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Calculus, EXT1* C1 2018 HSC 13c

The population of a country grew exponentially between 1910 and 2010. This population can be modelled by the equation  , where    is the population of the country in millions, is the time in years after 1910 and is a positive constant. The population of the country in 1960 was 184 million.

  1. Show that the value of is 0.0139, correct to 4 decimal places.  (2 marks)

  2. Assuming that this model continues to be valid after 2010, estimate the population of the country in 2020 to the nearest million.  (2 marks)

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i.   

 
 

 

ii. 

Calculus, EXT1* C1 2004 HSC 7b

At the beginning of 1991 Australia’s population was 17 million. At the beginning of 2004 the population was 20 million.

Assume that the population is increasing exponentially and satisfies an equation of the form  , where    and    are constants, and    is measured in years from the beginning of 1991.

  1. Show that    satisfies  .  (1 mark)

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

  3. Find the value of  .  (2 marks)

  4. Predict the year during which Australia’s population will reach 30 million.  (2 marks)

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i.  
 
   

 

ii. 

   
 

 

iii. 

 
 
 

 

iv. 

MARKER’S COMMENT: Many students had the correct calculations but didn’t answer the question by identifying the exact year and lost a valuable mark.
 
 

 
 

 

Calculus, EXT1* C1 2007 HSC 8a

One model for the number of mobile phones in use worldwide is the exponential growth model,

,

where is the estimate for the number of mobile phones in use (in millions), and is the time in years after 1 January 2008.

  1. It is estimated that at the start of 2009, when  , there will be 1600 million mobile phones in use, while at the start of 2010, when  , there will be 2600 million. Find and .  (3 marks)

  2. According to the model, during which month and year will the number of mobile phones in use first exceed 4000 million?  (2 marks)

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i.  

 

 
 
 
 

 

 
 

 

ii. 

 

 

Calculus, EXT1* C1 2010 HSC 8a

Assume that the population,  ,  of cane toads in Australia has been growing at a rate proportional to  .  That is,    where  is a positive constant.

There were 102 cane toads brought to Australia from Hawaii in 1935.

Seventy-five years later, in 2010, it is estimated that there are 200 million cane toads in Australia.

If the population continues to grow at this rate, how many cane toads will there be in Australia in 2035?   (4 marks)

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NOTE: Students should be comfortable converting 200 million into scientific notation. It can help noting that 200 million

 

MARKER’S COMMENT: This question differed from previous years in that it did not ask students to verify that that    is a solution to  .
 

 

 
 

 

Calculus, EXT1* C1 2011 HSC 10a

The intensity, ,  measured in  watt/m2,  of a sound is given by

,

where    is the loudness of the sound in decibels.

  1. If the loudness of a sound at a concert is 110 decibels, find the intensity of the sound. Give your answer in scientific notation.   (1 mark)

  2. Ear damage occurs if the intensity of a sound is greater than  watt/m2.

     

    What is the maximum loudness of a sound so that no ear damage occurs?    (2 marks)

  3. By how much will the loudness of a sound have increased if its intensity has doubled?     (2 marks)

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i.    

MARKER’S COMMENT: Note that is not correct scientific notation.
 
 
 

 

ii.   

 
 

 

 

iii. 

♦♦ Mean mark 32%
MARKER’S COMMENT: Actual values can help here. Calculate the intensity at (which equals ) and then find when this intensity level is doubled (to ).

 
 

 

Calculus, EXT1* C1 2012 HSC 14c

Professor Smith has a colony of bacteria. Initially there are 1000 bacteria. The number of bacteria,  ,  after    minutes is given by

. 

  1. After 20 minutes there are 2000 bacteria.

     

    Show that  correct to four decimal places.   (1 mark)

  2. How many bacteria are there when  ?    (1 mark)

  3. What is the rate of change of the number of bacteria per minute, when  ?     (1 mark)

  4. How long does it take for the number of bacteria to increase from 1000 to 100 000?    (2 marks)

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  4.  
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i.   

 

 

 ii.  

NOTE: Students could have used the exact value of    in parts (ii), (iii) and (iv), which would yield the answers (ii) 64,000, (iii) 2,218, and (iv) 133 minutes.
 
 

 

 

iii.  

MARKER’S COMMENT: This part proved challenging for many students. Differentiation is required to find the rate of change in these type of questions.
 

 

 
 

 

 

iv. 

 
 

 

Calculus, EXT1* C1 2013 HSC 16b

Trout and carp are types of fish. A lake contains a number of trout. At a certain time, 10 carp are introduced into the lake and start eating the trout. As a consequence, the number of trout,  ,  decreases according to

,

where is the time in months after the carp are introduced.

The population of carp,  ,  increases according to  .

  1. How many trout were in the lake when the carp were introduced?    (1 mark)

  2. When will the population of trout be zero?    (1 mark)

  3. Sketch the number of trout as a function of time.     (1 marks)

  4. When is the rate of increase of carp equal to the rate of decrease of trout?    (3 marks)

  5. When is the number of carp equal to the number of trout?    (2 marks)

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  3.  
     
    2UA 2013 HSC 16b Answer
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i.   

MARKER’S COMMENT: A number of students did not equate in this part.

 

ii.   

NOTE: The last line of the solution isn’t necessary but is included as good practice as a check that the answer matches the exact question asked.
 
 

 

 

iii.   2UA 2013 HSC 16b Answer

♦♦ Mean mark 33% for part (iii)

 

iv.   


  

♦♦♦ Mean mark 20%
COMMENT: Students who progressed to received 2 full marks in this part. Show your working!

 
 

 

 
 

 

v.   

♦♦♦ Mean mark 4%!
MARKER’S COMMENT: Correctly applying substitution to exponentials to form a solvable quadratic proved very difficult for almost all students.