Mathematics

Question

5500 trees were planted in a forest on 1st October 2018.
Only 5060 of these trees were still alive on 1st October 2019.

It is assumed that the number of trees still alive is given by N = ar^t
where N is the number of trees still alive t years after 1" October 2018.

a) Write down the value of a.

b) Work out the value of r.

c) Work out the predicted percentage decrease in trees still alive between
1st October 2018 and 1st October 2030.

1 Answer

  • Answer:

      a)  a = 5500

      b)  r = 0.92

      c)  63.2%

    Step-by-step explanation:

    You want various values related to the exponential decrease in the number of living trees from 5500 to 5060 between 2018 and 2019, and you want the percentage decrease to 2030 at the same rate.

    a) Exponential function

    The function ...

      N = a·r^t

    has an initial value of 'a' when t=0. The problem statement tells you that is 5500.

      a = 5500

    b) Growth rate

    The value of r is the ratio of the values of N for t=1 and t=0:

      r = (a·r^1)/(a·r^0) = 5060/5500 = 23/25

      r = 0.92

    c) Decrease

    The fraction still alive after 12 years is predicted to be ...

      0.92^12 ≈ 0.3677

    So, the percentage decrease is ...

      (1 -0.3677) × 100% ≈ 63.2%

    The predicted percentage decrease in living trees is 63.2% by 2030.

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