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Financial Maths, 2ADV M1 2016 HSC 14b

A gardener develops an eco-friendly spray that will kill harmful insects on fruit trees without contaminating the fruit. A trial is to be conducted with 100 000 insects. The gardener expects the spray to kill 35% of the insects each day and that exactly 5000 new insects will be produced each day.

The number of insects expected at the end of the `n`th day of the trial is `A_n.`

  1. Show that  `A_2 = 0.65 (0.65 xx 100\ 000 + 5000) + 5000`.  (2 marks)

  2. Show that  `A_n = 0.65^n xx 100\ 000 + 5000 ((1 - 0.65^n))/0.35`.  (1 mark)

  3. Find the expected insect population at the end of the fourteenth day, correct to the nearest 100.  (1 mark)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `text(Proof)\ \ text{(See Worked Solutions)}`
  3. `14\ 500\ text{(nearest 100)}`
Show Worked Solution
i.   `A_1` `= 0.65 xx 100\ 000 + 5000`
  `A_2` `= 0.65 xx A_1 + 5000`
    `= 0.65 (0.65 xx 100\ 000 + 5000) + 5000`
    `qquad qquad text(… as required)`

 

ii.  `A_2` `= 0.65^2 xx 100\ 000 + 0.65 xx 5000 + 5000`
  `A_3` `= 0.65^3 xx 100\ 000 + 0.65^2 xx 5000 + 0.65 xx 5000 + 5000`
  `vdots`  
  `A_n` `= 0.65^n xx 100\ 000 + 0.65^(n – 1) xx 5000 + 0.65^(n – 2) xx 5000 + … + 5000`
    `= 0.65^n xx 100\ 000 + 5000 (1 + 0.65 + … + 0.65^(n – 1))`
    `qquad qquad => text(GP where)\ \ a = 1,\ \ r = 0.65`
    `= 0.65^n xx 100\ 000 + 5000 ({(1 – r^n)}/(1 – r))`
    `= 0.65^n xx 100\ 000 + 5000 ((1 – 0.65^n))/0.35`

 

iii.  `A_14` `= 0.65^14 xx 100\ 000 + 5000 ((1 – 0.65^14)/0.35)`
    `= 14\ 491.70…`
    `= 14\ 500\ text{(nearest 100)}`