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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

`C(t) = Ae^(kt),`

where `A` and `k` are constants, and `t` is the number of years since the kangaroo died.

  1. Show that `C(t)` satisfies  `(dC)/(dt) = kC`.  (1 mark)

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

     

    Show that the value of `k`, 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)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `text(Proof)\ \ text{(See Worked Solutions)}`
  3. `870\ text{years (2 sig. fig.)}`
Show Worked Solution
i.   `C` `= Ae^(kt)`
   `(dC)/(dt)` `= k * Ae^(kt)`
    `= kC …\ text(as required)`

 

ii.  `text(When)\ \ t = 5730, qquad A = 0.5 A_0`

`0.5 A_0` `= A_0 * e^(5730 k)`
`e^(5730 k)` `= 0.5`
`text(ln)\ e^(5730 k)` `= text(ln)\ 0.5`
`5730 k` `= text(ln)\ 0.5`
`k` `= {text(ln)\ 0.5}/5730`
  `= -0.0001209…`
  `= -0.00012\ text{(2 sig fig) … as required}`

 

iii.  `text(Find)\ t\ text(when)\ A = 0.9 A_0`

`0.9 A_0` `= A_0 e^(kt)`
`e^(kt)` `= 0.9`
`kt` `= text(ln)\ 0.9`
`t` `= (text(ln)\ 0.9)/k`
  `= (5730 xx text(ln)\ 0.9)/(text(ln)\ 0.5)`
  `= 870.97…`
  `= 870\ text{years (2 sig.fig.)}`

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 `M(t)` in milligrams of Curium, after `t` days, is given by

`M(t) = Ae^(-kt),`

where `A` and `k` are constants.

  1. State the value of `A`.  (1 mark)

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

Show Answers Only
  1. `10`
  2. `(ln 2)/163`
Show Worked Solution

i.  `text(When)\ \ t = 0,\ \ M = 10`

`10` `= Ae°`
`:. A` `= 10`

 

ii.  `text(When)\ \ t = 163,\ \ M = 5`

`5` `= 10 e^(-163k)`
`e^(-163k)` `= 1/2`
`e^(163k)` `= 2`
`163 k` `= ln 2`
`:. k` `= (ln 2)/163`

Calculus, EXT1* C1 2014 HSC 13b

A quantity of radioactive material decays according to the equation 

`(dM)/(dt) = -kM`,

where  `M`  is the mass of the material in kg,  `t`  is the time in years and  `k`  is a constant.

  1. Show that  `M = Ae^(–kt)`  is a solution to the equation, where  `A`  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)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `1.98\ text(kg)\ text{(2 d.p.)}`
Show Worked Solution
i. `M` `= Ae^(-kt)`
  `(dM)/(dt)` `= -k * Ae^(-kt)`
    `= -kM\ \ text(… as required)`

 

ii.    `text(At)\ \ t = 0,\ M = 20`
`=> 20` `= Ae^0`
`A` `= 20`
`:.\ M` `= 20 e^(-kt)`

`text(At)\ \ t = 300,\ M = 10`

TIP: Many students find it efficient to save the exact value of `k` in the memory function of their calculator for these questions.
`=> 10` `= 20 e^(-300 xx k)`
`e^(-300k)` `= 10/20`
`ln e^(-300k)` `= ln 0.5`
`-300 k` `= ln 0.5`
`k` `= – ln 0.5/300`
  `= 0.00231049…`

 
`text(Find)\ M\ text(when)\ t = 1000`

`M` `= 20 e^(-1000k)`
  `= 20 e^(-1000 xx 0.00231049…)`
  `= 20 xx 0.099212…`
  `= 1.9842…`
  `= 1.98\ text(kg)\ text{(2 d.p.)}`

Calculus, EXT1* C1 2009 HSC 6b

Radium decays at a rate proportional to the amount of radium present. That is, if  `Q(t)`  is the amount of radium present at time  `t`,  then  `Q=Ae^(-kt)`,  where  `k`  is a positive constant and  `A`  is the amount present at  `t=0`. It takes 1600 years for an amount of radium to reduce by half.

  1. Find the value of  `k`.   (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)

Show Answers Only
  1. `(-ln(1/2))/1600\ \ text(or 0.000433)`
  2. `2536\ text(years)`
Show Worked Solution

i.   `Q=Ae^(-kt)`

MARKER’S COMMENT: Students must be familiar with “half-life” and the algebra required. i.e. using `Q=1/2 A` within their calculations.

`text(When)\ \ t=0,\ \ Q=A`

`text(When)\ \ t=1600,\ \ Q=1/2 A`

`:.1/2 A` `=A e^(-1600xxk)`
`e^(-1600xxk)` `=1/2`
`lne^(-1600xxk)` `=ln(1/2)`
`-1600k` `=ln(1/2)`
`k` `=(-ln(1/2))/1600`
  `=0.0004332\ \ text{(to 4 sig. figures)}`

 

ii.   `text(Find)\ \ t\ \ text(when)\ \ Q=1/3 A :`

IMPORTANT: Know how to use the memory function on your calculator to store the exact value of `k` found in part (i) to save time.
`1/3 A` `=A e^(-kt)`
`e^(-kt)` `=1/3`
`lne^(-kt)` `=ln(1/3)`
`-kt` `=ln(1/3)`
`:.t` `=(-ln(1/3))/k,\ \ \ text(where)\ \ \ k=(-ln(1/2))/1600`
  `=(ln(1/3) xx1600)/ln(1/2)`
  `=2535.940…`

 
`:.\ text(It will take  2536  years.)`