1. **Stating the problem:** We are given a table showing the amount of a drug in a person's system over time and asked to determine if the relationship is linear or exponential, and then write an equation for the amount of drug $y$ in terms of hours $x$. 2. **Analyzing the data:** The table is: | Hours ($x$) | Amount ($y$ mg) | |-------------|-----------------| | 0 | 500.0 | | 1 | 350.0 | | 2 | 245.0 | | 3 | 171.5 | | 4 | 120.0 | | 5 | 84.0 | 3. **Checking if linear:** In a linear relationship, the amount decreases by a constant difference each hour. Calculate differences: $350 - 500 = -150$ $245 - 350 = -105$ $171.5 - 245 = -73.5$ $120 - 171.5 = -51.5$ $84 - 120 = -36$ The differences are not constant, so the relationship is not linear. 4. **Checking if exponential:** In an exponential relationship, the amount decreases by a constant ratio each hour. Calculate ratios: $\frac{350}{500} = 0.7$ $\frac{245}{350} = 0.7$ $\frac{171.5}{245} \approx 0.7$ $\frac{120}{171.5} \approx 0.7$ $\frac{84}{120} = 0.7$ The ratios are approximately constant at 0.7, indicating an exponential decay. 5. **Writing the equation:** The general form for exponential decay is: $$y = y_0 \cdot r^x$$ where $y_0$ is the initial amount and $r$ is the decay factor per hour. Here, $y_0 = 500$ mg and $r = 0.7$. So the equation is: $$y = 500 \cdot 0.7^x$$ 6. **Summary:** The relationship is exponential because the amount decreases by about 30% each hour (multiplied by 0.7), not by a fixed amount. **Final answer:** $$\boxed{y = 500 \cdot 0.7^x}$$