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The question involves determining the production levels that make the average profit, \(A(x)\), greater than zero. We have the revenue function \(R(x) = -x^2 + 10x\) and the cost function \(C(x) = 4x + 5\). The average profit function is defined as:

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

First, let's calculate the profit function \(P(x)\):

\[
P(x) = R(x) - C(x) = (-x^2 + 10x) - (4x + 5) = -x^2 + 6x - 5
\]

Now, we can find \(A(x)\):

\[
A(x) = \frac{-x^2 + 6x - 5}{x}
\]

To determine where \(A(x) > 0\), we solve for \(x\) when \(P(x) > 0\):

\[
-x^2 + 6x - 5 > 0
\]

We can solve this quadratic inequality using a sign chart after finding the roots. First, we find the roots by setting the equation to zero and solving for \(x\):

\[
-x^2 + 6x - 5 = 0
\]

Let's solve this quadratic equation.The roots of the equation \(-x^2 + 6x - 5 = 0\) are \(x = 1\) and \(x = 5\).

Now, to find the intervals where the profit \(P(x)\) is positive, consider the factorization based on the roots:

\[
P(x) = -(x - 1)(x - 5)
\]

This expression will be positive where the factors \((x-1)\) and \((x-5)\) have opposite signs. We analyze the intervals \(x < 1\), \(1 < x < 5\), and \(x > 5\):

- For \(x < 1\), both factors are negative, making \(P(x)\) positive.
- For \(1 < x < 5\), one factor is positive and the other is negative, making \(P(x)\) negative.
- For \(x > 5\), both factors are positive, making \(P(x)\) positive.

Thus, \(P(x) > 0\) when \(x < 1\) or \(x > 5\). However, since production cannot be negative, we only consider \(x > 5\).

**Conclusion:** The production levels that make the average profit \(A(x)\) greater than zero are when \(x > 5\) thousands of shoes.

Would you like further details or have any other questions? Here are some related topics you might consider:

1. How does the average profit change as production increases beyond 5,000 shoes?
2. What is the maximum average profit and at what production level does it occur?
3. How does modifying the cost or revenue functions affect the break-even points?
4. What is the effect of changing the fixed cost in the cost function on the profit?
5. How would adding a volume discount to the revenue function affect the analysis?
6. Analyze the behavior of the profit function for large values of \(x\).
7. Determine the total revenue and total cost at the production level where average profit is zero.
8. Discuss the implications of a negative average profit at certain production levels.
9. Tip: When assessing business production models, it's crucial to consider both fixed and variable costs to understand overall profitability thresholds.

Learn how to maximize average profit using revenue and cost functions. Solve the quadratic inequality to determine production levels for positive average profit. Suitable for advanced high school students and those interested in business analytics.

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