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We can break down the problem step-by-step to find the profit function $P(x)$ and the break-even point for Stinky's perfume business.

### Step 1: Define the Profit Function $ P(x) $

Stinky's profit is the difference between his revenue and his costs. Let's break these down:

1. **Revenue**: Stinky sells the perfume for $900 per gram. Therefore, for $x$ grams of perfume, the revenue is:
   \[
   \text{Revenue} = 900x
   \]

2. **Costs**:
   - **Overheads**: These are fixed costs, amounting to $20,000 per day.
   - **Perfume cost**: He buys imitation perfume at $20 per gram. So for $x$ grams, the cost of the perfume is:
     \[
     20x
     \]
   - **Transportation cost**: He pays an additional $30 per 100 grams, which translates to $0.30 per gram. So for $x$ grams, the transportation cost is:
     \[
     0.30x
     \]
   - Total variable costs (for perfume and transportation):
     \[
     \text{Total variable cost} = (20 + 0.30)x = 20.30x
     \]
   - Therefore, his total costs are:
     \[
     \text{Total cost} = 20,000 + 20.30x
     \]

3. **Profit function**: Profit is revenue minus total costs:
   \[
   P(x) = 900x - (20,000 + 20.30x)
   \]
   Simplifying this:
   \[
   P(x) = 900x - 20.30x - 20,000
   \]
   \[
   P(x) = 879.70x - 20,000
   \]

### Step 2: Calculate the Break-Even Point

To find the break-even point, we need to set the profit function equal to zero:
\[
P(x) = 0 = 879.70x - 20,000
\]
Solve for $x$:
\[
879.70x = 20,000
\]
\[
x = \frac{20,000}{879.70} \approx 22.74
\]

So, Stinky needs to handle approximately **22.74 grams** of perfume per day to break even.

### Summary:

- The profit function is: 
  \[
  P(x) = 879.70x - 20,000
  \]
- Stinky needs to sell about **22.74 grams** of perfume per day to break even.

Would you like further clarification or details on any step? Here are some additional related questions for deeper understanding:

1. How would the profit function change if the overhead costs increased by 10%?
2. What would happen to the break-even point if the selling price per gram was reduced to $850?
3. How would transportation costs affect the profit if it increased to $40 per 100 grams?
4. What if Stinky could sell the perfume at $1,000 per gram instead of $900—how would that affect his profit and break-even point?
5. What is the maximum profit Stinky can earn if he sells 50 grams per day?

**Tip**: Always keep track of both fixed and variable costs to ensure accurate profit calculations.

The organized crime boss and perfume king Butch (Stinky) Rose has daily overheads (bribes to corrupt officials, etc.) amounting to $20,000 per day. He buys imitation French perfume at $20 per gram, pays an additional $30 per 100 grams for transportation, and sells the perfume via his street thugs for $900 per gram. Specify Stinky's profit function P(x), where x is the quantity (in grams) of perfume he buys and sells. Calculate how much perfume (in grams) should pass through Stinky's hands per day to break even.

Learn how to calculate the profit function and break-even point for a perfume business run by Stinky Rose, with daily overheads of $20,000 and selling perfume at $900 per gram. A detailed step-by-step algebraic breakdown suitable for high school students.

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