First, identify the type of problem you are being asked to solve. Is it a constrained optimization problem requiring a Lagrangian setup? Would solving the problem requiring using a definite integral? Is the problem a straightforward optimization problem like maximizing profits or revenues, or minimizing costs? Please show every detail you can. I need the math written out with as much detail as possible.

A new restaurant has recently opened in your community and it is a huge success with an average of 520-540 customers daily. The owner has had to hire additional workers at $20 per hour to help with the dishwashing as each customer uses on average 5 dishes. The owner has investigated purchasing a large dishwashing machine which costs $260,000, but is uncertain as to whether it is worth it as this would be a huge expenditure for a new business owner and the purchase feels a bit daunting. The machine has an expected useful life of 10 years. The owner has hired you to determine whether the investment would be wise. Restaurant Facts 1. There are 3 restaurant shifts (breakfast, lunch, dinner) where each shift lasts 3 hours 2. On average, 900 dishes need washing per shift 3. The restaurant is open 360 days per year Market conditions: 1. Wages are expected to increase 1% each year 2. Interest rates are 8% 3. Interest is compounded quarterly Manual labor for dishwashing 1. For each of the 3 shifts, dishwashers need to setup and get ready for the rush. Setup takes 1 hour per shift. 2. A dishwasher can wash 5 dishes per minute. Machine labor requirements 1. The machine needs to be prepared to run for the day and it takes 1 hour. The owner needs to hire a maintenance worker who can maintain the machine. This worker would cost the owner $40 per day. 2. The labor requirements to load and empty the machine are 2 hours per shift