1. **Problem Statement:** We want to compare the New York and Massachusetts speeding fine systems based on miles per hour (mph) over the speed limit. 2. **New York System:** - For 1-10 mph over: fine ranges from $45 to $150. - For 11-30 mph over: fine ranges from $90 to $300. - For 31+ mph over: fine ranges from $180 to $600. 3. **Massachusetts System:** - For 1-10 mph over: flat fine of $100. - For 11-30 mph over: $100 plus $10 for each mph above 10. 4. **Formulas:** - New York fines are ranges, so no single formula. - Massachusetts fine for $x$ mph over speed limit where $x > 10$ is: $$\text{Fine} = 100 + 10(x - 10)$$ 5. **Example Calculation:** Calculate Massachusetts fine for 15 mph over: $$\text{Fine} = 100 + 10(15 - 10) = 100 + 10 \times 5 = 100 + 50 = 150$$ 6. **Explanation:** - Massachusetts system uses a flat rate for the first 10 mph and then adds $10 for each additional mph. - New York system uses ranges, allowing judges to assign fines within minimum and maximum limits. 7. **Summary:** - Massachusetts fine function for $x$ mph over (where $x \geq 1$) is: $$f(x) = \begin{cases} 100 & \text{if } 1 \leq x \leq 10 \\ 100 + 10(x - 10) & \text{if } 11 \leq x \leq 30 \end{cases}$$ - New York fines are intervals, not a single function. This completes the comparison and explanation of the speeding fine systems.