## Exam-Style Question on Exponential Models## A mathematics exam-style question with a worked solution that can be revealed gradually |

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.

The worked solutions to these exam-style questions are only available to those who have a Transum Subscription. Subscribers can drag down the panel to reveal the solution line by line. This is a very helpful strategy for the student who does not know how to do the question but given a clue, a peep at the beginnings of a method, they may be able to make progress themselves. This could be a great resource for a teacher using a projector or for a parent helping their child work through the solution to this question. The worked solutions also contain screen shots (where needed) of the step by step calculator procedures. A subscription also opens up the answers to all of the other online exercises, puzzles and lesson starters on Transum Mathematics and provides an ad-free browsing experience. |

Drag this panel down to reveal the solution

The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

The solutions to the questions on this website are only available to those who have a Transum Subscription.

Exam-Style Questions Main Page

To search the **entire Transum website** use the search box in the grey area below.

Do you have any comments about these exam-style questions? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

©1997 - 2023 Transum Mathematics :: For more exam-style questions and worked solutions go to Transum.org/Maths/Exam/