Initial Cost and Material Cost

Oct 13, 2025 by thesweetdesigns.com 145 views

Introduction

The school band is selling t-shirts for a fundraiser, and the organizers are looking to maximize their profits. To do this, they need to consider the initial cost of a steamer to apply the decals, the cost of material for each shirt, and the selling price of the t-shirts. In this article, we will analyze the situation mathematically and provide a function that represents the total cost of producing and selling t-shirts.

Initial Cost and Material Cost

The initial cost for a steamer to apply the decals is $100. This is a fixed cost that does not change regardless of the number of t-shirts sold. The cost of material for each shirt is $12. This is a variable cost that changes depending on the number of t-shirts produced.

Selling Price and Profit

The selling price of each t-shirt is not specified, but we can assume it to be $x. The profit made from selling each t-shirt is the selling price minus the cost of material, which is $x - $12.

Total Cost Function

Let's denote the number of t-shirts sold as t. The total cost of producing and selling t-shirts can be represented by the function:

C(t) = 100 + 12t

This function represents the total cost of producing and selling t-shirts, where 100 is the initial cost of the steamer and 12t is the cost of material for t shirts.

Total Revenue Function

The total revenue from selling t-shirts is the selling price of each t-shirt multiplied by the number of t-shirts sold. Let's denote the selling price as x. The total revenue function is:

R(t) = xt

Profit Function

The profit made from selling t-shirts is the total revenue minus the total cost. The profit function is:

P(t) = R(t) - C(t) = xt - (100 + 12t) = xt - 100 - 12t

Maximizing Profit

To maximize profit, we need to find the value of t that maximizes the profit function P(t). We can do this by taking the derivative of P(t) with respect to t and setting it equal to zero.

dP/dt = x - 12

Setting dP/dt = 0, we get:

x - 12 = 0 x = 12

This means that the selling price of each t-shirt should be $12 to maximize profit.

Optimal Number of T-Shirts

To find the optimal number of t-shirts to sell, we need to substitute x = 12 into the profit function P(t) and find the value of t that maximizes profit.

P(t) = 12t - 100 - 12t = -100

This means that the profit is maximized when the number of t-shirts sold is zero. However, this is not a feasible solution, as the school band needs to sell some t-shirts to raise funds.

Conclusion

Recommendations

Based on the analysis, the school band should consider the following recommendations:

Sell t-shirts at a price of $12 to maximize profit.
Determine the optimal number of t-shirts to sell based on demand and cost of production.
Consider offering discounts or promotions to increase sales and maximize profit.

Limitations

This analysis has some limitations. For example, it assumes that the cost of material for each shirt is $12, which may not be the case in reality. Additionally, it assumes that the selling price of each t-shirt is $x, which may not be the case in reality. Therefore, the school band should consider these limitations when making decisions about t-shirt sales.

Future Research

Future research could involve analyzing the demand for t-shirts and the cost of production in more detail. This could involve collecting data on the number of t-shirts sold, the cost of material, and the selling price of each t-shirt. Additionally, future research could involve analyzing the impact of discounts and promotions on sales and profit.

Conclusion

In conclusion, the school band should sell t-shirts at a price of $12 to maximize profit. However, the optimal number of t-shirts to sell is not zero, but rather a value that needs to be determined based on demand and cost of production. The school band should consider the recommendations and limitations outlined in this article when making decisions about t-shirt sales.