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Calculus, EXT1* C1 2019 HSC 12c

The number of leaves, , on a tree days after the start of autumn can be modelled by

  1. What is the number of leaves on the tree when  ?  (1 mark)

  2. What is the rate of change of the number of leaves on the tree when  ?  (2 marks)

  3. For what value of are there 100 leaves on the tree?  (2 marks)

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i. 

 
 

 

ii.   
 
   

 

 

 

 

iii.   

 
 

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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ii. 

 

 

Calculus, EXT1* C1 2015 HSC 15a

The amount of caffeine, , in the human body decreases according to the equation

 

where is measured in mg and is the time in hours.

  1. Show that    is a solution to  where is a constant.

     

    When , there are 130 mg of caffeine in Lee’s body.  (1 mark)

  2. Find the value of   (1 mark)

  3. What is the amount of caffeine in Lee’s body after 7 hours?   (1 mark)

  4. What is the time taken for the amount of caffeine in Lee’s body to halve?  (2 marks)

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Calculus, EXT1* C1 2008 HSC 5c

Light intensity is measured in lux. The light intensity at the surface of a lake is 6000 lux. The light intensity,    lux, a distance    metres below the surface of the lake is given by

where  ,  and    are constants.

  1. Write down the value of  .   (2 marks)

  2. The light intensity 6 metres below the surface of the lake is 1000 lux. Find the value of  .    (2 marks)

  3. At what rate, in lux per metre, is the light intensity decreasing 6 metres below the surface of the lake?    (2 marks)

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i.   

 

ii.   

MARKER’S COMMENT: Many students used the “log” function on their calculator rather than the function. BE CAREFUL!
 
 

 

iii. 

ALGEBRA TIP: Tidy your working by calculating using and then only substituting for in part (ii) at the final stage.

 

 
 

 

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 ).