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) =

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.)
g

Let's start by defining the profit function $ P(x) $, where $ x $ represents the amount of perfume in grams that Butch (Stinky) Rose buys and sells daily.

### Costs:
1. **Cost of buying perfume:** He buys imitation perfume at $30 per gram, so the cost for $ x $ grams is:
   \[
   30x
   \]

2. **Transportation cost:** He pays an additional $40 per 100 grams for transportation, which is equivalent to $0.40 per gram. So the transportation cost for $ x $ grams is:
   \[
   0.4x
   \]

3. **Daily overheads:** His fixed daily overhead is $25,000.

Thus, the total daily cost for $ x $ grams of perfume is:
\[
\text{Total cost} = 30x + 0.4x + 25,000 = 30.4x + 25,000
\]

### Revenue:
1. **Revenue from selling perfume:** He sells the perfume at $800 per gram, so the revenue for $ x $ grams is:
   \[
   800x
   \]

### Profit function:
The profit function $ P(x) $ is the revenue minus the total costs:
\[
P(x) = 800x - (30.4x + 25,000)
\]
Simplifying:
\[
P(x) = 800x - 30.4x - 25,000
\]
\[
P(x) = 769.6x - 25,000
\]

### Breaking even:
To find the break-even point, set the profit function $ P(x) = 0 $ and solve for $ x $:
\[
0 = 769.6x - 25,000
\]
\[
769.6x = 25,000
\]
\[
x = \frac{25,000}{769.6} \approx 32.49
\]

So, Stinky needs to buy and sell approximately **32.49 grams** of perfume per day to break even.

Would you like further details on any step?

Here are some related questions for deeper understanding:
1. What would be Stinky's daily profit if he sold 40 grams of perfume?
2. How would an increase in transportation costs affect the break-even point?
3. What is Stinky's profit for selling 50 grams of perfume per day?
4. How would a decrease in perfume price to $25 per gram change his profit function?
5. What if the overheads increased by $5,000 daily? How would this affect his business?

**Tip:** Always analyze both fixed and variable costs when determining a business's profit function.

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.

Explore how to calculate the profit function for Butch (Stinky) Rose's perfume business. Learn to model daily costs, revenue, and determine the break-even point using algebraic formulas. Solve the break-even problem to find that Stinky needs to sell approximately 32.49 grams of perfume per day. This problem covers linear equations, profit functions, and break-even analysis, suitable for Grades 9-12.

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