Exam-Style Question on Exponential Models

In a remote lake it was noticed by conservationists that a disease was rapidly spreading amongst two species of fish, R and S, which is reducing their numbers. The conservationists calculated that the numbers of each type of fish can be modelled by the functions:

$$ r(t) = 9000e^{-\frac{1}{10}t} $$

and

$$ s(t) = 6000e^{-\frac{1}{20}t} $$

respectively where t is the time in weeks after the disease was first detected on the 2nd August 2019.

(a) Use the two models to find the number of species R and S on 2nd August 2019.

(b) Find the number of species S after 24 weeks from 2nd August 2019, giving your answer to the nearest 10.

(c) After how many whole weeks will the number of species R first fall below 4500?

(d) Use logarithms and the two models to calculate the value of t when the number of species S will be three times that of species R. Give your answer to the nearest whole number.

(e) When \(t = T\) the number of species \(S\) first exceeds that of species R by 500. Use this information and the two models to derive a quadratic equation in \(x\) where:

$$ x=e^{-\frac{1}{20}T} $$

(f) Hence find the number of days after 2nd August 2019 when this difference of 500 fish will first occur. Give your answer to the nearest day.