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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.

Each unit of a product can be made on either machine A or machine B. The cost functions for each machine are given. The goal is to find the number of units to produce on each machine to minimize the total cost, given that a total of 12,210 units are required.

This problem involves finding the optimal number of units to produce on two machines, A and B, to minimize total production costs. The solution uses cost functions and optimization techniques, including differentiation, with a total of 12,210 units required.

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