The total cost for a product is C(x) = 100(.02x+4)^3 dollars, where represents the number of hundreds of units produced. Find the minimum average cost

To find the minimum average cost for the product, we'll follow these steps:

1. **Identify the average cost function,** \( A(x) \), by dividing the total cost function \( C(x) \) by \( x \):
   \[
   A(x) = \frac{C(x)}{x} = \frac{100(0.02x + 4)^3}{x}
   \]

2. **Simplify the average cost function** by rewriting the expression:
   \[
   A(x) = 100 \cdot \frac{(0.02x + 4)^3}{x}
   \]

3. **Differentiate** \( A(x) \) with respect to \( x \) to find \( A'(x) \), which helps us locate the minimum points:
   \[
   A'(x) = \frac{d}{dx} \left(100 \cdot \frac{(0.02x + 4)^3}{x}\right)
   \]

4. **Set \( A'(x) = 0 \)** and solve for \( x \) to find the critical points.

5. **Evaluate** these critical points to determine the minimum average cost.

I'll go through the differentiation and solving steps to find the critical points.The derivative \( A'(x) \) is:

\[
A'(x) = \frac{96(0.005x + 1)^2}{x} - \frac{6400(0.005x + 1)^3}{x^2}
\]

Setting \( A'(x) = 0 \), we find critical points at \( x = -200 \) and \( x = 100 \).

Since \( x \) represents the number of hundreds of units produced, only \( x = 100 \) is relevant in this context.

6. **Evaluate \( A(x) \) at \( x = 100 \)** to find the minimum average cost.The minimum average cost occurs at \( x = 100 \) (or 10,000 units produced), and the minimum average cost is:

\[
A(100) = 216 \text{ dollars}
\]

Would you like further details or have questions on this process?

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### Related Questions

1. How would the minimum average cost change if the cost function had a different coefficient?
2. What is the significance of finding the minimum average cost in production?
3. How do you verify if a critical point indeed represents a minimum?
4. Can this approach be applied if \( C(x) \) were a different function, like a polynomial?
5. What happens to the average cost as \( x \) approaches very large values?

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**Tip:** For cost functions, minimizing average cost can help determine the most efficient production level.

The total cost for a product is C(x) = 100(.02x+4)^3 dollars, where x represents the number of hundreds of units produced. Find the minimum average cost.

Learn how to find the minimum average cost for a product with the cost function C(x) = 100(.02x + 4)^3, where x is the number of hundreds of units produced. This solution involves calculus concepts like average cost, optimization, and differentiation, providing a step-by-step approach. Ideal for undergraduate calculus students.

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