2. A drug is given to a patient. At time zero, there are 50 mg of the drug in the patient's body. At the end of each hour, of the amount of the drug that was in the patient's system at the end of the hour, only $$ 75 \% $$ remains. Let A be the amount of the drug (in mg ) in the patient's system and let t be time in hours since the drug was administered. We will examine the relationship between A and t . Is this a linear or exponential relationship? Explain. . On the graph at the right, ot the A versus $$ t $$ relationship $$ r $$ the 10 hours after the iministration of the drug. $$ \square $$

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1. Since at time $$ t = 0 $$ there are 50 mg of the drug, and at the end of each hour only $$ 75\% $$ of the drug remains, the amount of drug in the patient’s system can be modeled by

$$
A(t) = 50 \times (0.75)^t.
$$

2. In a linear relationship, the amount of change per unit time would be constant. For a linear model, we expect a form such as

$$
A(t) = mt + b,
$$

where $$ m $$ is the constant rate of change and $$ b $$ is the initial amount. Here, however, the change is multiplicative ($$ 0.75 $$ multiplied every hour) rather than additive or subtractive by a constant amount. The fact that the remaining drug is always $$ 75\% $$ of the previous amount indicates an exponential decay.

3. Therefore, the relationship between $$ A $$ and $$ t $$ is exponential, not linear.

4. The graph of $$ A $$ versus $$ t $$ for the first 10 hours would show a curve that starts at 50 mg at $$ t = 0 $$ and decreases rapidly at first before gradually approaching zero, showing the typical decay shape of an exponential function.
The relationship between the amount of drug $$ A $$ and time $$ t $$ is exponential, not linear. This is because the drug decreases by $$ 75\% $$ each hour, leading to a multiplicative decrease over time, which is characteristic of exponential decay.

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