1. **State the problem:** Determine if the graph showing the relationship between time (hours) and distance (miles) is a proportional relationship. 2. **Recall the definition of proportional relationship:** A relationship is proportional if the graph is a straight line passing through the origin $(0,0)$ and the ratio of the dependent variable to the independent variable is constant. 3. **Check the graph:** The graph is a straight line passing through the origin, which is a key indicator of proportionality. 4. **Calculate the ratio of distance to time for given points:** - At time $\frac{1}{2}$ hours, distance = 15 miles, ratio = $\frac{15}{\frac{1}{2}} = 15 \times 2 = 30$ - At time 1 hour, distance = 30 miles, ratio = $\frac{30}{1} = 30$ - At time $\frac{3}{2}$ hours, distance = 45 miles, ratio = $\frac{45}{\frac{3}{2}} = 45 \times \frac{2}{3} = 30$ - At time 2 hours, distance = 60 miles, ratio = $\frac{60}{2} = 30$ 5. **Interpretation:** The ratio $\frac{\text{distance}}{\text{time}}$ is constant at 30 miles per hour. 6. **Conclusion:** Since the graph is a straight line through the origin and the ratio of distance to time is constant, the graph shows a proportional relationship. **Final answer:** Yes, the graph shows a proportional relationship with a constant of proportionality 30 miles per hour.