Problem 63 - Calculus with Concepts in Calculus

Question

The revenue function \(R\) for a certain product is given by $$ R(x)=5 x^{2}-\frac{x^{4}}{10} $$ The cost function \(C\) is given by $$ C(x)=4 x^{2}-24 x+38 $$ The profit function \(P\) is defined as the difference \(R-C\). Find the equation that describes \(P\). Then find \(P(1)\) and \(P(2)\), and show that it is possible to lose money and also possible to make a profit.

Step-by-Step Solution

Answer

The profit function is \(P(x) = -\frac{x^4}{10} + x^2 + 24x - 38\). \(P(1) = -13.1\) (a loss) and \(P(2) = 12.4\) (a profit).

Step 1: Define Profit Function

The profit function \(P(x)\) is the difference between the revenue function \(R(x)\) and the cost function \(C(x)\). Thus, we have: \[ P(x) = R(x) - C(x) \] We substitute the given functions: \[ P(x) = \left(5x^2 - \frac{x^4}{10}\right) - \left(4x^2 - 24x + 38\right) \]

Step 2: Simplify the Profit Function

To simplify \(P(x),\) we distribute the negative sign to the cost function and combine like terms: \[ P(x) = 5x^2 - \frac{x^4}{10} - 4x^2 + 24x - 38 \] Combining like terms gives: \[ P(x) = -\frac{x^4}{10} + x^2 + 24x - 38 \]

Step 3: Calculate P(1)

Substitute \(x = 1\) into the profit function: \[ P(1) = -\frac{1^4}{10} + 1^2 + 24(1) - 38 \] Simplifying further: \[ P(1) = -\frac{1}{10} + 1 + 24 - 38 = -\frac{1}{10} - 13 \] Thus, \(P(1) = -13.1\)

Step 4: Calculate P(2)

Substitute \(x = 2\) into the profit function: \[ P(2) = -\frac{2^4}{10} + 2^2 + 24(2) - 38 \] Simplifying further: \[ P(2) = -\frac{16}{10} + 4 + 48 - 38 \] \[ P(2) = -1.6 + 4 + 48 - 38 = 12.4 \] Thus, \(P(2) = 12.4\)

Step 5: Interpret the Results

From the calculations, \(P(1) = -13.1\) indicating a loss, and \(P(2) = 12.4\) indicating a profit. Therefore, it is possible for the company to both incur a loss and make a profit, depending on the production level.

Key Concepts

Revenue Function

In calculus, the revenue function is a crucial concept that tells us how much money a company earns from selling a certain number of products or services. It's defined in terms of another variable, often represented as \(x\), which usually stands for the number of units sold. For this exercise, the revenue function is:\[R(x) = 5x^2 - \frac{x^4}{10}\]This function features two terms:

• The term \(5x^2\) suggests the revenue grows at an accelerating rate as more units are sold because the revenue is proportional to the square of units sold.
• The term \(-\frac{x^4}{10}\) implies there's a decline or inefficiency at higher production scales, as the negative quartic term reduces revenue at even larger \(x\) values.

Understanding these components helps companies to plan better, aiming to maximize revenue by calculating the optimal number of products to produce without overstretching resources.

Cost Function

The cost function, another important concept in calculus, represents the total cost of producing a certain number of goods or services. Like the revenue function, it is often expressed in terms of \(x\), the number of units produced. In this case, the given cost function is:\[C(x) = 4x^2 - 24x + 38\]Let's break it down:

• The quadratic term \(4x^2\) shows how the cost rises quickly with increased production, reflecting the additional expenses of producing more units.
• The linear term \(-24x\) suggests a reduction in cost per unit, possibly indicating savings due to efficiencies at increased production levels.
• The constant term \(38\) indicates fixed costs, which are expenses the company incurs regardless of how much they produce, like rent or salaries.

By examining each component, businesses can better predict production costs and identify how different factors affect overall expenditure.

Polynomial Functions

Polynomial functions are a key mathematical concept used to model relationships and predict results in a variety of real-world scenarios, including economics. They come in various degrees, and can be identified by terms with non-negative integer exponents. A polynomial function takes the form:\[a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\]where \(a_i\) are constants, and \(n\) indicates the polynomial's degree.In our exercise, both the revenue function \(R(x) = 5x^2 - \frac{x^4}{10}\) and the cost function \(C(x) = 4x^2 - 24x + 38\) are polynomial functions.

• The highest degree term is \(-\frac{x^4}{10}\) from the revenue function, making it a quartic polynomial.
• The cost function is a quadratic polynomial due to its highest degree term \(4x^2\).

Polynomial functions can present complex behavior through their combination of terms, which allows us to model intricate relationships like those between cost, revenue, and profit in business contexts.