Problem 50 - Algebra 2

Question

A company estimates that it costs \(0.03 x^{2}+4 x+1000\) dollars to produce \(x\) units of a product. Find an expression for the average cost per unit.

Step-by-Step Solution

Answer

The average cost per unit is \(0.03x + 4 + \frac{1000}{x}\).

Step 1: Understand the Cost Function

The cost function given is \(0.03x^2 + 4x + 1000\). This function represents the total cost in dollars to produce \(x\) units of a product.

Step 2: Define the Average Cost Formula

The average cost per unit is defined as the total cost divided by the number of units. Mathematically, this is represented as: \[ \text{Average Cost} = \frac{\text{Total Cost}}{x} \] where \(x\) is the number of units produced.

Step 3: Substitute the Total Cost into the Average Cost Formula

Substituting the given cost function into the average cost formula gives: \[ \text{Average Cost} = \frac{0.03x^2 + 4x + 1000}{x} \]

Step 4: Simplify the Expression

Simplify the expression by dividing each term in the numerator by \(x\): \[ 0.03x^2 \div x + 4x \div x + 1000 \div x = 0.03x + 4 + \frac{1000}{x} \]

Step 5: Present the Simplified Expression

The expression for the average cost per unit is \(0.03x + 4 + \frac{1000}{x}\). This formula can be used to compute average cost for any number of units \(x\).

Key Concepts

Understanding the Cost Function

A cost function is a mathematical equation that describes the total cost involved in producing a certain number of items. For this exercise, the cost function is given as \(0.03x^2 + 4x + 1000\), where \(x\) represents the number of units produced and each part of the equation contributes to the total production cost.

• \(0.03x^2\) can be thought of as a variable cost that depends on the square of the number of units produced. This might represent the increased costs related to scaling up production.
• \(4x\) is a linear cost associated directly with the number of units, suggesting a consistent cost per item produced.
• The \(1000\) is a fixed cost, independent of the production volume. It could cover expenses that do not change, such as facility rent.

Understanding these components helps in grasping how costs relate to production levels.

Expressing and Understanding Algebraic Expressions

An algebraic expression involves numbers, variables, and arithmetic operations. Understanding how to work with these expressions is crucial for solving problems like finding the average cost. In this case, you've been provided with the algebraic expression \(0.03x^2 + 4x + 1000\), which models the cost function.
Breaking down the expression:

• "Variable terms" like \(0.03x^2\) and \(4x\) indicate parts of the cost that change with different production levels.
• The "constant term" \(1000\) is fixed, no matter the value of \(x\).

By plugging this expression into another formula, such as average cost, you learn how variables and constants affect overall costs.
This skill is essential for making predictions and economic decisions in realistic scenarios.

Applying Polynomial Simplification

Polynomial simplification involves reducing expressions to their simplest form by cancelling out like terms and minimizing complexity. This is especially important when calculating average costs to make the formula more understandable and applicable.
For the average cost, we use the expression \(\frac{0.03x^2 + 4x + 1000}{x}\). Consider dividing each term by \(x\):

• \(0.03x^2 \div x = 0.03x\)
• \(4x \div x = 4\)
• \(1000 \div x = \frac{1000}{x}\)

After simplification, the expression becomes \(0.03x + 4 + \frac{1000}{x}\).
This makes it easy to calculate the average cost for varying production quantities and reflect on how changes in production affect cost efficiency.Simplifying polynomials is a critical math skill, helping in both academic and real-world contexts, by transforming complex equations into more manageable forms.