Given the cost function $C(x) = 7600 + 21x$ and the revenue function $R(x) = 40x$, let's explore key aspects of these functions.

1. Break-even Point

The break-even point occurs when cost equals revenue:

$C(x) = R(x)$

Substitute the given functions:

$7600 + 21x = 40x$

Solve for $x$:

$7600 = 40x - 21x$

$7600 = 19x$

$x = \frac{7600}{19} \approx 400$

Thus, the break-even quantity is approximately 400 units.

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2. Profit Function

The profit function $P(x)$ is defined as:

$P(x) = R(x) - C(x)$

Substitute the given functions:

$P(x) = 40x - (7600 + 21x)$

Simplify:

$P(x) = 40x - 7600 - 21x$

$P(x) = 19x - 7600$

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3. Key Points to Analyze

• Break-even Revenue: At the break-even point ($x = 400$):

$R(400) = 40 \cdot 400 = 16000$

• Profit: Profit starts being positive when $P(x) > 0$, i.e., $19x - 7600 > 0$:

$19x > 7600$

$x > 400$

Thus, profit is positive when $x > 400$.

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Would you like a detailed analysis of any specific aspect or a graph of these functions? Here are 5 related questions to expand the discussion:

1. How is the profit at specific quantities (e.g., $x = 500$) calculated?
2. What happens to the cost and revenue if the production cost changes?
3. How can we find the maximum profit, if applicable?
4. What is the interpretation of the slope of the cost and revenue functions?
5. How does the fixed cost $7600$ affect the break-even point?

Tip: To analyze profitability effectively, always compare the marginal cost (change in cost per unit) with the marginal revenue (change in revenue per unit).