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Calculus, EXT1* C1 2017 HSC 14c

Carbon-14 is a radioactive substance that decays over time. The amount of carbon-14 present in a kangaroo bone is given by

where and are constants, and is the number of years since the kangaroo died.

  1. Show that satisfies  .  (1 mark)

  2. After 5730 years, half of the original amount of carbon-14 is present.

     

    Show that the value of , correct to 2 significant figures, is – 0.00012.  (2 marks)

  3. The amount of carbon-14 now present in a kangaroo bone is 90% of the original amount.

     

    Find the number of years since the kangaroo died. Give your answer correct to 2 significant  figures.  (2 marks)

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i.  
   
   

 

ii. 

 
 

 

iii. 

 
 
 

Calculus, EXT1* C1 2016 HSC 13c

A radioactive isotope of Curium has a half-life of 163 days. Initially there are 10 mg of Curium in a container.

The mass in milligrams of Curium, after days, is given by

where and are constants.

  1. State the value of .  (1 mark)

  2. Given that after 163 days only 5 mg of Curium remain, find the value of .  (2 marks)

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i. 

 

ii. 

Calculus, EXT1* C1 2014 HSC 13b

A quantity of radioactive material decays according to the equation 

,

where    is the mass of the material in kg,    is the time in years and    is a constant.

  1. Show that    is a solution to the equation, where    is a constant.   (1 mark)

  2. The time for half of the material to decay is 300 years. If the initial amount of material is 20 kg, find the amount remaining after 1000 years.   (3 marks)

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i.
 
   

 

ii.    

TIP: Many students find it efficient to save the exact value of in the memory function of their calculator for these questions.
 

 

 
 
 
 

Calculus, EXT1* C1 2009 HSC 6b

Radium decays at a rate proportional to the amount of radium present. That is, if    is the amount of radium present at time  ,  then  ,  where    is a positive constant and    is the amount present at  . It takes 1600 years for an amount of radium to reduce by half.

  1. Find the value of  .   (2 marks)

  2. A factory site is contaminated with radium. The amount of radium on site is currently three times the safe level.

     

    How many years will it be before the amount of radium reaches the safe level.    (2 marks)

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i.   

MARKER’S COMMENT: Students must be familiar with “half-life” and the algebra required. i.e. using within their calculations.

 

 

ii.   

IMPORTANT: Know how to use the memory function on your calculator to store the exact value of found in part (i) to save time.