Problem 17 - Algebra A Combined Function

Question

The polynomial function \(P(x)=45 x-100,000\) models the relationship between the number of computer briefcases \(x\) that a company sells and the profit the company makes, \(P(x)\). Find \(P(4000),\) the profit from selling 4000 computer briefcases.

Step-by-Step Solution

Answer

The profit from selling 4000 briefcases is $80,000.

Step 1: Identify the Polynomial Function

The given polynomial function is \( P(x) = 45x - 100,000 \). This represents the profit as a function of the number of computer briefcases sold, \( x \).

Step 2: Substitute the Value of x

We need to find \( P(4000) \), so substitute \( x = 4000 \) into the polynomial function. This gives us \( P(4000) = 45(4000) - 100,000 \).

Step 3: Perform the Multiplication

Calculate the product of 45 and 4000. \[ 45 \times 4000 = 180,000 \]

Step 4: Calculate the Profit

Subtract 100,000 from 180,000 to find the profit. \[ 180,000 - 100,000 = 80,000 \]

Step 5: Interpret the Result

The profit from selling 4000 computer briefcases is \( P(4000) = 80,000 \).

Key Concepts

Profit Calculation

Profit calculation is an essential concept, especially for businesses seeking to understand their financial performance. In this context, profit is defined as the revenue generated from sales minus the costs associated with producing or procuring these goods. Here, we have a simple model for determining profit:

• The profit function for a company selling computer briefcases is captured by the polynomial equation: \( P(x) = 45x - 100,000 \).
• In this function, 45 represents the profit contribution per briefcase sold. Meanwhile, -100,000 is a fixed cost, often referring to expenses unrelated to the quantity of items sold.

To find the profit for a specific number of sales, substitute the number of briefcases into the function and evaluate the equation to determine profit.

Substitution Method

The substitution method is a straightforward technique in mathematics used to find a specific value of a function. Here’s how it applies in our context:

• Start with the given polynomial function, which is in a general form of \( P(x) = 45x - 100,000 \).
• Substitute the known value of \( x \) into the function. In our exercise, \( x = 4000 \) stands for selling 4000 briefcases.

By replacing \( x \) with 4000, the equation becomes \( P(4000) = 45(4000) - 100,000 \). This method allows for direct computation of specific outcomes from a general equation.

Algebraic Expressions

Algebraic expressions form the foundation of polynomial functions and consist of numbers, variables, and operations. In the context of our polynomial function:

• \( 45x \) is an algebraic term where 45 is the coefficient, and \( x \) is the variable.
• The term -100,000 is a constant term in the expression.

Algebraic expressions permit manipulation, such as substitution and simplification. This is key to calculating profit in our problem by plugging in specific values for \( x \) and solving accordingly.

Polynomials

Polynomials are mathematical expressions that involve sums of powers of variables with coefficients. They are foundational to understanding algebraic relationships. In our example:

• The polynomial is \( P(x) = 45x - 100,000 \), which is a linear polynomial.
• It consists of one variable \( x \) and the polynomial degree is one, indicating a straight line when graphed.

Understanding polynomials is crucial since they model relationships and scenarios in standardized forms, such as profit calculations, making complex real-world matters more approachable.