Calculating Profit: A Retail Math Problem

Hey everyone, let's dive into a cool math problem! So, we've got an online retail company, and we're trying to figure out when they'll finally start making some money. We're given two key functions here: the cost function and the revenue function. Our goal is to figure out how many days it'll take for the company to become profitable. It's kinda like a real-world application of algebra, which makes it super interesting, right? This is something that you might actually encounter when you are starting a business. Understanding how costs, revenues, and profits relate is crucial to running a successful business.

Understanding the Problem: Costs, Revenue, and Profit

Okay, before we jump into the calculations, let's break down the basics. The problem gives us two important pieces of information: the cost function, C(x) = 2x² - 20x + 80, and the revenue function, R(x) = -x² + 5x + 58. Each of these functions is a quadratic function, which means they'll create parabolas when graphed. The variable 'x' represents the number of days. The cost function, C(x), tells us the total cost the company incurs for a given number of days. This could include things like the cost of goods sold, marketing expenses, and other operational costs. The revenue function, R(x), represents the total income the company generates from sales over a given number of days. Profit, the ultimate goal, is the difference between revenue and cost. To make a profit, the company's revenue needs to be higher than its costs. So, our core idea here is: Profit = Revenue - Cost. In other words, to find the profit function, we need to subtract the cost function from the revenue function. This is the first step in finding out when the company starts making money.

Finding the Profit Function

Now that we understand the context, let's calculate the profit function. We know that profit is revenue minus cost, so we'll subtract the cost function from the revenue function. Here's how it looks:

P(x) = R(x) - C(x)

P(x) = (-x² + 5x + 58) - (2x² - 20x + 80)

When subtracting the cost function, we need to be very careful with the signs. Every term in the cost function must be subtracted. So, distribute that negative sign carefully.

P(x) = -x² + 5x + 58 - 2x² + 20x - 80

Now, let's combine like terms. This simplifies our equation and gets us closer to finding out the profit function.

P(x) = (-x² - 2x²) + (5x + 20x) + (58 - 80)

P(x) = -3x² + 25x - 22

So, our profit function is P(x) = -3x² + 25x - 22. This new function tells us the profit or loss the company experiences on any given day.

Determining the Break-Even Points

Now that we have the profit function, we need to figure out when the company starts making a profit. This means we need to find out when P(x) > 0. However, the easiest way to determine this is to find the break-even points, where P(x) = 0. The company breaks even when its revenue equals its costs. So we need to solve the quadratic equation: -3x² + 25x - 22 = 0. We can solve this using the quadratic formula. For a quadratic equation of the form ax² + bx + c = 0, the quadratic formula is:

x = (-b ± √(b² - 4ac)) / 2a

In our profit function, a = -3, b = 25, and c = -22. Let's plug these values into the quadratic formula:

x = (-25 ± √(25² - 4(-3)(-22))) / (2(-3))*

x = (-25 ± √(625 - 264)) / -6

x = (-25 ± √361) / -6

x = (-25 ± 19) / -6

So now we have two possible solutions:

x₁ = (-25 + 19) / -6 = -6 / -6 = 1

x₂ = (-25 - 19) / -6 = -44 / -6 = 7.33

We get two values for x: approximately 1 and 7.33. These are the points where the profit is zero. The company breaks even on day 1 and around day 7.33. To figure out when the company makes a profit, we now need to test the regions between and outside these points.

Finding the Profitable Region

So, we've calculated the break-even points, now let's determine the days where the company will make a profit. The profit function forms a parabola that opens downward (because the coefficient of x² is negative). This means the function is positive (profit is made) between the roots. We know that the company breaks even at approximately 1 day and approximately 7.33 days. The company makes a profit between these two points. We can test this by plugging a value between 1 and 7.33 into the profit function P(x) = -3x² + 25x - 22. Let's try x = 4:

P(4) = -3(4²) + 25(4) - 22 = -48 + 100 - 22 = 30

Since P(4) = 30, and 30 > 0, the company is making a profit between the break-even points! The company earns a profit when the number of days is between 1 and approximately 7.33 days. We're looking for when the company starts to make a profit, so the answer to our problem is the smallest integer value greater than 1, which would be on day 2. But, since the question is how many days will it take to earn a profit, we need to consider the values between the break-even points.

Selecting the Correct Answer

Now that we've analyzed the profit function and identified the break-even points, let's look at the given answer choices:

• A. 32: This value is far outside the profitable range (1 to 7.33). So, the company wouldn't start earning a profit by day 32. This is incorrect.
• B. 30: Similar to A, this value is too large. The company's profit would likely be negative (a loss) at this point. Incorrect.
• C. 5: This value falls within the profitable range (between 1 and 7.33 days). Therefore, this is the correct option.
• D. 1: This is a break-even point, not a point of profit. Incorrect.

Therefore, the correct answer is C. 5.

Final Thoughts

And there you have it, guys! We've successfully used math to solve a real-world business problem. This demonstrates how understanding profit, revenue, and cost functions can help businesses make informed decisions. Remember, the key is to take the time to understand the problem, break it down into manageable steps, and apply the appropriate mathematical tools. Keep practicing, and you'll ace these kinds of problems every time! If you have any further questions, feel free to ask! This question highlights how we can apply our math skills to understand and analyze financial scenarios. It's a pretty cool blend of algebra and basic economics. Remember, understanding these concepts is crucial for anyone looking to start a business, manage finances, or just make smart decisions with money in general. So keep exploring, keep learning, and always question how math can explain the world around you. Feel free to reach out if you have more questions or want to discuss this further. Happy calculating!