# Exam-Style Question on Exponential Models

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 434. This question is similar to one that appeared on an IB AI Standard paper (specimen) for 2021. The use of a calculator is allowed.

Doctor Octothorpe investigated the decreasing population of a colony of ants in a remote province of China. His investigation took place in 1958.

He found that during the summer season their population, $$P$$, could be modelled by the exponential equation

$$P = 560 + 9560(1.3)^{-t} \quad \text{where} \quad t \ge 0$$

where $$t$$ is the number of days into the season ($$t = 1$$ represents the beginning of 1st June).

(a) Find the population of the ants at the beginning of 31st May 1958.

(b) Find the population of the ants at the beginning of 10th June.

(c) Calculate the date when the population first fell below 1000.

(d) According to this model, find the smallest possible population of ants.

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