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Algebra, STD2 A4 2024 HSC 22

The graph shows the populations of two different animals, and , in a conservation park over time. The -axis is the size of the population and the -axis is the number of years since 1985 .

Population is modelled by the equation  .

Population is modelled by the equation  .
 

Complete the table using the information provided.   (3 marks)

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Mean mark 51%.

Algebra, STD2 A4 2021 HSC 24

A population,  ,  is to be modelled using the function  ,  where is the time in years.

  1. What is the initial population?  (1 mark)

  2. Find the population after 5 years.  (1 mark)

  3. On the axes below, draw the graph of the population against time, showing the points at    and at  .  (2 marks)
      

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a.   
 
   

 

b.   

 
   
   

 

♦ Mean mark (c) 48%.

c. 

Algebra, STD2 A4 2020 HSC 33

The graph shows the number of bacteria, , at time minutes. Initially (when ) the number of bacteria is 1000.
 


 

  1. Find the number of bacteria at 40 minutes.   (1 mark)

  2. The number of bacteria can be modelled by the equation  , where and are constants.
    Use the guess and check method to find, to two decimal places, an upper and lower estimate for the value of . The upper and lower estimates must differ by 0.01. (2 marks)

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a.   


 

b.   

♦♦♦ Mean mark 14%.


 

Algebra, STD2 A4 2011 HSC 26b

Jack needs to find the number of years, , it will take for a population of bats to first exceed 18 000.

He uses a ‘guess-and-check’ method to estimate in the following equation

Here is his working:

  1. Jack’s next guess is  . Show Jack’s correct working for this guess, including the calculation and conclusion.  (1 mark)

  2. Continue using the ‘guess-and-check’ method to find the number of years, , it will take for the population to first exceed 18 000, if is a whole number. Include the calculations and conclusions.  (2 marks)

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i.   

 

ii.   


 

Algebra, STD2 A4 2016 HSC 29b

The mass kg of a baby pig at age days is given by    where is a constant. The graph of this equation is shown.
 

2ug-2016-hsc-q29_1

  1. What is the value of ?   (1 mark)

  2. What is the daily growth rate of the pig’s mass? Write your answer as a percentage.   (1 mark)

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i.   

♦ Mean mark (i) 48%.
♦♦♦ Mean mark part (ii) 6%!

 
ii.   

Algebra, STD2 A4 2004 HSC 26a

  1. The number of bacteria in a culture grows from 100 to 114 in one hour.

     

    What is the percentage increase in the number of bacteria?   (1 mark)

  2. The bacteria continue to grow according to the formula  , where    is the number of bacteria after    hours.

     

    What is the number of bacteria after 15 hours?   (1 mark)

  1. Use the values of from    to    to draw a graph of  .

     

    Use about half a page for your graph and mark a scale on each axis.   (4 marks)

  2. Using your graph or otherwise, estimate the time in hours for the number of bacteria to reach 300.   (1 mark)

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i.   

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii. 

 
 

 

iii. 

 

iv. 

Algebra, STD2 A4 2012 HSC 30c

In 2010, the city of Thagoras modelled the predicted population of the city using the equation

.

That year, the city introduced a policy to slow its population growth. The new predicted population was modelled using the equation

.

In both equations, is the predicted population and is the number of years after 2010.  

The graph shows the two predicted populations.
 

  1. Use the graph to find the predicted population of Thagoras in 2030 if the population policy had NOT been introduced.   (1 mark)

  2. In each of the two equations given, the value of is 3 000 000.

     

    What does represent?   (1 mark)

  3. The guess-and-check method is to be used to find the value of , in  .

     

    (1) Explain, with or without calculations, why 1.05 is not a suitable first estimate for .  (1 mark)

     

    (2) With    and  , use the guess-and-check method and the equation    to estimate the value of to two decimal places. Show at least TWO estimate values for , including calculations and conclusions.  (2 marks)

  4. The city of Thagoras was aiming to have a population under 7 000 000 in 2050. Does the model indicate that the city will achieve this aim?

     

    Justify your answer with suitable calculations.  (2 marks)

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  1.  
  2.  
  3.  

     

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i.   

COMMENT: Common ADV/STD2 content in new syllabus.

 

ii.    


 

♦ Mean mark part (ii) 48%

iii. (1) 

   
 

iii. (2) 

 

 

 

 

 

iv.