T-Shirt Profit: Calculate Sales At $15 | Math Explained

Hey guys! Let's dive into a cool math problem that's super relevant to real-world scenarios, like figuring out the profit from selling T-shirts. We're going to break down the function $p(x)=-2(x-9)^2+100$, which tells us the profit $p(x)$ when T-shirts are sold for $x$ dollars. The big question we're tackling today is: What's the profit if we sell these awesome T-shirts for $15 apiece? So, grab your thinking caps, and let's get started!

Understanding the Profit Function

Before we jump straight into plugging in numbers, let's take a moment to really understand what this function is telling us. The function $p(x)=-2(x-9)^2+100$ is a quadratic function, and it's written in vertex form. This form is super helpful because it gives us some key information right away. The general form of a quadratic function in vertex form is $p(x) = a(x-h)^2 + k$, where:

• a tells us whether the parabola opens upwards (if a is positive) or downwards (if a is negative). In our case, a = -2, which means the parabola opens downwards. This tells us that there's a maximum profit we can achieve.
• (h, k) represents the vertex of the parabola. The vertex is the highest or lowest point on the graph. In our function, h = 9 and k = 100. This means the vertex is at the point (9, 100). In the context of our problem, this tells us that the maximum profit of $100 is achieved when the T-shirts are sold for $9 apiece.

Now, why is this important? Well, understanding the shape of the profit function gives us a better picture of how the price affects our profit. If we price the T-shirts too low, we might not cover our costs. If we price them too high, we might not sell enough. The vertex represents the sweet spot where we maximize our profit. The negative coefficient -2 in front of the squared term means that the parabola opens downwards, indicating that there's a maximum profit. This is crucial for businesses because it helps them understand that there's a point beyond which increasing the price will actually decrease profit due to lower sales volume.

Thinking critically about the function's components is super important. The $(x-9)^2$ part tells us how far the selling price is from the optimal price of $9. The further away from $9 the price is, the larger the squared term becomes. Since this term is multiplied by -2, the profit decreases as the price moves away from $9 in either direction. The +100 at the end shifts the entire parabola upwards, setting the maximum profit at $100. This means that even if we sell the T-shirts at the optimal price, our profit won't exceed $100. Understanding these components allows us to make informed decisions about pricing strategies.

Moreover, considering the practical implications of this mathematical model is key. In a real-world scenario, factors like production costs, marketing expenses, and competitor pricing would influence the actual profit. This function provides a simplified view, but it's a powerful tool for initial analysis. For instance, if the production cost per T-shirt is $5, the function helps determine the price point that not only covers the cost but also maximizes the profit margin. By analyzing the function, a business can set a pricing strategy that attracts customers while ensuring profitability. Furthermore, understanding the function helps in planning sales and promotions. If the current selling price yields a profit lower than the maximum, the business might consider temporary discounts to boost sales and move closer to the optimal profit point. In essence, a deep understanding of the profit function empowers businesses to make data-driven decisions, optimizing their pricing strategies for maximum profitability.

Plugging in the Price: $x = 15$

Okay, now that we've dissected the profit function, let's get to the heart of the question. We want to find the profit when the T-shirts are sold for $15 apiece. That means we need to plug in $x = 15$ into our function $p(x)=-2(x-9)^2+100$. Here's how it looks:

$$p(15) = -2(15-9)^2 + 100$$

Let's break this down step by step. First, we tackle the parentheses:

$$15 - 9 = 6$$

So, our equation becomes:

$$p(15) = -2(6)^2 + 100$$

Next, we square the 6:

$$6^2 = 36$$

Now we have:

$$p(15) = -2(36) + 100$$

Multiply -2 by 36:

$$-2 * 36 = -72$$

Our equation is now:

$$p(15) = -72 + 100$$

Finally, add -72 and 100:

$$p(15) = 28$$

So, when the T-shirts are sold for $15 apiece, the profit is $28. It's awesome to see how we can use a mathematical function to predict real-world outcomes! This step-by-step calculation not only provides the answer but also illustrates the practical application of quadratic functions in business and economics.

The process of substitution and simplification highlights the importance of order of operations in mathematics. By following the correct order, we ensure accurate results, which is crucial in business decisions where precision can significantly impact profitability. Moreover, this calculation underscores the concept of marginal returns. While the function peaks at a selling price of $9, selling at $15 still yields a profit, albeit a smaller one. This suggests that there's a trade-off between price and sales volume. Selling at a higher price reduces the potential profit per shirt but might still be beneficial if the volume of sales remains sufficiently high.

Furthermore, this calculation can serve as a basis for further analysis. For instance, we could compare the profit at $15 with the profit at other price points, like $10 or $12, to identify the most profitable selling price within a specific range. This comparative analysis is essential for dynamic pricing strategies, where prices are adjusted based on market demand and other factors. Additionally, the calculated profit can be used in broader financial forecasting. Businesses can use this information to estimate total revenue and project future profitability, helping them in strategic planning and investment decisions. In essence, plugging in the price and calculating the profit is not just a mathematical exercise; it's a practical tool for informed business management.

Interpreting the Result: What Does $28 Mean?

Okay, we've crunched the numbers and found that the profit is $28 when the T-shirts are sold for $15. But what does this $28 figure really tell us? It's not just a random number; it's a crucial piece of information for making business decisions. This $28 represents the profit earned for each T-shirt sold at a price of $15. This is after all the costs associated with making and selling the T-shirt have been taken into account. Remember, the profit function encapsulates all these factors, giving us a clear picture of our financial gain at this price point.

This result indicates that while selling the T-shirts at $15 generates a profit, it's likely not the maximum profit we could achieve. We know from our earlier analysis of the function that the maximum profit occurs when the selling price is $9. This means that selling at $15, while profitable, is less efficient in terms of profit maximization. The fact that the profit is lower than the potential maximum suggests that there may be a trade-off at play. A higher price point might reduce the number of sales, impacting the overall profit.

Understanding this trade-off is essential for effective pricing strategy. The business needs to consider the price elasticity of demand, which measures how the quantity demanded changes in response to a change in price. If demand is highly elastic, a small increase in price could lead to a significant decrease in sales volume, potentially reducing overall profit. In this context, selling at $15 might deter some customers, resulting in fewer sales compared to selling at a lower price closer to the optimal $9. However, the higher price per shirt compensates to some extent, leading to the $28 profit.

Moreover, the $28 profit can be used to evaluate the financial viability of selling T-shirts at this price. If the business has specific profit targets or margins, this figure can be compared against those benchmarks. If the profit is below the desired level, the business might consider adjusting the price, reducing costs, or implementing marketing strategies to boost sales. Additionally, the profit per T-shirt can be factored into inventory management decisions. Understanding the profitability of each item helps in optimizing stock levels and preventing overstocking or stockouts. In broader financial planning, the $28 profit contributes to the overall revenue projections, influencing budget allocations and investment decisions. In essence, the $28 figure is a key metric for assessing the financial health and performance of the T-shirt sales, guiding strategic actions to enhance profitability and sustainability.

Conclusion: Math in Action

So, there you have it! We've taken a mathematical function, plugged in a value, and interpreted the result to understand the profit from selling T-shirts at $15 apiece. We found that the profit would be $28 per T-shirt. This example shows how math isn't just about numbers and equations; it's a powerful tool for making informed decisions in real-world situations. Whether you're running a business, managing your finances, or just trying to understand the world around you, math can provide valuable insights. Keep those thinking caps on, and keep exploring the amazing world of mathematics! This journey through the profit function illustrates the broader applicability of mathematical models in various fields, from business and economics to engineering and science.

The ability to translate real-world scenarios into mathematical expressions and interpret the results is a crucial skill in the modern world. In business, understanding concepts like profit functions, cost curves, and demand elasticity enables managers to make data-driven decisions, optimizing operations and maximizing profitability. In finance, mathematical models are used for risk assessment, portfolio management, and financial forecasting. In engineering, mathematical equations describe physical systems, enabling engineers to design and build structures, machines, and technologies. In science, mathematical models are used to describe natural phenomena, from the movement of celestial bodies to the behavior of subatomic particles.

Moreover, the problem-solving approach we used in this example is transferable to a wide range of situations. We started by understanding the function, then we plugged in the given value, and finally, we interpreted the result in the context of the problem. This systematic approach can be applied to various challenges, whether in academics, professional life, or personal endeavors. Breaking down a problem into smaller, manageable steps, identifying the relevant information, and using appropriate tools and techniques are key to successful problem-solving.

In conclusion, the exploration of the T-shirt profit function serves as a microcosm of the broader role of mathematics in our lives. It demonstrates how mathematical concepts can be used to model real-world scenarios, make informed decisions, and solve complex problems. By embracing mathematical thinking, we empower ourselves to navigate the world more effectively and contribute meaningfully to our communities and beyond. As we continue to learn and grow, let's remember that mathematics is not just a subject to be studied; it's a powerful lens through which we can understand and shape the world around us.