When no more than 110 units are​ produced, the cost of producing y units is given by​ C(x).
​C(x)equals0.2 x cubed minus 24 x squared plus 1511 x plus 33 comma 436
How many units should be produced in order to have the lowest possible average​ cost?

The goal is to minimize the **average cost** of producing \( x \) units, which is the total cost function \( C(x) \) divided by \( x \). The average cost function, \( A(x) \), is given by:

\[
A(x) = \frac{C(x)}{x}
\]

where \( C(x) = 0.2x^3 - 24x^2 + 1511x + 33,436 \). To find the minimum of the average cost, we'll follow these steps:

### 1. Find the Average Cost Function:
\[
A(x) = \frac{0.2x^3 - 24x^2 + 1511x + 33,436}{x}
\]

Simplifying:
\[
A(x) = 0.2x^2 - 24x + 1511 + \frac{33,436}{x}
\]

### 2. Differentiate the Average Cost Function:
To minimize the average cost, take the derivative of \( A(x) \) with respect to \( x \) and set it equal to zero:

\[
A'(x) = \frac{d}{dx} \left( 0.2x^2 - 24x + 1511 + \frac{33,436}{x} \right)
\]

\[
A'(x) = 0.4x - 24 - \frac{33,436}{x^2}
\]

### 3. Set the Derivative Equal to Zero:
\[
0.4x - 24 - \frac{33,436}{x^2} = 0
\]

### 4. Solve for \( x \):
To solve this equation, multiply through by \( x^2 \) to eliminate the fraction:

\[
0.4x^3 - 24x^2 - 33,436 = 0
\]

We can solve this equation either numerically or using approximation methods to find the value of \( x \) that minimizes the average cost.

Let me compute the exact value for \( x \).The real solution to the equation is approximately:

\[
x \approx 74.9
\]

This means that the lowest possible average cost occurs when about **75 units** are produced.

Would you like further details on this, or do you have any questions?

Here are five related questions for deeper exploration:
1. How can the total cost function \( C(x) \) be interpreted in practical terms?
2. What happens to the average cost if production exceeds 110 units?
3. How is the derivative of the average cost function related to optimization?
4. What is the significance of the other complex solutions?
5. How could this model be adjusted for different cost behaviors?

**Tip:** Always verify if the units calculated align with any constraints given in the problem to ensure the solution is realistic.

When no more than 110 units are produced, the cost of producing y units is given by C(x). C(x) = 0.2x^3 - 24x^2 + 1511x + 33,436. How many units should be produced in order to have the lowest possible average cost?

Discover how to minimize the average cost of production for up to 110 units, using calculus and optimization techniques. This example explores a cost function C(x) = 0.2x^3 - 24x^2 + 1511x + 33,436, suitable for advanced high school or college-level students.

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