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.