Problem 60 - Calculus and its applications

Question

Cruzin' Boards has found that the cost, in dollars, of producing \(x\) skateboards is given by $$ C(x)=900+18 x^{0.7} $$ If the revenue from the sale of \(x\) skateboards is given by \(R(x)=75 x^{0.8},\) find the rate at which average profit per skateboard is changing when 20 skateboards have been built and sold.

Step-by-Step Solution

Answer

The average profit per skateboard is increasing at approximately $2.04 per skateboard when 20 skateboards have been made and sold.

Step 1: Understand the Average Profit

The profit is given by the revenue minus the cost. We have to find the average profit per skateboard, which is the profit function divided by the number of skateboards, \(x\). Therefore, the average profit per skateboard, \( AP(x) \), is given by \( AP(x) = \frac{R(x) - C(x)}{x} \).

Step 2: Write Down the Functions

The functions provided are:\[R(x) = 75x^{0.8}\] and\[C(x) = 900 + 18x^{0.7}\] The profit function is:\[P(x) = R(x) - C(x) = 75x^{0.8} - (900 + 18x^{0.7})\] which simplifies to:\[P(x) = 75x^{0.8} - 900 - 18x^{0.7}\] Now, find the average profit per skateboard:\[AP(x) = \frac{75x^{0.8} - 900 - 18x^{0.7}}{x}\] which simplifies to:\[AP(x) = 75x^{-0.2} - \frac{900}{x} - 18x^{-0.3}\]

Step 3: Differentiate the Average Profit Function

To find how the average profit per skateboard is changing, differentiate \( AP(x) \) with respect to \( x \). The derivative \( AP'(x) \) is:\[AP'(x) = \frac{d}{dx} \left( 75x^{-0.2} - \frac{900}{x} - 18x^{-0.3} \right)\] Using the power rule, we find:\[AP'(x) = -15x^{-1.2} + \frac{900}{x^2} + 5.4x^{-1.3}\]

Step 4: Evaluate the Derivative at x = 20

Substitute \( x = 20 \) into the derivative:\[AP'(20) = -15 \cdot 20^{-1.2} + \frac{900}{20^2} + 5.4 \cdot 20^{-1.3}\]Calculate each term separately:\[-15 \cdot 20^{-1.2} \approx -15 \cdot 0.0187 = -0.2805\]\[\frac{900}{400} = 2.25\]\[5.4 \cdot 20^{-1.3} \approx 5.4 \cdot 0.0135 = 0.0729\]Combine the values:\[AP'(20) \approx -0.2805 + 2.25 + 0.0729 = 2.0424\]

Step 5: Conclusion on the Rate of Change

The rate at which the average profit per skateboard is changing, when 20 skateboards are built and sold, is approximately 2.04 dollars per skateboard.

Key Concepts

Understanding Average Profit

Average profit is a crucial concept in business and economics, as it helps determine the profitability of each unit of a product. When you calculate the average profit, you divide the total profit by the number of units sold, providing insight into how much profit each unit contributes. This becomes useful when deciding whether production levels should increase or decrease.

• The average profit per unit can assist in pricing strategies, resource allocation, and investment decisions.
• To find the average profit, you first need to calculate the total profit, which is the difference between total revenue and total cost.
• The formula for average profit per skateboard in this exercise is: \[ AP(x) = \frac{R(x) - C(x)}{x} \]

In simple terms, average profit gives you a straightforward glance at how well each unit performs financially.

Derivatives and Their Role in Optimization

Derivatives are a fundamental tool in calculus used for finding the rate at which things change. In optimization problems, they help us determine how functions behave as we tweak their inputs. By finding the derivative of a function, we can explore its growth and decay patterns, discover local maxima and minima, and understand its sensitivity to change.

• When applied to a profit function, derivatives reveal how the profit changes with a change in the number of units produced or sold.
• For average profit, specifically, the derivative tells us how the average profit per product shifts as production levels change. This is the key to selecting optimal production quantities.
• In this problem, the derivative of the average profit function, \( AP'(x) \), shows the change of average profit per skateboard as the number of skateboards sold varies.

Taking derivatives allows businesses to make strategic decisions, ensuring that operations are both efficient and profitable.

Revenue and Cost Functions

The revenue and cost functions are two of the most important components in understanding a business's financial health. Revenue is the total income generated from sales, while cost considers all expenses incurred. Understanding and differentiating these functions can significantly influence decision-making processes.

• Revenue function, like \(R(x) = 75x^{0.8}\), describes the relationship between the number of units sold and the income generated from those sales. Consider it as the front-end picture of financial performance.
• Cost function, such as \(C(x) = 900 + 18x^{0.7}\), details the expenses associated with producing goods, including both fixed costs (like initial investments) and variable costs that change with production levels.
• Together, these functions form the backbone of the profit equation: Profit = Revenue - Cost.

Evaluating and balancing these functions helps businesses forecast and enhance profitability, ensuring sustainable growth in competitive markets.