Exam-Style Question on Exponential Models

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Question id: 392. This question is similar to one that appeared on an A-Level paper. The use of a calculator is allowed.

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.

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