Problem 88 - College Algebra

Question

The weekly cost, in thousands of dollars, for producing \(x\) stereo headphones is \(30 x+50 .\) The weekly revenue, in thousands of dollars, for selling \(x\) stereo headphones is \(90 x^{2}-x .\) Write a polynomial in standard form for the weekly profit, in thousands of dollars, for producing and selling \(x\) stereo headphones.

Step-by-Step Solution

Answer

The polynomial for the weekly profit, in thousands of dollars, for producing and selling \(x\) stereo headphones is \(90x^{2} - 31x - 50 .\)

Step 1: Understand the definitions

Recall that the profit is the difference between the revenue (the money collected from sales) and the cost (the money spent on production). This means we should subtract the cost function from the revenue function.

Step 2: Subtract cost from revenue

Subtract the cost function \(30x + 50\) from the revenue function \(90x^{2} - x\). Now, you have \( (90x^{2} - x) - (30x + 50) \) for calculation.

Step 3: Simplify the function

Combine like terms, distribute the negative sign and simplify the expression to its simplest form. The simplified form is \(90x^{2} - 31x - 50 \).

Key Concepts

Cost and Revenue Functions

In the realm of business and economics, cost and revenue functions are foundational concepts that must be understood before one can dive into more advanced topics like profit analysis.

A cost function, typically denoted as C(x), is an equation that describes the total cost of producing a certain number of goods. In the given exercise, the cost function is expressed as C(x) = 30x + 50. This tells us that for every unit of product made, in our case, stereo headphones, the production costs increase linearly by \(30,000, and there is an additional fixed cost of \)50,000, irrespective of the quantity produced.

Conversely, a revenue function, denoted as R(x), calculates the total revenue generated from the sale of a number of goods. In our exercise, the revenue when x stereo headphones are sold is described by R(x) = 90x^2 - x. This indicates that the revenue is not simply a multiple of the number of headphones sold. Instead, it varies as a quadratic function of x, which might represent effects like bulk discounts or increased demand at higher volumes.

The important point is that these functions, when computed accurately, can provide crucial insights into the financial health of a business. They help owners and managers to forecast and optimize for different scenarios, aiding in strategic decision-making.

Simplifying Polynomials

When working with polynomial functions in algebra, one often needs to manipulate and simplify these expressions to make them more usable for further calculations or to interpret their meaning more easily.

Simplifying polynomials includes several steps.

• The first is combining like terms, which are terms in the polynomial that share the same variable raised to the same power.
• The second step typically involves distributing any coefficients across terms that are inside parentheses, especially when a polynomial is subtracted from another, as in our profit function example.
• Finally, simplifying may also mean factoring the polynomial or dividing it by another polynomial, if needed.

For example, in our exercise, once the cost function 30x + 50 is subtracted from the revenue function 90x^2 - x, we distribute the negative sign and combine like terms. Through these steps, the initially more complex expression simplifies down to the profit function 90x^2 - 31x - 50, which is much more straightforward to use in further calculations or analyses.

Standard Form Polynomials

The standard form of a polynomial is an essential concept in algebra, providing a uniform way to write polynomials so they can be easily compared, added, or subtracted. It involves arranging the terms of a polynomial in descending order of their degree, with the coefficients in front of each term.

Polynomials in standard form make it clear what the highest power of the variable is, which is known as the degree of the polynomial. The coefficient of this term with the highest power is called the leading coefficient. For instance, the profit function 90x^2 - 31x - 50, from our exercise, is already in standard form. The highest degree is two, shown by the term with x^2, and the leading coefficient here is 90.

Understanding and using standard form is critical for graphing polynomials, finding their roots, and performing polynomial division. It provides a consistent approach to dealing with these expressions, particularly when they are part of complex equations, enabling one to more clearly see the structure and key characteristics of the polynomial.