1. **Problem statement:** John received quotes from two companies for repairing his oil tank. Company A charges a call out fee plus a rate per hour, and Company B does the same with different rates. We want to find the cost for 4 hours 15 minutes, write formulas for total cost, and find when costs are equal. 2. **Given:** - Company A: call out charge = 100, hourly rate = 60 - Company B: call out charge = 80, hourly rate = 70 - Repair time = 4 hours 15 minutes = 4.25 hours 3. **Calculating cost for 4 hours 15 minutes:** Since charges apply for each hour or part thereof, 4 hours 15 minutes counts as 5 hours. Company A cost = 100 + 60 \times 5 = 100 + 300 = 400 Company B cost = 80 + 70 \times 5 = 80 + 350 = 430 4. **Formulas for total cost for n hours (where n \in \mathbb{N}):** Company A: $$C_A = 100 + 60n$$ Company B: $$C_B = 80 + 70n$$ 5. **Finding when costs are equal:** Set $$C_A = C_B$$ $$100 + 60x = 80 + 70x$$ Subtract 80 from both sides: $$100 - 80 + 60x = 70x$$ $$20 + 60x = 70x$$ Subtract 60x from both sides: $$20 = 70x - 60x$$ $$20 = 10x$$ Divide both sides by 10: $$\cancel{10}x = \frac{20}{\cancel{10}}$$ $$x = 2$$ So, after 2 hours, both companies charge the same amount. **Final answers:** - Cost for 4 hours 15 minutes: Company A = 400, Company B = 430 - Formulas: Company A: $$100 + 60n$$, Company B: $$80 + 70n$$ - Equal cost at $$x = 2$$ hours.