Maximize Revenue: Backpack Sales & Pricing Strategies

Hey there, math enthusiasts and business-minded individuals! Today, we're diving into a fascinating real-world problem: how to maximize revenue for a store selling backpacks. We'll be using a mathematical model to figure out the perfect price point to boost those sales figures. Buckle up, because we're about to explore the sweet spot of pricing! This is a core concept that blends mathematics with business strategy, making it super relevant for anyone interested in understanding how pricing decisions impact profit. We will break down the problem step-by-step, ensuring you grasp the underlying principles. Get ready to flex your math muscles and learn how to make data-driven decisions!

Understanding the Backpack Sales Model

So, let's get down to business. A store is using the expression -2p + 50 to model the number of backpacks it sells each day. Here, p represents the price of a backpack. The price can range from $9 to $15. This model helps us understand how the number of backpacks sold changes as the price fluctuates. Intuitively, we know that as the price of a backpack increases, the number of backpacks sold is likely to decrease. The given expression captures this relationship, quantifying the change in sales for every dollar increase in price. This means we are dealing with a linear model, a concept you might have encountered in algebra. Now, let's clarify what this means in terms of the number of backpacks sold. If the price p is set at $9, the number of backpacks sold would be -2 * 9 + 50 = 32. Conversely, if the price p is $15, the number of backpacks sold would be -2 * 15 + 50 = 20. This decrease in sales with an increase in price illustrates the relationship the model seeks to represent. The concept is straightforward. The store's aim is to find the price that generates the highest possible revenue. Revenue is a key financial metric; it's the total amount of money a business earns from selling its goods or services. But what price maximizes the revenue? That's what we aim to find.

Now, how do we use this information? The beauty of mathematics is its ability to model real-world situations, enabling predictions and informed decision-making. Here, the expression models the quantity sold at a certain price. Revenue, in this case, is calculated by multiplying the price of each backpack by the number of backpacks sold. Our task is to find the perfect price within the given range ($9 to $15) that will maximize the store's revenue. So, let’s dig a bit deeper into what these mathematical concepts represent, so you can have a full understanding!

We know the expression, but what does it all mean? It is a function, right? Precisely! The expression -2p + 50 represents a linear function. The function describes a relationship between the price of the backpacks (p) and the quantity of backpacks sold per day. As the price changes, the quantity sold also changes, in accordance with this function. The expression acts as a sort of a demand curve, which is a fundamental concept in economics. A demand curve illustrates how the quantity of a good or service consumers are willing to buy changes at different price points. In this instance, the demand curve is linear. So, it shows the change in demand as the price increases or decreases. The linear nature means that for every dollar increase in price, the quantity sold decreases by a consistent amount (in this case, by 2 units, given the coefficient of p is -2). The -2 indicates that each time we increase the price by $1, the number of backpacks sold decreases by 2. The 50 represents the quantity of backpacks that would be sold if the price was $0 (though this is theoretical, as the price range is $9-$15). We are going to use the core concept: Revenue = Price * Quantity to solve the problem.

Setting Up the Revenue Equation

Alright, let's get mathematical. The revenue generated by selling backpacks is calculated by multiplying the price per backpack (p) by the number of backpacks sold. We already have the model for the number of backpacks sold per day, which is -2p + 50. Therefore, our revenue equation becomes: Revenue = p * (-2p + 50). Let's simplify this equation to make it easier to work with. Multiplying p through the parentheses, we get Revenue = -2p^2 + 50p. This is a quadratic equation, and its graph is a parabola that opens downward, meaning it has a maximum point. The maximum point of this parabola represents the maximum revenue achievable. We now have a revenue equation that is defined as a quadratic equation, which opens downward. The maximum point on this curve represents the maximum revenue we are trying to find. This gives us a systematic way to solve the problem by considering the mathematical properties of the revenue function and the constraints placed on our pricing. Let's dig deeper into the world of quadratic equations! Now, let's explore what this quadratic equation represents.

We've got Revenue = -2p^2 + 50p. It's a quadratic equation, which means it forms a parabola when graphed. Because the coefficient of p^2 is negative (-2), the parabola opens downward. This shape is crucial. The peak of the parabola, also known as the vertex, represents the maximum value of the function. In our context, this vertex tells us the price p that results in the highest revenue and the actual maximum revenue. To find the vertex, we can use a couple of methods. We will use the vertex formula: the x-coordinate (in our case, the p-coordinate) of the vertex is given by -b / 2a, where a and b are the coefficients in the quadratic equation ax^2 + bx + c. In our case, a = -2 and b = 50. Thus, p = -50 / (2 * -2) = 12.5. This value is within our acceptable range of $9 to $15. Hence, the price that gives the maximum revenue is $12.5. To find the maximum revenue, we substitute this value back into our revenue equation: Revenue = -2 * (12.5)^2 + 50 * 12.5 = -312.5 + 625 = 312.5. So, the maximum revenue is $312.5.

Finding the Price for Maximum Revenue

To find the price that maximizes revenue, we need to understand the properties of the revenue equation Revenue = -2p^2 + 50p. As mentioned, this is a quadratic equation, and its graph is a downward-opening parabola. The vertex of the parabola represents the maximum point on the graph. The x-coordinate (in this case, the price, p) of the vertex is found using the formula -b / 2a. In our equation, a = -2 and b = 50. So, the x-coordinate of the vertex (the price that maximizes revenue) is -50 / (2 * -2) = 12.5. However, we have to consider the price range constraint. Since the price can only be between $9 and $15, the price of $12.5 is a valid price within this range. So, the best price to maximize revenue is $12.50. This is pretty cool, right? We've managed to pin down the exact price point that should generate the most revenue for the store. Finding the vertex of the parabola is a straightforward mathematical process that offers valuable insights into the behavior of the equation and its implications for the business.

Now, how do we find the maximum revenue? Remember that we have a range of prices that we can play with. Now, the vertex helps us to pinpoint the best price to be chosen. The price is $12.5 within the price range of $9 to $15. To find the maximum revenue, substitute this price into the revenue equation. Revenue = -2 * (12.5)^2 + 50 * 12.5. This gives us Revenue = -312.5 + 625 = 312.5. Therefore, the maximum revenue is $312.50.

Calculating the Maximum Revenue

Now that we know the optimal price, let's calculate the maximum revenue. We found that the price p = 12.5 maximizes revenue. We plug this price back into our revenue equation Revenue = -2p^2 + 50p. Substituting p = 12.5, we get Revenue = -2 * (12.5)^2 + 50 * 12.5. This calculation simplifies to Revenue = -312.5 + 625. Therefore, the maximum revenue is $312.50. This means that by setting the price of each backpack to $12.50, the store can achieve the highest possible daily revenue, according to our model. Keep in mind that this is a model, and real-world factors may influence the actual revenue. However, by using this model, the store has a powerful tool to make informed pricing decisions. It's awesome to see how math can directly influence business strategies and outcomes! With the model and its findings, the store can adjust its pricing strategies, and predict revenue based on these price changes, which can provide them with valuable decision-making tools.

Summary and Conclusion

To wrap things up, we've successfully navigated the math behind maximizing revenue in our backpack sales scenario. We learned that by using the model -2p + 50 for the number of backpacks sold, we were able to create a revenue equation: Revenue = -2p^2 + 50p. We then found that a price of $12.50 maximizes the revenue, resulting in a maximum revenue of $312.50. The value of this exercise goes far beyond the specific numbers. The real win here is understanding the process of using mathematical models to analyze and optimize real-world business situations. This is a skill that will serve you well in various aspects of life, from personal finance to career choices. The process can be applied to many different scenarios. In business, it's about making data-driven decisions that can lead to greater profitability and efficiency. So, the next time you are trying to make a business decision, keep this in mind! By combining mathematical concepts with business strategies, we can make informed decisions. Keep up the awesome work, and keep exploring the amazing world of math!