Problem 7 - Algebra A Combined Function

Question

The average cost per DVD, in dollars, for a company to produce \(x\) DVDs on exercising is given by the formula \(A=\frac{3 x+400}{x},\) where \(A\) is the average cost per DVD and \(x\) is the number of DVDs produced. a. Find the cost for producing \(1 \mathrm{DVD}\). b. Find the average cost for producing 100 DVDs. c. Does the cost per DVD decrease or increase when more DVDs are produced? Explain your answer.

Step-by-Step Solution

Answer

a. 403 dollars; b. 7 dollars; c. Cost per DVD decreases as more DVDs are produced.

Step 1: Insert the value of x in the formula for 1 DVD

To find the cost of producing 1 DVD, substitute \(x = 1\) into the formula: \[A = \frac{3 \times 1 + 400}{1} = \frac{3 + 400}{1} = 403\] Thus, the cost for producing 1 DVD is 403 dollars.

Step 2: Insert the value of x in the formula for 100 DVDs

For producing 100 DVDs, substitute \(x = 100\) in the formula: \[A = \frac{3 \times 100 + 400}{100} = \frac{300 + 400}{100} = \frac{700}{100} = 7\] The average cost for producing 100 DVDs is 7 dollars.

Step 3: Analyze the behavior of the average cost as x increases

Observe the expression for \(A\) which is \(A = \frac{3x + 400}{x}\). To understand the behavior as \(x\) increases, simplify to: \[A = 3 + \frac{400}{x}\]As \(x\) increases, the term \(\frac{400}{x}\) decreases, thus making \(A\) decrease. Thus, the cost per DVD decreases as more DVDs are produced.

Key Concepts

Cost Function

Understanding the cost function is crucial in determining how expenses change relative to production levels. In this scenario, the cost function is represented by the formula \(A=\frac{3x+400}{x}\). This formula is used to calculate the average cost per DVD when producing \(x\) DVDs.
The function has two parts:

• The variable component \(3x\) indicates a linear increase in total costs with each additional DVD produced. This reflects the cost related to each unit, assumed here to be $3 per DVD.
• The fixed component, 400, represents costs that do not change with production levels, such as machinery or rent.

When calculating the cost function, substituting different values of \(x\) into the formula provides the average cost per DVD for that quantity. For instance, calculating for \(x = 1\) results in a higher average cost than for \(x = 100\), showing how fixed costs are spread more efficiently over larger production volumes.
By recognizing both fixed and variable costs, businesses can optimize production and pricing strategies effectively.

Economies of Scale

Economies of scale describe how larger production volumes can lead to lower costs per unit. This occurs because fixed costs, like the initial setup or administrative expenses, are distributed over more units.
In the given formula \(A=\frac{3x+400}{x}\), the average cost \(A\) explores the economies of scale when more DVDs are produced. As production increases, the term \(\frac{400}{x}\) decreases, indicating reduced contribution from fixed costs to the overall cost per DVD.
For example:

• At \(x = 1\), 400 is still divided by 1, resulting in a $400 contribution from fixed costs alone.
• However, at \(x = 100\), the contribution reduces to 4 dollars per DVD.

This illustrates that producing more DVDs reduces the weight of fixed costs in each unit's price, a key principle of economies of scale. Recognizing these cost efficiencies allows firms to reduce prices, increase competitiveness, or enhance profitability by expanding production.

Algebraic Expressions

Algebraic expressions like \(A=\frac{3x+400}{x}\) are vital in representing real-world scenarios involving costs and production. They allow us to simplify, rearrange, and interpret the relationships between different variables quantitatively.
This expression simplifies to \(A = 3 + \frac{400}{x}\), which highlights the interaction between variable costs (3 per DVD) and the influence of fixed costs expressed through \(\frac{400}{x}\).
Using algebraic manipulation, we can:

• Identify how average costs behave as we produce more DVDs.
• Predict outcomes for various production levels by substituting different \(x\) values.
• Understand the diminishing impact of fixed costs as production scales up.

This illustrates not only how algebra connects theoretical math to tangible business insights but also how it provides a framework to make informed decisions by recognizing patterns and calculating costs effectively.