T-Shirt Profit: Price For Max Sales!

Are you trying to figure out the sweet spot for pricing your T-shirts to maximize your profits? Let's dive into how we can use a mathematical function to help us do just that. We'll break down the function $p(x)=-2(x-9)^2+100$, which models the profit from selling T-shirts at a price of $x$ dollars. We will explore how to calculate the profit when the price is set at $15 apiece, and we'll also discuss the broader implications of this model for your pricing strategy.

Understanding the Profit Function

The profit function, $p(x)=-2(x-9)^2+100$, might look a bit intimidating at first, but let's break it down to understand what each part means. This is a quadratic function, which means it graphs as a parabola—a U-shaped curve. In our case, because of the negative sign in front of the $2$, the parabola opens downwards. This is crucial because it tells us that there's a maximum point on the curve, representing the price at which we'll achieve the highest profit.

Let's dissect the function piece by piece:

• $x$: This represents the price of each T-shirt in dollars. It's the variable we're controlling to see how it affects our profit.
• $(x-9)$: This part shows how the price $x$ deviates from the value $9$. The number $9$ is significant because it's related to the vertex of the parabola, which we'll discuss shortly.
• $(x-9)^2$: Squaring the difference $(x-9)$ means we're looking at the magnitude of the deviation from $9$, regardless of whether $x$ is greater or smaller than $9$. This is important because prices both above and below a certain optimal price will typically result in lower profits.
• $-2(x-9)^2$: The negative sign and the multiplication by $2$ tell us that the profit decreases as the price moves away from $9$. The '$-2$' scales the impact of the deviation—the further $x$ is from $9$, the more the profit decreases, and the profit decreases more rapidly than it would with a smaller coefficient.
• $+100$: This is the constant term and represents the maximum possible profit. It's the vertex (highest point) of the parabola. This is the peak profit you can achieve if you price your T-shirts just right.

In essence, the function is designed to tell you that there's an optimal price point (which we'll find is around $9), and any deviation from this price will reduce your profit. The function gracefully handles the common business reality that pricing too high or too low can hurt your bottom line.

Why This Shape?

The parabolic shape of the profit function is not arbitrary; it reflects economic principles. If you price your T-shirts too low, you might sell a lot, but your profit margin on each shirt is small, leading to a lower overall profit. On the other hand, if you price them too high, you won't sell as many, and again, your overall profit suffers. The maximum profit is achieved at the sweet spot where the price balances demand and margin effectively.

Calculating Profit at $15 Apiece

Now, let's get to the specific question: What would the profit be if the price of the T-shirts were $15 apiece? This is a straightforward calculation. We substitute $x = 15$ into our profit function:

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

First, we calculate the value inside the parentheses:

$15 - 9 = 6$

Then, we square that result:

$6^2 = 36$

Next, we multiply by $-2$:

$-2 imes 36 = -72$

Finally, we add $100$:

$-72 + 100 = 28$

So, the profit from sales would be $28 if the price of the T-shirts were $15 apiece. It's crucial to understand what this number means in the context of our profit-maximizing goal.

Interpreting the Result

A profit of $28 when pricing T-shirts at $15 tells us something important: while you're making a profit, you're likely not maximizing your potential earnings. Remember, the function suggests that the peak profit occurs when $x$ is around $9. Pricing at $15 is significantly higher than this optimal value, leading to a reduced profit.

This result illustrates the trade-off between price and volume. At $15, you might be making a higher margin on each shirt, but you're probably selling fewer shirts compared to what you would sell at a price closer to $9. The function helps quantify this trade-off, showing us that the decrease in volume outweighs the increase in margin at this price point.

Finding the Optimal Price

Okay, so we know $15 isn't the best price. But how do we find the optimal price? This is where the vertex form of the quadratic equation comes in handy. Our profit function is already in vertex form:

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

In the vertex form of a quadratic equation, $f(x) = a(x-h)^2 + k$, the vertex of the parabola is the point $(h, k)$. The vertex represents the maximum (or minimum, if $a$ is positive) value of the function. In our case:

• $h = 9$
• $k = 100$

So, the vertex of our parabola is $(9, 100)$. This tells us that the maximum profit ($100) is achieved when the price ($x$) is $9. This is a crucial insight for setting our pricing strategy.

Why is the Vertex the Optimum?

The vertex is the optimum because of the symmetrical nature of a parabola. The squared term $(x-9)^2$ ensures that deviations from $9$ in either direction (higher or lower prices) will reduce the profit by the same amount. The maximum profit, therefore, occurs at the point where this deviation is zero—at the vertex.

Implications for Pricing Strategy

Understanding the profit function and its vertex has significant implications for your pricing strategy. Here are some key takeaways:

1. Optimal Price Point: The function suggests that pricing T-shirts at $9 apiece will yield the maximum profit of $100. This is your target price.
2. Price Sensitivity: The function highlights the sensitivity of profit to price changes. Pricing too high ($15 in our example) or too low will reduce your overall profit. You need to be mindful of this sensitivity when making pricing decisions.
3. Market Research: While the function provides a mathematical model, it's essential to complement it with market research. Real-world factors such as competitor pricing, customer demand, and the perceived value of your T-shirts can influence the optimal price. The model gives you a starting point, but market dynamics will help you fine-tune your strategy.
4. Cost Considerations: The profit function doesn't explicitly include the cost of producing the T-shirts. In a real-world scenario, you'd need to factor in your costs to ensure that the $9 price point covers your expenses and provides a reasonable profit margin. A more comprehensive model would incorporate cost as a variable.

Beyond the Model

While the mathematical model is a powerful tool, it's not the whole story. Other factors can influence your pricing strategy:

• Brand Perception: If you're selling high-quality, designer T-shirts, customers might be willing to pay a premium. Your brand perception can allow you to price higher than the model suggests.
• Promotional Pricing: You might choose to temporarily lower your price to run promotions, clear inventory, or attract new customers. These short-term strategies can deviate from the optimal price point suggested by the model.
• Bundling and Discounts: Offering discounts for bulk purchases or bundling T-shirts with other products can influence your pricing strategy. These tactics can help increase volume and overall profit, even if the per-shirt price is lower.

Conclusion

The function $p(x)=-2(x-9)^2+100$ provides a valuable framework for understanding how price affects profit in your T-shirt business. By analyzing the function, we determined that pricing T-shirts at $15 apiece yields a profit of $28, but the optimal price, according to the model, is $9, which would maximize profit at $100. Remember, guys, that this mathematical model is a tool to guide your decisions, but it should be combined with market research and real-world business considerations to develop a comprehensive pricing strategy. Happy pricing, and may your profits be maximized!