To solve the problem, let's address each part step by step:

(a) Find the equation of the profit function $P(x)$.

The profit function is calculated as:

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

Given:

• The total cost function $C(x) = 85x + 9504$
• The total revenue function $R(x) = 382x$

Now, substitute these into the profit function:

$P(x) = 382x - (85x + 9504)$

Simplify the equation:

$P(x) = 382x - 85x - 9504$

$P(x) = 297x - 9504$

So, the equation of the profit function is:

$P(x) = 297x - 9504$

(b) Find the profit on 253 items.

To find the profit when $x = 253$, substitute $x = 253$ into the profit function:

$P(253) = 297(253) - 9504$

Now, calculate the value:

$P(253) = 75141 - 9504 = 65637$

So, the profit on 253 items is:

$P(253) = 65637 \text{ dollars}$

(c) How many items must be sold to avoid losing money?

To avoid losing money, the profit must be at least 0, i.e., $P(x) \geq 0$. Set the profit function to 0 and solve for $x$:

$297x - 9504 = 0$

Solve for $x$:

$297x = 9504$

$x = \frac{9504}{297} = 32$

So, the manufacturer must sell at least 32 items to avoid losing money.

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If you have any questions or need more details, feel free to ask! Here are some related questions:

1. How would the profit function change if the total cost per item increases to $100?
2. What is the break-even point for this scenario?
3. What is the revenue when selling 253 items?
4. How would the profit change if the fixed cost increases to $10,000?
5. How does the total cost function compare to a scenario with a different fixed cost?

Tip: To better understand profit calculations, always keep track of fixed costs and variable costs separately.