Problem 162 - Calculus Volume 1

Question

A profit is earned when revenue exceeds cost. Suppose the profit function for a skateboard manufacturer is given by \(P(x)=30 x-0.3 x^{2}-250,\) where \(x\) is the number of skateboards sold. a. Find the exact profit from the sale of the thirtieth skateboard. b. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard.

Step-by-Step Solution

Answer

Exact profit for the 30th skateboard is 380. Marginal profit method estimates an additional 12, but does not calculate total profit directly.

Step 1: Understand the Problem

We are given a profit function \(P(x) = 30x - 0.3x^2 - 250\) where \(x\) is the number of skateboards sold. We need to find the profits at specific quantities of skateboards sold.

Step 2: Calculate Exact Profit for 30 Skateboards

Substitute \(x = 30\) into the profit function \(P(x)\): \(P(30) = 30(30) - 0.3(30)^2 - 250.\) Calculate each term: \(30(30) = 900\), \(0.3(30)^2 = 0.3 \times 900 = 270\). Thus, \(P(30) = 900 - 270 - 250 = 380.\) The exact profit for 30 skateboards is 380.

Step 3: Calculate Marginal Profit Function

The marginal profit function is the derivative of the profit function \(P(x)\). Differentiate \(P(x) = 30x - 0.3x^2 - 250\). The derivative is \(P'(x) = 30 - 0.6x\). This represents the change in profit as sales increase by one skateboard.

Step 4: Estimate Profit for 30 Skateboards Using Marginal Profit

Substitute \(x = 30\) into the marginal profit function \(P'(x) = 30 - 0.6x\). \(P'(30) = 30 - 0.6(30) = 30 - 18 = 12.\) The estimated additional profit for the 30th skateboard is approximately 12. However, to find total profit at 30 using marginal profit, we consider the profit function around \(P(29)\) plus \(P'(30)\).

Key Concepts

Marginal Profit

Marginal profit helps us understand the change in profit resulting from selling one additional unit of product. It is particularly useful for determining the profitability of scaling production. To explore marginal profit, we need to relate it to the concept of derivatives, which is a fundamental part of calculus. The marginal profit function is essentially the derivative of the total profit function.

For the skateboard manufacturer, the marginal profit is represented as the derivative of their profit function, which is calculated as:

• The profit function: \(P(x) = 30x - 0.3x^2 - 250\)
• The marginal profit function (derivative): \(P'(x) = 30 - 0.6x\)

This equation, \(P'(x)\), helps estimate how each sale affects total profit, letting us know that the profit will increase by $12 when the 30th skateboard is sold. This shows that analyzing marginal profit is crucial for businesses keen on optimizing production and sales decisions.

Calculus

Calculus plays a vital role in analyzing many economic and business situations, especially when change is involved. Marginal profit, and indeed the calculations we are discussing, depend heavily on concepts from calculus. It is a branch of mathematics that deals with continuous change.

The core components of calculus that we focus on here are derivatives and how they describe the rate of change. Through this mathematical tool, businesses can better manage production levels and predict the outcomes of specific actions, such as producing one more skateboard. By understanding calculus, you gain insights into dynamic decision-making processes.

Derivatives

Derivatives are the backbone of finding the marginal profit function. They're used to measure how a function's output changes as its input changes. More simply, they help us understand the rate of change. In terms of a profit function, the derivative tells us how the profit will change with each additional unit sold.

When calculating the derivative for our skateboard manufacturer's profit function, we consider the following:

• Differentiate constant terms to zero, such as \(-250\), since it does not change with respect to \(x\).
• The derivative of \(30x\) is simply 30, showing a linear increase in profit.
• For \(-0.3x^2\), apply the power rule of derivatives to get \(-0.6x\), indicating how the rate of profit changes slows down as more units are sold.

Together, these make up the function \(P'(x) = 30 - 0.6x\), a tool for predicting how profit will adjust with incremental sales. By mastering derivatives, students can effectively analyze profit behaviors in various economic scenarios, leading to smarter business strategies.