We are given the cost function $C(n) = 75n^2 - 1800n + 60,000$, where $C(n)$ represents the cost of producing $n$ thousand items. To determine the number of items that minimize the cost, we need to find the value of $n$ that minimizes this quadratic function.

Step 1: Identify the structure of the quadratic function

The given function is a quadratic function of the form: $C(n) = 75n^2 - 1800n + 60,000$ Quadratic functions are parabolas, and since the coefficient of $n^2$ (i.e., 75) is positive, the parabola opens upwards. This means the function has a minimum point, which occurs at the vertex of the parabola.

Step 2: Use the formula for the vertex

For a quadratic function $ax^2 + bx + c$, the vertex occurs at: $n = -\frac{b}{2a}$ In this case, $a = 75$ and $b = -1800$.

Step 3: Calculate the value of $n$

Substituting the values of $a$ and $b$ into the vertex formula: $n = -\frac{-1800}{2(75)} = \frac{1800}{150} = 12$

Thus, the number of items that should be produced to minimize the cost is 12 thousand items.

Would you like a detailed explanation of the cost calculation at $n = 12$ or any other clarification?

Follow-up Questions:

1. What is the minimum cost at $n = 12$?
2. How do changes in the coefficient of $n^2$ affect the shape of the cost function?
3. What is the significance of the constant term $60,000$ in this function?
4. How would the cost function change if storage costs were reduced?
5. Can this model be applied to real-world business scenarios, and how might you adjust it?

Tip: When analyzing quadratic functions in business, the vertex formula $-\frac{b}{2a}$ is a quick way to find optimal solutions!