Exam-Style Question on Exponential Decay

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.