# Exam-Style Question on Exponential Decay

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

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Question id: 692. This question is similar to one that appeared on an IB AA Higher paper in 2021. The use of a calculator is allowed.

In medical imaging, the radioactive isotope Technetium-99m is used due to its short half-life of 6 hours. After being introduced into the body, the isotope decays, and the amount of Technetium-99m present can be tracked over time to study the function of specific organs.

The amount, $$A$$, of Technetium-99m present in a patient's body $$t$$ hours after injection can be modelled by the equation $$A = A_0e^{-kt}$$ where $$t \geq 0$$ , $$A_0$$ and $$k$$ are positive constants.

At the time of injection, the patient is defined to have 1000 units of Technetium-99m.

(a) Show that $$A_0 = 1000$$.

The time taken for half the original amount of Technetium-99m to decay is known to be 6 hours.

(b) Show that $$k = \frac{\ln 2}{6}$$.

(c) Medical guidelines suggest that a scan should be performed when the activity of Technetium-99m is between 25% to 50% of the initial dose to obtain a clear image. Calculate the time window after the injection during which the scans should be performed.

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