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Algebra, STD2 A1 2024 HSC 24

Sarah, a 60 kg female, consumes 3 glasses of wine at a family dinner over 2.5 hours.

Note: there are 1.2 standard drinks in one glass of wine.

The blood alcohol content \((BAC)\) for females can be estimated by

\(B A C_{\text {female}}=\dfrac{10 N-7.5 H}{5.5 M},\)

  where \(N\)  = number of standard drinks
  \(H\)  = number of hours drinking
  \(M\)  = mass in kilograms
 
  1. Calculate Sarah's \(BAC\) at the end of the dinner, correct to 3 decimal places.   (2 marks)

  2. The time it takes a person's BAC to reach zero is given by   

                  \(\text {Time}=\dfrac{B A C}{0.015}.\)

    Calculate the time it takes for Sarah's BAC to return to zero, assuming she stopped drinking after 2.5 hours. Give your answer to the nearest minute.  (2 marks)

Show Answers Only

a.   \(BAC=0.052\)

b.   \(\text{3 h 28 m}\)

 

Show Worked Solution

a.     \(N=3 \times 1.2=3.6, H=2.5, M=60\)

  \(\therefore B A C\) \(=\dfrac{10 \times 3.6-7.5 \times 2.5}{5.5 \times 60}\)
    \(=0.052 \ \text{(3 d.p.)}\)

 

b.     \(T\) \(=\dfrac{0.052}{0.015}\)
    \(=3.466\)
    \(=3 \text{ h 28 m}\)
COMMENT: Use calculator degrees/minutes function for part (b).

Algebra, STD2 A1 2024 HSC 11 MC

A train left Richmond at 6:42 am and arrived at Central Station at 8:04 am. The distance travelled by the train from Richmond to Central was 61 km.

What was the average speed of this train, correct to the nearest km/h ?

  1. 38
  2. 45
  3. 50
  4. 74
Show Answers Only

\(B\)

Show Worked Solution

\(\text {Time of travel:  8:04 less 6:42 = 82 minutes.}\)

  \(\text{Speed (avg)}\) \(=\dfrac{\text{distance}}{\text{time}}\)
    \(=\dfrac{61}{82}\ \ \text{km/min}\)
    \(=\dfrac{61}{82} \times 60\ \ \text{km/hr}\)
    \(=44.6\ \ \text{km/hr}\)

 
\(\Rightarrow B\)

Measurement, STD1 M4 2023 HSC 14

The distance-time graph shows the first two stages of a car journey from home to a holiday house.
  

  1. At what speed, in kilometres per hour, did the car travel during stage \(A\) of the journey?   (1 mark)

  2. For how long did the car stop during stage \(B\) of the journey?   (1 mark)

  3. After stage \(B\), the car continues to travel towards the holiday house at a constant speed of \(50\ \text{km/h}\) for 2 hours. Graph this part of the journey on the grid above.   (2 marks)

Show Answers Only

a.    \(100\text{ km/h}\)

b.    \(30\text{ minutes}\)

c.   

Show Worked Solution

a.   \(S=\dfrac{D}{T}=\dfrac{150}{1.5}=100\ \text{km/h}\)
 

b.    \(\text{Stage}\textit{ B}=1.5\rightarrow 2\text{ hours}=30\text{ minutes}\)
 

c.    \(\text{Position after 2 hours at 50km/h}=(150+2\times 50 , 2 + 2) = (250 , 4)\)


♦ Mean mark (c) 50%.

Algebra, STD2 A1 2023 HSC 36

The following formula can be used to calculate an estimate for blood alcohol content (`BAC`) for males.
 

`BAC_text{male}=(10N-7.5H)/(6.8M)`

`N` is the number of standard drinks consumed

`M` is the person's weight in kilograms

`H` is the number of hours of drinking
 

Cameron weighs 75 kg. His `BAC` was zero when he began drinking alcohol. At 9:00 pm, after consuming 3 standard drinks, his `BAC` was 0.02.

Using the formula, estimate at what time Cameron began drinking alcohol, to the nearest minute.  (4 marks)

Show Answers Only

`text{6:22 pm}`

Show Worked Solution
`BAC` `=(10N-7.5H)/(6.8M)`  
`0.02` `=(10xx3-7.5xxH)/(6.8 xx 75)`  
`0.02xx510` `=30-7.5H`  
`7.5H` `=30-10.2`  
`H` `=19.8/7.5`  
  `=2.64\ text{hours}`  
  `=2\ text{hours}\ 38\ text{minutes (nearest minute)}`  

 
`text{Time Cameron began drinking}`

`=\ text{9 pm less 2 h 38 m}`

`=\ text{6:22 pm}`

Mean mark 56%.

Algebra, STD1 A1 2022 HSC 19

Fried's formula is used to calculate the dosage of medication for children aged 1-2 years based on the adult dosage. The formula is

`text{Dosage}=(text{age (in months)} xx\ text{adult dosage})/(150)`.

The adult dosage of a particular medication is 200 mg.

Betty's dosage is calculated to be 24 mg.

How old is Betty in months?  (2 marks)

Show Answers Only

18 months

Show Worked Solution

`text{Substituting into the formula:}`

`24` `=( \ text{age} xx 200)/150`  
`text{age}` `=(24 xx 150)/200`   
  `=360/20`  
  `= 18\ text(months)`  

♦ Mean mark 52%.

Algebra, STD2 A1 2020 HSC 13 MC

When Jake stops drinking alcohol at 10:30 pm, he has a blood alcohol content (BAC) of 0.08375.

The number of hours required for a person to reach zero BAC after they stop consuming alcohol is given by the formula

`text(Time) = frac{ text(BAC)}{0.015}`.

At what time on the next day should Jake expect his BAC to be 0.05?

  1.  12:45 am
  2.  1:50 am
  3.  2:15 am
  4.  4:05 am
Show Answers Only

`A`

Show Worked Solution

♦♦♦ Mean mark 17%.
COMMENT: The rates aspect of this question proved extremely challenging.

`text(Time from  0.08375 → 0  BAC)`

`= frac(0.08375)(0.015)`

`approx 5.58 \ text(hours)`
 

`text(Time from  0.08375 → 0.05  BAC)`

`approx frac{(0.08375 – 0.05)}{0.08375} xx 5.58` 

`approx 0.4 xx 5.58`

`approx 2.25\ text(hours)`
 

`therefore \ text(Time)` `approx \ 10:30 \ text(pm) \ + 2 \ text(hr) \ 15 \ text(min)`
  `approx \ 12:45 \ text(am)`

 
`=> \ A`

Algebra, STD2 A1 2020 HSC 3 MC

The distance between Bricktown and Koala Creek is 75 km. A person travels from Bricktown to Koala Creek at an average speed of 50 km/h.

How long does it take the person to complete the journey?

  1.  40 minutes
  2.  1 hour 25 minutes
  3.  1 hour 30 minutes
  4.  1 hour 50 minutes
Show Answers Only

`C`

Show Worked Solution
`text(Time)` `= frac(text(Distance))(text(Speed))`
  `= frac(75)(50)`
  `=1.5 \ text(hours)`
  `= 1 \ text(hour) \ 30 \ text(minutes)`

 
`=> \ C`

Algebra, STD2 A1 SM-Bank 15

Fabio drove 300 km in  `4 1/2`  hours.

His average speed for the first 210 km was 70 km per hour.

How long did he take to travel the last 90 km?  (2 marks)

Show Answers Only

`1 1/2`

Show Worked Solution

`text(Time for 1st 210 km)`

`= 210/70`

`= 3\ text(hours)`

 

`:.\ text(Time for last 90 km)`

`= 4 1/2 – 3`

`= 1 1/2\ text(hours)`

Algebra, STD2 A1 SM-Bank 2 MC

Fleur lives 15 kilometres from her work.

On Wednesday, she drove to work and averaged 60 kilometres per hour.

On Thursday, she took the bus which averaged 15 kilometres per hour.

What was the extra time of the bus journey, in minutes, compared to when she drove on Wednesday?

  1.  15
  2.  45
  3.  60
  4.  75
Show Answers Only

`B`

Show Worked Solution
`text(Time on Wednesday)` `= 15/60`
  `= 0.25\ text(hour)`
  `= 15\ text(minutes)`

 

`text(Time on Thursday)` `= 15/15`
  `= 1\ text(hour)`
  `= 60\ text(minutes)`

 

`:.\ text(The extra time taking the bus)\ =60-15 = 45\ text(minutes)`

`=>B`

Algebra, STD2 A1 2019 HSC 28

The formula below is used to calculate an estimate for blood alcohol content `(BAC)` for females.

`BAC_text(female) = (10N - 7.5H)/(5.5M)`

The number of hours required for a person to reach zero `BAC` after they stop consuming alcohol is given by the following formula.

`text(Time) = (BAC)/0.015`

The number of standard drinks in a glass of wine and a glass of spirits is shown.
 

Hannah weighs 60 kg. She consumed 3 glasses of wine and 4 glasses of spirits between 6:15 pm and 12:30 am the following day. She then stopped drinking alcohol.

Using the given formulae, calculate the time in the morning when Hannah's `BAC` should reach zero.  (4 marks)

Show Answers Only

`text(6:23 am)`

Show Worked Solution

`text(Standard drinks consumed)\ (N) = 3 xx 1.2 + 4 = 7.6`

`text(Hours drinking)\ (H) = text(6 h 15 min = 6.25 hours)`

`BAC(text(Hannah))` `= (10 xx 7.6 – 7.5 xx 6.25)/(5.5 xx 60)`
  `= 0.08825…`

COMMENT: Convert a decimal answer into hours and minutes using the calculator degree/minute function.

`text(Time(to zero))` `= (0.08825…)/0.015`
  `= 5.883…\ text(hours)`
  `= 5\ text(hours 53 minutes)`

 
`:.\ text(Hannah should reach zero)\ BAC\ text(at 6:23 am)`

Algebra, STD2 A1 2018 HSC 28e

Sophie is driving at 70 km/h. She notices a branch on the road ahead and decides to apply the brakes. Her reaction time is 1.5 seconds. Her braking distance (`D` metres) is given by  `D = 0.01v^2`, where  `v ` is speed in km/h.

Stopping distance can be calculated using the following formula
 

`text(stopping distance = {reaction time distance} + {braking distance})`
 

What is Sophie’s stopping distance, to the nearest metre?  (3 marks)

Show Answers Only

`78\ text{m  (nearest m)}`

Show Worked Solution
`text(70 km/hr)` `= 70\ 000\ text(m/hr)`
  `= (70\ 000)/(60 xx 60)\ text(m/sec)`
  `= 19.44…\ text(m/sec)`

 

`:.\ text(Total stopping distance)`

♦ Mean mark 46%.

`=\ text(reaction time distance + braking distance)`

`= 1.5 xx 19.44… + 0.01 xx 70^2`

`= 78.166…`

`= 78\ text{m  (nearest m)}`

Algebra, STD2 A1 2018 HSC 26b

Clark’s formula, given below, is used to determine the dosage of medicine for children.
 

`text(Dosage) = (text(weight in kg × adult dosage))/70`

 
For a particular medicine, the adult dosage is 325 mg and the correct dosage for a specific child is 90 mg.

How much does the child weigh, to the nearest kg?  (2 marks)

Show Answers Only

`19\ text(kg)`

Show Worked Solution

`90 = (text(weight) xx 325)/70`

`:.\ text(weight)` `= (70 xx 90)/325`
  `= 19.38…`
  `= 19\ text(kg  (nearest kg))`

Algebra, STD2 A1 SM-Bank 5

Fried's formula is used to calculate the medicine dosages for children aged 1-2 years.
 

`text(Child dosage) = {text{Age(in months)}\ xx\ text(adult dosage)}/150`
 

Ben is 1.5 years old and receives a daily dosage of 450 mg of a medicine.

According to Fried's formula, what would the appropriate adult daily dosage of the medicine be?  (2 marks)

Show Answers Only

`3750\ text(mg)`

Show Worked Solution

`text(Substituting into the formula:)`

`450` `= (18 xx text{adult dosage})/150`
`:.\ text(Adult dosage)` `= (450 xx 150)/18`
  `= 3750\ text(mg)`

Algebra, STD2 A1 SM-Bank 4

Doris is driving in a school zone at a speed of 35 kilometres per hour and needs to stop immediately to avoid an accident.

It takes her 1.25 seconds to react and her breaking distance is 5.3 metres.

Stopping distance can be calculated using the following formula
 

`text(stopping distance = {reaction time distance} + {braking distance})`
 

What is Doris' total stopping distance? Give your answer to 1 decimal place.  (2 marks)

Show Answers Only

`17.5\ text{metres  (to 1 d.p.)}`

Show Worked Solution
`35\ text(km/hr)` `= 35\ 000\ text(m/hr)`
  `= (35\ 000)/(60 xx 60)\ text(m/sec)`
  `= 9.722…\ text(m/sec)`

 

`:.\ text(Total stopping distance)`

`=\ text(Reaction time distance) + text(braking distance)`

`= 1.25 xx 9.722… +5.3`

`= 17.452…`

`= 17.5\ text{metres  (to 1 d.p.)}`

Algebra, STD2 A1 2017 HSC 27e

Rhys is drinking low alcohol beer at a party over a five-hour period. He reads on the label of the low alcohol beer bottle that it is equivalent to 0.8 of a standard drink.

Rhys weighs 90 kg.

The formula below  can be used to calculate a Rhys's blood alcohol content.
 

`BAC_text(Male) = (10N - 7.5H)/(6.8M)`

where    `N` is the number of standard drinks consumed

`H` is the number of hours drinking

`M` is the person's mass in kilograms
 

What is the maximum number of complete bottles of the low alcohol beer he can drink to remain under a Blood Alcohol Content (BAC) of 0.05?  (4 marks)

Show Answers Only

`8`

Show Worked Solution
`text(BAC)_text(male)` `= (10N – 7.5H)/(6.8M)`
`0.05` `= (10N – 7.5 xx 5)/(6.8 xx 90)`
`10N` `= (0.05 xx 6.8 xx 90) + 7.5 xx 5`
  `= 68.1`
`N` `= 6.81\ \ text(standard drinks)`

 

`:.\ text(Number of low alcohol bottles)`

`= 6.81/0.8`

`= 8.51`
 

`:.\ text(Max complete bottles to stay under 0.05)`

`= 8`

Algebra, STD2 A1 2017 HSC 19 MC

Young’s formula, shown below, is used to calculate the dosage of medication for children aged 1−12 years based on the adult dosage.

`D = (yA)/(y + 12)`

where     `D`   = dosage for children aged 1−12 years
`y`   = age of child (in years)
`A`   = Adult dosage

 
A child’s dosage is calculated to be 20 mg, based on an adult dosage of 40 mg.

How old is the child in years?

A.     `6`

B.     `8`

C.     `10`

D.     `12`

Show Answers Only

`text(D)`

Show Worked Solution
`D` `= (yA)/(y + 12)`
`20` `= (40y)/(y + 12)`
`20(y + 12)` `= 40y`
`20y + 240` `= 40y`
`20y` `= 240`
`y` `= 12`

`=>\ text(D)`

Algebra, STD2 A1 2017 HSC 2 MC

A car is travelling at 95 km/h.

How far will it travel in 2 hours and 30 minutes?

A.     `38\ text(km)`

B.     `41.3\ text(km)`

C.     `218.5\ text(km)`

D.     `237.5\ text(km)`

Show Answers Only

`text(D)`

Show Worked Solution
`text(Distance)` `= 95 xx 2.5`
  `= 237.5\ text(km)`

`=>\ text(D)`

Algebra, STD2 A1 2016 HSC 10 MC

Caroline drinks two small bottles of wine over a three-hour period. Each of these bottles contains 2.3 standard drinks. Caroline weighs 53 kg.

Using the formula below, what is Caroline's approximate blood alcohol content (BAC) at the end of this period?
 

`BAC_text(Female) = (10N - 7.5H)/(5.5M)`
 

where    `N` is the number of standard drinks consumed

`H` is the number of hours drinking

`M` is the person's mass in kilograms
 

  1. `0.081`
  2. `0.065`
  3. `0.0017`
  4. `0.0014` 
Show Answers Only

`A`

Show Worked Solution
`text(BAC)_f` `= (10N – 7.5H)/(5.5M)`
  `= (10(2 xx 2.3) – 7.5(3))/(5.5 xx 53)`
  `= 0.0806…`

 
`=> A`

Algebra, STD2 A1 2015 HSC 30d

Claire is driving on a motorway at a speed of 110 kilometres per hour and has to brake suddenly. She has a reaction time of 2 seconds and a braking distance of 59.2 metres.

Stopping distance can be calculated using the following formula
 

`text(stopping distance = {reaction time distance} + {braking distance})`

 
What is Claire's stopping distance.  (2 marks)

Show Answers Only

`120.3\ text{metres  (to 1 d.p.)}`

Show Worked Solution
♦ Mean mark 34%.
`110\ text(km/hr)` `= 110\ 000\ text(m/hr)`
  `= (110\ 000)/(60 xx 60)\ text(m/sec)`
  `= 30.555…\ text(m/sec)`

 

`text(Reaction time distance)` `=2 xx 30.555…`
  `= 61.11…\ text(metres)`

 

`:.\ text(Stopping distance)`

`=\ text(Reaction time distance + braking distance)`

`= 61.11… + 59.2`

`= 120.311…`

`= 120.3\ text{metres  (to 1 d.p.)}`

Algebra, STD2 A1 2015 HSC 26b

Clark’s formula is used to determine the dosage of medicine for children.
 

`text(Dosage) = text(weight in kg × adult dosage)/70`
 

The adult daily dosage of a medicine contains 3150 mg of a particular drug.

A child who weighs 35 kg is to be given tablets each containing 525 mg of this drug.

How many tablets should this child be given daily?  (2 marks)

Show Answers Only

`3`

Show Worked Solution
`text(Dosage)` `= (35 xx 3150)/70`
  `= 1575\ text(mg)`

 

`text(# Tablets per day)`

`= text(Dosage)/text(mg per tablet)`

`= 1575/525`

`= 3`

 

`:.\ text(The child should be given 3 tablets per day.)`

Algebra, STD2 A1 2015 HSC 23 MC

The number of ‘standard drinks’ in various glasses of wine is shown.
 

A woman weighing 62 kg drinks three small glasses of white wine and two large glasses of red wine between 8 pm and 1 am.

Using the formula for calculating blood alcohol below, what would be her blood alcohol content (BAC) estimate at 1 am, correct to three decimal places?
 

`BAC_text(Female) = (10N-7.5H)/(5.5M)`
 

where    `N` is the number of standard drinks consumed

`H` is the number of hours drinking

`M` is the person's mass in kilograms
 

  1. `0.030`
  2. `0.037`
  3. `0.046`
  4. `0.057`
Show Answers Only

`D`

Show Worked Solution
`N` `= 3 xx 0.9 + 2 xx 1.5`
  `= 5.7\ text(standard drinks)`
`H` `=\ text(5 hours)`
`M` `=\ text(62 kg)`
   
`:.BAC_f` `= (10 xx 5.7-7.5 xx 5)/ (5.5 xx 62)`
  `= 0.05718…`

`⇒ D`

Algebra, STD2 A1 2005 HSC 24b

The formula  `D = (2A)/15`  is used to calculate the dosage of Hackalot cough medicine to be given to a child.

    • D is the dosage of Hackalot cough medicine in millilitres (mL).
    • A is the age of the child in months.
  1. If George is nine months old, what dosage of Hackalot cough medicine should he be given?  (1 mark)

The correct dosage of Hackalot cough medicine for Sam is 4 mL.

  1. What is the difference in the ages of Sam and George, in months?  (3 marks)

Show Answers Only
  1. `text(1.2 mL)`
  2. `30`
Show Worked Solution
i. `D` `= (2A)/15`
    `= (2 × 9)/15`
    `=1.2\ text(mL)`

 

`:.\ text(George should be given a dosage of 1.2 mL)`

 

ii.   `text(Find)\ A\ text(when)\ D = text(4 mL)`

`4` `= (2 × A)/15`
 `2A` `= 60`
`A`  `= 30` 

 

`:.\ text(Sam is 30 months old and is 21 months)`

`text(older than George.)`

Algebra, STD2 A1 2014 HSC 29b

Blood alcohol content of males can be calculated using the following formula

`BAC_text(Male) = (10N - 7.5H)/(6.8M)`

where    `N` is the number of standard drinks consumed

`H` is the number of hours drinking

`M` is the person's mass in kilograms 

What is the maximum number of standard drinks that a male weighing 84 kg can consume over 4 hours in order to maintain a blood alcohol content (BAC) of less than 0.05?   (3 marks)

Show Answers Only

`5`

Show Worked Solution

`text(BAC)_text(male) = (10N\ – 7.5H)/(6.8M)`

`text(Find)\ \ N\ \ text(for BAC)<0.05,\ \ text(given)\ \ H = 4\ \ text(and)\ \ M = 84`
 

` (10N – 7.5(4))/(6.8(84))` `< 0.05`
`10N – 30` `< 0.05 (571.2)`
`10N` `< 28.56 + 30`
  `< 58.56`
`N` `< 5.856`

 

`:.\ text(Max number of drinks is 5.)`

Algebra, STD2 A1 2014 HSC 4 MC

Young’s formula below is used to calculate the required dosages of medicine for children aged 1–12 years.

 `text(Dosage) = (text(age of child)\ text{(in years)} xx text(adult dosage))/(text(age of child)\ text{(in years)} + 12)`

How much of the medicine should be given to an 18-month-old child in a 24-hour period if each adult dosage is 45 mL? The medicine is to be taken every 6 hours by both adults and children.

  1.       5 mL
  2.    20 mL
  3.    27 mL
  4.    30 mL
Show Answers Only

`B`

Show Worked Solution
♦ Mean mark 42%
`text(Dosage)` `= (1.5 xx 45)/(1.5 + 12)`
  `= 5\ text(mL)`

 
`text(S)text(ince 1 dosage every 6 hrs)`

`text(In 24 hours,)`

`text(Medicine given) = 4 xx 5 = 20\ text(mL)`

`=>  B`

Algebra, STD2 A1 SM-Bank 1 MC

The blood alcohol content (`BAC`) of a male's blood is given by the formula;

`BAC_text(male) = (10N - 7.5H)/(6.8M)`  , where

`N` is the number of standard drinks consumed,

`H` is the number of hours drinking and 

`M` is the person's mass in kgs. 

Calculate the  `BAC`  of a male who consumed 4 standard drinks in 3.5 hours and weighs 68 kgs, correct to 2 decimal places. 

  1.    1.03
  2.    0.03
  3.    0.04
  4.    0.01
Show Answers Only

`B`

Show Worked Solution
`BAC_text(male)` `= ( (10 xx 4)\ – (7.5 xx 3.5) )/( (6.8 xx 68) )`
  `= 13.75/462.4`
  `=0.0297…`

`=>  B`

Algebra, STD2 A1 2011 HSC 21 MC

A train departs from Town A at 3.00 pm to travel to Town B. Its average speed for the
journey is 90 km/h, and it arrives at 5.00 pm. A second train departs from Town A at
3.10 pm and arrives at Town B at 4.30 pm.

What is the average speed of the second train?

  1.   135 km/h
  2. 150 km/h
  3.   216 km/h
  4.   240 km/h
Show Answers Only

`A`

Show Worked Solution

 `text(1st train)`

♦ Mean mark 49%

`text(Travels 2hrs at 90km/h)`

`text(Distance)` `=text(Speed)xxtext(Time)`
  `=90xx2`
  `=180\ text(km)`

 
`text(2nd train)`

`text(Travels 180 km in 1 hr 20 min)\ (4/3\ text(hrs))`

`text(Speed)` `=text(Distance)/text(Time)`
  `=180-:4/3`
  `=180xx3/4`
  `=135\ text(km/h)`

`=> A`

Algebra, STD2 A1 2012 HSC 15 MC

A car takes 6 hours to complete a journey when travelling at 60 km/h.

How long would the same journey take if the car were travelling at 100 km/h?

  1.    36 minutes
  2.      1 hour and 40 minutes
  3.      3 hours and 6 minutes
  4.      3 hours and 36 minutes
Show Answers Only

`D`

Show Worked Solution

`T = D/S`

`text(S)text(ince)\ \ \ T = 6\ \ \ text(when)\ \ \ S = 60`

`6` `= D/60`
`D` `= 360\ text(km)`

 

`text(Find)\ \ T\ \ text (when)\ \ \ S = 100\ \ text(and)\ \ \ D = 360`

`T` `= 360/100`
  `= 3.6\ text(hours)`
  `= 3\ text(hrs)\ \ 36\ text(minutes)`

`=>  D`