Final answer:
The plumber's function has a greater initial value of $50 compared to Aaron's Word Processing Service, which has an initial value of $31.50. For production methods based on labor and capital costs, the best method is determined by calculating the total cost for each and comparing them.
Step-by-step explanation:
To determine which of the following functions has the greater initial value, we'll look at the functions described. For the scenario with the plumber, the initial value is the charge to come out, which is $50. The function representing the plumber's total charge would be represented as y = 50 + 30x, where y is the total cost and x is the number of hours of labor.
In the case of Aaron's Word Processing Service, the initial charge is $31.50, and the charge per hour is $32. Therefore, the function for the total cost of Aaron's service is y = 31.50 + 32x.
Comparing the initial values of the two functions (also known as the y-intercepts), we can see that the plumber's initial charge ($50) is greater than Aaron's initial charge ($31.50).
When it comes to the question about the best production method based on the cost of labor and capital, we would calculate the total cost for each method, considering the changes in labor costs from $100 to $200 per unit. For example, the total cost for Method 1 at $100 per labor unit is (50 × 100) + (10 × 400) = $9000, and it would change accordingly if the labor cost increases.