Problem 7 - Calculus Concepts

Question

Shirt Profit The function \(P\) gives the profit in dollars that a fraternity makes selling \(x\) T-shirts. a. Is it possible for \(P(30)\) to be negative? Explain. b. Is it possible for \(P^{\prime}(100)\) to be negative? Explain. c. If \(P^{\prime}(200)=-1.5,\) is the fraternity losing money? Explain.

Step-by-Step Solution

Answer

a. Yes, if costs exceed revenues. b. Yes, it means profit decreases with more sales. c. No, but additional sales decrease profit.

Step 1: Understanding Profit Function

The function \( P(x) \) represents the profit based on selling \( x \) T-shirts. The profit can be either positive, zero, or negative, depending on costs and revenues from selling the T-shirts.

Step 2: Analyzing P(30)

A negative \( P(30) \) indicates that the costs exceed the revenue from selling 30 T-shirts. Thus, it is possible for \( P(30) \) to be negative if the total cost of making and selling 30 T-shirts is more than the income generated from those sales.

Step 3: Analyzing the Meaning of P'(x)

The derivative \( P'(x) \) represents the rate of change of profit with respect to the number of T-shirts sold. It indicates how profit changes as more shirts are sold.

Step 4: Interpreting P'(100) being Negative

If \( P'(100) \) is negative, it suggests that selling the next shirt (i.e., the 101st shirt) will decrease the profit. This can occur when the additional cost of producing and selling one more T-shirt exceeds the revenue gained from selling it.

Step 5: Understanding P'(200) = -1.5

The value \( P'(200) = -1.5 \) means the profit decreases by $1.50 for each additional shirt sold at around 200 shirts. Even though the rate of profit is decreasing, it doesn't imply the fraternity is losing money overall; it just means selling more shirts may not be profitable at this point.

Key Concepts

Derivative of Profit

In the world of business and economics, understanding the changes in profit is crucial. The derivative of the profit function, denoted by \( P'(x) \), is an important concept that helps in analyzing these changes. It essentially tells us the "rate of change" of profit with respect to the number of items sold, in this case, T-shirts.

The derivative captures how much the profit is changing as an additional T-shirt is sold. If \( P'(x) \) is positive, it implies that selling more T-shirts is increasing the profit. Conversely, if \( P'(x) \) is negative, the profit decreases with each extra T-shirt sold. This negative change could happen when additional costs surpass the extra revenue generated by selling more shirts. Thus, understanding \( P'(x) \) is key to making informed financial decisions.

For example, if \( P'(200) = -1.5 \), it indicates that selling the 201st shirt will reduce the profit by $1.50. By keeping track of the derivative, businesses can determine the most profitable number of items to sell, or identify when increasing production might not be wise.

Rate of Change

The concept of rate of change is a fundamental idea in calculus that applies directly to profits from sales. When applied to the profit function, the rate of change helps businesses understand how the profit grows or shrinks as more items are sold. Calculating this rate involves using the derivative, \( P'(x) \), to measure how much profit changes with each additional unit sold.

When \( P'(x) \) is zero, it indicates that selling one more shirt does not affect the profit at all; the company neither gains nor loses profit from one more sale. On the other hand, if \( P'(x) \) is far from zero, it signals significant changes in profit for each item sold. A positive \( P'(x) \) implies that more sales lead to an increase in profit, while a negative \( P'(x) \) suggests a decrease in profit.

Understanding the rate of change can help businesses determine optimal pricing, production levels, and sales quantities to maximize profit. It allows them to anticipate how profit will evolve as they adjust their strategies in response to market conditions.

Costs and Revenues

Profit is intrinsically linked to costs and revenues. Simply put, profit is what remains after subtracting total costs from total revenues. To deeply understand the function \( P(x) \), firms need to analyze both these components.

Costs include all expenditures associated with producing and selling a product, such as materials, labor, and overhead. If the costs per T-shirt are high, they may outweigh revenue, leading to negative profit, as indicated by \( P(30) \). This scenario emphasizes the importance of managing costs efficiently.

Revenues are earned through sales. They represent the income a company makes from selling its products. For a profit to be positive, revenues must exceed costs. Thus, there's a direct correlation between how efficiently a company can sell (and price) its products and its eventual profit.

The delicate balance between costs and revenues determines the overall profit function \( P(x) \). A thorough understanding of these elements allows businesses to plan effectively, optimize production, reduce unnecessary expenses, and enhance profitability.