The managers of a business are examining costs. It is more cost-effective for them
to produce more items. However, if too many items are produced, their costs
will rise because of factors such as storage and overstock. Suppose that
they model the cost, C, of producing
n thousand items with the function
C(n) = 75n2 - 1800n + 60 000. Determine the number of items produced that will minimize their costs.

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!

The managers of a business are examining costs. It is more cost-effective for them to produce more items. However, if too many items are produced, their costs will rise because of factors such as storage and overstock. Suppose that they model the cost, C, of producing n thousand items with the function C(n) = 75n^2 - 1800n + 60,000. Determine the number of items produced that will minimize their costs.

Learn how to minimize production costs using the quadratic function C(n) = 75n^2 - 1800n + 60,000. This optimization problem demonstrates the application of algebra and quadratic functions to real-world business scenarios. Suitable for students in Grades 10-12.

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