Job Earnings Analysis Bc7C75

1. **Problem Statement:** We analyze four jobs with linear earnings given by: - Marine: $$y = 1000x + 9000$$ - Teacher: $$y = -2000x + 1000$$ - Garbage Collector: $$y = 2000x + 4000$$ - Business Owner: $$y = 4000x - 12000$$ We answer questions about starting balances, slopes, earnings at specific months, and intersections. 2. **Starting Balances:** The starting balance is the y-intercept $$b$$ in $$y = mx + b$$. - Marine: $$9000$$ (positive) - Teacher: $$1000$$ (positive) - Garbage Collector: $$4000$$ (positive) - Business Owner: $$-12000$$ (negative) **Answer:** The Business Owner started with a negative balance. 3. **Jobs with Positive Starting Balance:** Marine, Teacher, and Garbage Collector all started with positive balances. 4. **Greatest Slope (Rate of Change):** Slopes $$m$$ are: - Marine: $$1000$$ - Teacher: $$-2000$$ (negative slope) - Garbage Collector: $$2000$$ - Business Owner: $$4000$$ Greatest positive slope is $$4000$$ (Business Owner). 5. **Least Slope:** Least slope means smallest rate of change (considering absolute value or actual value depending on context). - Teacher has $$-2000$$ (negative slope, decreasing) - Marine has $$1000$$ (smallest positive slope) If considering absolute value, smallest magnitude is Marine's $$1000$$. 6. **Earnings after 6 months:** Calculate $$y$$ at $$x=6$$ for each: - Marine: $$y = 1000(6) + 9000 = 6000 + 9000 = 15000$$ - Teacher: $$y = -2000(6) + 1000 = -12000 + 1000 = -11000$$ - Garbage Collector: $$y = 2000(6) + 4000 = 12000 + 4000 = 16000$$ - Business Owner: $$y = 4000(6) - 12000 = 24000 - 12000 = 12000$$ 7. **Earnings after 20 months:** Calculate $$y$$ at $$x=20$$: - Marine: $$1000(20) + 9000 = 20000 + 9000 = 29000$$ - Teacher: $$-2000(20) + 1000 = -40000 + 1000 = -39000$$ - Garbage Collector: $$2000(20) + 4000 = 40000 + 4000 = 44000$$ - Business Owner: $$4000(20) - 12000 = 80000 - 12000 = 68000$$ 8. **Time when Teacher and Garbage Collector earn the same:** Set equal: $$-2000x + 1000 = 2000x + 4000$$ Move terms: $$-2000x - 2000x = 4000 - 1000$$ $$-4000x = 3000$$ Divide: $$x = \frac{3000}{-4000} = -0.75$$ Negative time is not meaningful here, so no positive intersection. 9. **Time when Marine and Business Owner earn the same:** $$1000x + 9000 = 4000x - 12000$$ $$9000 + 12000 = 4000x - 1000x$$ $$21000 = 3000x$$ $$x = \frac{21000}{3000} = 7$$ months 10. **Time when Teacher and Marine earn the same:** $$-2000x + 1000 = 1000x + 9000$$ $$1000 - 9000 = 1000x + 2000x$$ $$-8000 = 3000x$$ $$x = \frac{-8000}{3000} = -\frac{8}{3} \approx -2.67$$ (negative, no positive intersection) 11. **Time when Garbage Collector and Business Owner earn the same:** $$2000x + 4000 = 4000x - 12000$$ $$4000 + 12000 = 4000x - 2000x$$ $$16000 = 2000x$$ $$x = \frac{16000}{2000} = 8$$ months 12. **Months until earning 100,000:** Solve $$y = 100000$$ for each: - Teacher: $$-2000x + 1000 = 100000$$ $$-2000x = 99000$$ $$x = \frac{99000}{-2000} = -49.5$$ (no positive solution) - Marine: $$1000x + 9000 = 100000$$ $$1000x = 91000$$ $$x = 91$$ months - Garbage Collector: $$2000x + 4000 = 100000$$ $$2000x = 96000$$ $$x = 48$$ months - Business Owner: $$4000x - 12000 = 100000$$ $$4000x = 112000$$ $$x = 28$$ months **Summary:** - Business Owner starts negative. - Marine, Teacher, Garbage Collector start positive. - Business Owner has greatest slope $$4000$$. - Marine has least positive slope $$1000$$. - Earnings at 6 months: Marine 15000, Teacher -11000, Garbage Collector 16000, Business Owner 12000. - Earnings at 20 months: Marine 29000, Teacher -39000, Garbage Collector 44000, Business Owner 68000. - Teacher and Garbage Collector never meet at positive time. - Marine and Business Owner meet at 7 months. - Teacher and Marine never meet at positive time. - Garbage Collector and Business Owner meet at 8 months. - Months to 100000: Teacher no positive solution, Marine 91, Garbage Collector 48, Business Owner 28.