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The problem gives the total cost function \( C(x) = 0.5x^2 + 15x + 5000 \), where \( x \) represents the number of units produced. We are tasked with finding the level of production where the **average cost per unit** is minimized. The average cost per unit is given by \( \text{AC}(x) = \frac{C(x)}{x} \).

### Steps to solve:

1. **Calculate the Average Cost Function**:
   \[
   \text{AC}(x) = \frac{C(x)}{x} = \frac{0.5x^2 + 15x + 5000}{x}
   \]
   Simplifying:
   \[
   \text{AC}(x) = 0.5x + 15 + \frac{5000}{x}
   \]

2. **Minimize the Average Cost**:
   To minimize the average cost function, we need to take its derivative with respect to \( x \), set it equal to zero, and solve for \( x \).

   The derivative of \( \text{AC}(x) \) is:
   \[
   \frac{d}{dx} \left( 0.5x + 15 + \frac{5000}{x} \right) = 0.5 - \frac{5000}{x^2}
   \]

3. **Set the Derivative Equal to Zero**:
   \[
   0.5 - \frac{5000}{x^2} = 0
   \]
   Solving for \( x \):
   \[
   \frac{5000}{x^2} = 0.5
   \]
   \[
   x^2 = \frac{5000}{0.5} = 10000
   \]
   \[
   x = \sqrt{10000} = 100
   \]

Thus, the level of production that minimizes the average cost per unit is \( x = 100 \) units.

Would you like further details or have any questions? 

Here are 5 related questions you might find interesting:
1. How would changes in fixed costs (e.g., 5000) affect the minimum average cost?
2. Can we find the minimum value of the average cost at \( x = 100 \)?
3. What if the quadratic term had a negative coefficient (i.e., if costs decreased as production increased)?
4. How do changes in the coefficient of \( x^2 \) affect the shape of the cost curve?
5. How would this method change if the cost function included logarithmic or exponential terms?

**Tip**: When minimizing functions, always remember to check if your solution corresponds to a minimum by using the second derivative test.

A manufacturer has determined that the total cost C of operating a factory is C = 0.5x^2 + 15x + 5000, where x is the number of units produced. At what level of production will the average cost per unit be minimized?

Find out how to minimize the average cost per unit for a factory with the total cost function C(x) = 0.5x^2 + 15x + 5000. This problem involves key optimization techniques and solving using derivatives. Suitable for high school and college students.

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