Calculate Profit: Shirt Sales Example

Hey guys! Ever wondered how businesses figure out their profits? It's actually a pretty cool process involving some basic math. Let's dive into a scenario where we're calculating the profit from selling shirts. We'll break down the revenue, costs, and ultimately, the profit. So, grab your thinking caps, and let's get started!

Understanding Revenue, Cost, and Profit

Before we jump into the calculations, let's quickly define the key terms we'll be using:

• Revenue (r(x)): This is the total amount of money earned from selling a certain number of items. In our case, it's the money made from selling shirts.
• Cost (c(x)): This is the total expense incurred in producing or acquiring the items to be sold. For us, it's the cost of buying the shirts.
• Profit (p(x)): This is the difference between the revenue and the cost. It's the money left over after all expenses are paid. Basically, it's what you get to keep! So, the profit is what we really care about in the end.

In mathematical terms, the relationship is expressed as:

• p(x) = r(x) - c(x)

This simple equation is the foundation of understanding profitability in any business. Now, let's apply this to our shirt-selling scenario.

Diving into the Shirt-Selling Scenario

Let's imagine we're running a small business selling cool, custom-designed shirts. We need to figure out how much profit we're making for every shirt we sell. We're given the following information:

• The revenue from selling x shirts is given by the function r(x) = 15x. This means that for every shirt we sell, we earn $15. Makes sense, right? If we sell one shirt, we get $15. If we sell ten shirts, we get 15 * 10 = $150.
• The cost of buying x shirts is given by the function c(x) = 7x + 20. This tells us that each shirt costs us $7, and we have a fixed cost of $20 (maybe for printing the designs or setting up our online store). So, even if we don't sell any shirts, we still have to pay that $20. The $7 per shirt covers the material and production costs.
• The profit from selling x shirts is given by the function p(x) = r(x) - c(x). This is the equation we talked about earlier. It's the revenue minus the cost.

Our goal is to find the actual function for p(x). We want to know exactly how much profit we make based on the number of shirts we sell.

Calculating the Profit Function p(x)

Okay, so we know that profit is revenue minus cost. We have the functions for revenue and cost. Now, let's plug them into the profit equation and see what we get:

• p(x) = r(x) - c(x)

Substitute the given functions for r(x) and c(x):

• p(x) = (15x) - (7x + 20)

Now, we need to simplify this expression. Remember to distribute the negative sign to both terms inside the parentheses:

• p(x) = 15x - 7x - 20

Combine the like terms (the terms with x):

• p(x) = 8x - 20

And there we have it! The profit function, p(x), is 8x - 20. This function tells us the profit we make from selling x shirts. For every shirt we sell, we make $8, but we need to subtract the initial $20 cost.

Interpreting the Profit Function

Let's break down what this profit function, p(x) = 8x - 20, actually means in the context of our shirt-selling business.

• The 8x part tells us that for each shirt we sell, we make $8 in profit. This is the marginal profit – the additional profit we earn for each additional shirt sold. Think of it as the profit per shirt.
• The -20 part represents our initial fixed cost. This is the cost we incur even if we don't sell any shirts. It could be for things like setting up our online store, designing the shirts, or any other upfront expenses. This is sometimes called the startup cost or fixed overhead. So, before we start making any real profit, we need to sell enough shirts to cover this initial $20.

To illustrate, let's look at a few examples:

• If we sell 0 shirts (x = 0), our profit is p(0) = 8(0) - 20 = -$20. We have a loss of $20 because we haven't sold anything to cover our initial cost.
• If we sell 1 shirt (x = 1), our profit is p(1) = 8(1) - 20 = -$12. We're still at a loss, but we're getting closer to breaking even.
• If we sell 3 shirts (x = 3), our profit is p(3) = 8(3) - 20 = $4. Now we're finally making a profit!

Identifying the Correct Answer

Now that we've calculated the profit function, let's look back at the multiple-choice options:

• A. p(x) = 22x + 20
• B. p(x) = 8x - 20
• C. p(x) = 8x + 20
• D. p'(x) = 22x - 20

We found that p(x) = 8x - 20, so the correct answer is B. Option D is incorrect because it represents the derivative of a profit function, which is a concept in calculus. We're just looking for the profit function itself here.

Key Takeaways About Calculate Profit

Calculating profit is a fundamental concept in business and economics. Understanding the relationship between revenue, cost, and profit allows us to make informed decisions about pricing, production, and overall business strategy. Let's recap the key takeaways from our shirt-selling example:

• Profit is revenue minus cost: p(x) = r(x) - c(x)
• Revenue is the total income from sales.
• Cost includes both fixed costs (like setup fees) and variable costs (like the cost per item).
• The profit function tells us the profit for any given number of items sold.
• Analyzing the profit function helps us understand our break-even point (the number of items we need to sell to cover our costs) and our potential for profitability.

By understanding these concepts, you're well on your way to making smart financial decisions, whether you're selling shirts, lemonade, or anything else! Remember, knowing your numbers is crucial for success in any business venture.

Conclusion

So, there you have it! We've successfully calculated the profit function for our shirt-selling business. By understanding the relationship between revenue, cost, and profit, we can make informed decisions and maximize our earnings. I hope this breakdown has been helpful and has demystified the process of calculating profit. Keep these concepts in mind, and you'll be well-equipped to tackle any business scenario that comes your way. Now go out there and start making some profit!