The organized crime boss and perfume king Butch (Stinky) Rose has daily overheads (bribes to corrupt officials, motel photographers, wages for hit men, explosives, etc.) amounting to $20,000 per day. On the other hand, he has a substantial income from his counterfeit perfume racket: he buys imitation French perfume (Chanel No. 22.5) at $20 per gram, pays an additional $40 per 100 grams for transportation, and sells the perfume via his street thugs for $700 per gram. Specify Stinky's profit function, P(x), where x is the quantity (in grams) of perfume he buys and sells.
P(x) = 
  
Correct: Your answer is correct.
Calculate how much perfume (in grams) should pass through Stinky's hands per day in order that he break even. (Round your answer to two decimal places.)
 
Incorrect: Your answer is incorrect.
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We are tasked with finding Stinky's profit function $ P(x) $, where $ x $ is the quantity of perfume (in grams) that he buys and sells. Additionally, we want to determine how much perfume must pass through his hands per day for him to break even.

### Step 1: Define the total costs and revenues.

- **Daily overheads:** Stinky incurs a fixed cost of \$20,000 per day, regardless of how much perfume he buys or sells.
- **Cost per gram of perfume:**
  - He buys imitation French perfume at \$20 per gram.
  - He pays an additional \$40 per 100 grams for transportation, which is $ \frac{40}{100} = 0.40 $ per gram.
  - Therefore, his total cost per gram is $ 20 + 0.40 = 20.40 $ dollars per gram.
- **Revenue per gram of perfume:** He sells the perfume for \$700 per gram.

### Step 2: Express the total cost and revenue functions.

- **Total cost function:** The total cost for buying and transporting $ x $ grams of perfume, plus the fixed overheads, is:
  \[
  C(x) = (20.40 \times x) + 20,000
  \]
- **Revenue function:** The revenue from selling $ x $ grams of perfume at \$700 per gram is:
  \[
  R(x) = 700 \times x
  \]

### Step 3: Define the profit function $ P(x) $.

The profit function is given by the revenue minus the total cost:
\[
P(x) = R(x) - C(x)
\]
Substitute the expressions for $ R(x) $ and $ C(x) $:
\[
P(x) = 700x - \left(20.40x + 20,000\right)
\]
Simplify the expression:
\[
P(x) = 700x - 20.40x - 20,000
\]
\[
P(x) = 679.60x - 20,000
\]

### Step 4: Find the break-even point.

At the break-even point, the profit $ P(x) $ is zero. So, set $ P(x) = 0 $ and solve for $ x $:
\[
0 = 679.60x - 20,000
\]
\[
679.60x = 20,000
\]
\[
x = \frac{20,000}{679.60}
\]
\[
x \approx 29.43 \text{ grams}
\]

### Final Answer:
Stinky must sell approximately **29.43 grams** of perfume per day to break even.

Would you like more details or have any questions about the process?

### Related Questions:
1. How would the break-even point change if transportation costs doubled?
2. What would happen to Stinky’s profit if he increases the price to \$800 per gram?
3. How much profit would Stinky make if he sold 50 grams of perfume per day?
4. What would the total daily cost be if Stinky sold 100 grams of perfume?
5. How would adding a 10% sales tax on perfume affect Stinky’s break-even point?

### Tip:
To quickly solve break-even problems, remember that the break-even point occurs when total revenue equals total costs.

The organized crime boss and perfume king Butch (Stinky) Rose has daily overheads (bribes to corrupt officials, motel photographers, wages for hit men, explosives, etc.) amounting to $20,000 per day. On the other hand, he has a substantial income from his counterfeit perfume racket: he buys imitation French perfume (Chanel No. 22.5) at $20 per gram, pays an additional $40 per 100 grams for transportation, and sells the perfume via his street thugs for $700 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 in order that he break even.

Learn how to calculate the break-even point for a perfume racket. This problem involves finding the profit function P(x), where x is the amount of perfume sold, and determining how many grams are needed to break even. The solution includes key algebraic concepts such as linear functions and cost analysis, suitable for Grades 9-12.

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