The problem you have uploaded involves finding the minimum total cost by determining the number of units to be produced on two machines, A and B, given their respective cost functions and a constraint on the total number of units produced.

Step-by-step breakdown:

1. Cost Function of Machine A: $C_A(x) = 30 + \frac{x^2}{6}$ where $x$ is the number of units produced on machine A.

2. Cost Function of Machine B: $C_B(y) = 240 + \frac{y^3}{9}$ where $y$ is the number of units produced on machine B.

3. Total Cost Function: The total cost is given by: $C(x, y) = C_A(x) + C_B(y) = \left(30 + \frac{x^2}{6}\right) + \left(240 + \frac{y^3}{9}\right)$

4. Constraint: The total number of units required is $x + y = 12,210$.

Goal:

To minimize the total cost $C(x, y)$, we need to determine how many units $x$ should be produced on machine A and how many units $y$ should be produced on machine B.

Solving the problem:

To solve this, we need to apply optimization techniques. First, substitute $y = 12,210 - x$ into the total cost function, and then differentiate the cost with respect to $x$ and set the derivative equal to zero to find the minimum.

Let me calculate that for you.The minimum total cost is achieved when 12,100 units are produced on machine A and 110 units are produced on machine B.

Would you like further details on the steps taken to solve this?

Here are 5 related questions to explore:

1. How is the derivative used to minimize a function?
2. Why do we substitute $y = 12,210 - x$ into the total cost equation?
3. What are the conditions for a function to have a minimum?
4. Can the same technique be used if the cost functions were more complex?
5. How do different constraints affect the optimization process?

Tip: When minimizing or maximizing a function with constraints, substituting the constraint into the objective function helps reduce the number of variables to manage.