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. \( \square \)

The relationship between the amount of the drug \( A \) in the patient's system and time \( t \) is exponential. This is because, instead of a constant amount being added or subtracted over time (as in linear relationships), the amount of the drug decreases by a consistent percentage (25% loss in this case) every hour. This means that the drug amount is reduced in a manner that can be modeled by the equation \( A(t) = 50 \times (0.75)^t \), showcasing a rapid initial decrease that gradually slows down over time. To visualize this relationship, a graph of \( A \) versus \( t \) will curve downward, reflecting the decreasing amount of the drug with each passing hour. At \( t=0 \), the graph starts at 50 mg, and then it will continuously approach zero but never actually reach it, indicative of an exponential decay function. This shape contrasts sharply with a linear decline, which would produce a straight line downwards.