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Half-life is the time it takes for half of the radioisotope to decay. This concept is crucial because it tells us how long the radioisotope remains active. For example, if we start with 100 units of a substance, after one half-life, only 50 units remain active. After two half-lives, 25 units remain active, and so on. This decay process follows an exponential pattern. The half-life of the radioisotope in our problem is 30 days. This means after 30 days, the radioisotope’s activity is halved. To calculate the decay over time, we use the decay factor formula: \[ Decay Factor = 2^{-t/T_{half}} \] Here, \(T_{half}\) is the half-life time (30 days, converted to minutes), and \(t\) is the time the radioisotope has been in the river. Understanding this decay helps us account for the radioisotope’s decrease in activity as it travels downstream. This is essential for calculating the river’s flow rate.

Injection rate refers to how much radioisotope is added to the river per unit time. In this exercise, the injection rate of the radioisotope is 10 µCi/min. This means every minute, 10 microcuries of the substance are injected into the river. When calculating the total activity injected \(A_{injected}\), we need to consider the time \(t\) the radioisotope has been added to the river: \[ A_{injected} = 10 \times t \text{ (in µCi)} \] This simple formula helps us keep track of the total radioisotope added to the river, which is a crucial first step before considering the effects of decay. The injection rate is vital in understanding how much radiological material enters the environment, which impacts both the river’s flow rate and the environmental safety guidelines that must be followed.

Understanding river hydraulics is key to solving the problem. Hydraulics involves studying how water flows and how different variables, like flow rate and cross-sectional area, affect this flow. The flow rate of a river, often denoted as \(F\), is the volume of water moving through the river per unit time, usually expressed in m³/h. In the problem, we need to calculate the river’s flow rate downstream, where the water activity is given as 7.3 nCi/g. To find \(F\), we use a steady-state assumption where the rate at which radioisotope is injected is equal to the rate at which it is observed downstream: \[ 10 \text{ µCi/min} = F \times 0.0073 \text{ µCi/g} \times \text{density of water} \] Assuming the density of water is 1 g/cm³ (1000 kg/m³), we convert this to: \[ F = \frac{10}{0.0073 \times 1000} \text{ m³/min} \times 60 \text{ min/h} \] Solving this, we get approximately 82.19 m³/h. This calculation shows the importance of river hydraulics in determining how fast the water is flowing based on the concentration of substances it carries. Proper understanding of hydraulics ensures accurate environmental assessments and effective management of water resources.