

b. $\text{65% max heart rate}\ = 0.65\times 182 = 118.3\ \text{bpm}$

$\text{85% max heart rate}\ = 0.85\times 182 = 154.7\ \text{bpm}$

$\therefore\ \text{Rhonda should aim for between 118 and 155 bpm during exercise.}$

v1 Algebra, STD2 A4 2022 HSC 22

The formula $C=80n+b$ is used to calculate the cost of producing desktop computers, where $C$ is the cost in dollars, $n$ is the number of desktop computers produced and $b$ is the fixed cost in dollars.

Find the cost $C$ when 2458 desktop computers are produced and the fixed cost is $$18\ 230$. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Some desktop computers have extra features added. The formula to calculate the production cost for these desktop computers is
$C=80n+an+18\ 230$
where $a$ is the additional cost in dollars per desktop computer produced.
Find the number of desktop computers produced if the additional cost is $35 per desktop computer and the total production cost is $$103\ 330$. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

$$214\ 870$
$740\ \text{desktop computers}$

Show Worked Solution

a. $\text{Find}\ C,\ \text{given}\ n=2458\ \text{and}\ b=18\ 230$

$C$	$=80\times 2458+18\ 230$
	$=$214\ 870$

b. $\text{Find}\ n,\ \text{given}\ C=103\ 330\ \text{and}\ a=35$

$C$	$=80n+an+18\ 230$
$103\ 330$	$=80n+35n+18\ 230$
$115n$	$=85\ 100$
$n$	$=\dfrac{85\ 100}{115}$
	$=740\ \text{desktop computers}$

\($70\ \text{New Zealand}\)	\(=40\ \text{Euro}\)
\(40\ \text{Euro}\)	\(=$55\ \text{Australian}\)

\(\text{Initial Volume}\)	\(=50\ 000\ \text{L}\)
\(\text{Final volume}\)	\(=15\ 000\ \text{L}\)
\(\text{Decrease}\)	\(=50\ 000-15\ 000\)
	\(=35\ 000\ \text{L (over 6 months)}\)

\(\text{Loss per month}\)	\(=\dfrac{35\ 000}{6}\)
	\(=5833.33\dots\ \text{L per month}\)
\(\text{% loss per month}\)	\(=\dfrac{5833.33\dots}{50\ 000}\times 100\%\)
	\(=11.666\dots \%\)

\(\rightarrow\ $P (\text{at}\ B)\)	\(=2520\times 75+1570\times 325\)
	\(=189\ 000+510\ 250\)
	\(=$699\ 250\)

\(\rightarrow\ $P (\text{at}\ C)\)	\(=2520\times 270+1570\times 130\)
	\(=680\ 400+204\ 100\)
	\(=$884\ 500\)

\(\text{Gradient at}\ A\)	\(=\dfrac{y_2-y_1}{x_2-x_1}\)
	\(=\dfrac{10\ 000-3000}{33\ 000-18\ 000}\)
	\(=\dfrac{7000}{15\ 000}\)
	\(=\dfrac{7}{15}\ \ \ \ \text{… as required}\)

\(\therefore\ \text{Tax payable →}\ \ T\)	\(=3000+\dfrac{7}{15}(I-18\ 000)\)
	\(=3000+\dfrac{7}{15} I-8400\)
	\(=\dfrac{7}{15}I-5400\)

\(75\ \text{US}$\)	\(=100\ \text{A}$\)
\(\rightarrow\ 750\ \text{US}$\)	\(=1000\ \text{A}$\)

\(100\ \text{A}$\)	\(=60\ \text{€}\)
\(\therefore\ 3150\ \text{A}$\)	\(=\dfrac{3150}{100}\times 60\)
	\(=1890\ \text{€}\)