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 $25,000 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 $30 per gram, pays an additional $40 per 100 grams for transportation, and sells the perfume via his street thugs for $800 per gram. Specify Stinky's profit function, P(x), where x is the quantity (in grams) of perfume he buys and sells.
P(x) = 
  
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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.)


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

Let's break down the problem in terms of costs and revenue for Stinky's perfume business:

1. **Income per gram:** Stinky sells each gram of perfume for $800.
   - **Total Revenue**: $ 800x $, where $ x $ is the quantity (in grams) of perfume he sells.

2. **Cost per gram:**
   - He buys imitation perfume at $30 per gram.
   - Transportation costs are $40 for every 100 grams, which amounts to $0.40 per gram.
   - Therefore, the **total cost per gram** is $ 30 + 0.40 = 30.40 $.
   - **Total Costs for perfume**: $ 30.40x $.

3. **Daily Overheads:** Stinky's daily expenses are $25,000.

Thus, the profit function $ P(x) $ is the difference between total revenue and total costs, which include the overhead expenses:

\[
P(x) = 800x - (30.40x + 25,000)
\]

Simplifying the equation:

\[
P(x) = 800x - 30.40x - 25,000
\]
\[
P(x) = 769.60x - 25,000
\]

### Step 2: Break-Even Condition

To break even, the profit should be zero:

\[
P(x) = 0
\]
\[
769.60x - 25,000 = 0
\]

### Step 3: Solve for $ x $

Now, solve for $ x $:

\[
769.60x = 25,000
\]
\[
x = \frac{25,000}{769.60}
\]

Let's calculate the value of $ x $.The amount of perfume that must pass through Stinky's hands per day for him to break even is approximately **32.48 grams**.

Would you like more details or have any further questions?

Here are 5 related questions to expand on this:

1. What would Stinky's profit be if he sold 50 grams of perfume per day?
2. How would the profit function change if transportation costs increased by 10%?
3. How much more perfume would he need to sell if his daily overhead increased to $30,000?
4. What is the maximum daily loss Stinky could incur if he failed to sell any perfume?
5. If he reduced the sale price to $700 per gram, how much would he need to sell to break even?

**Tip:** In profit functions, understanding how to factor fixed and variable costs can help in forecasting financial outcomes.

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 $25,000 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 $30 per gram, pays an additional $40 per 100 grams for transportation, and sells the perfume via his street thugs for $800 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 define and solve a profit function for a counterfeit perfume business with daily overheads and transportation costs. Find out how much perfume must be sold daily to break even, using linear equations. Suitable for students in Grades 10-12.

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