Problem 110 - Algebra and Trigonometry

Question

Average Cost \(A\) clothing manufacturer finds that the cost of producing \(x\) shirts is \(500+6 x+0.01 x^{2}\) dollars. (a) Explain why the average cost per shirt is given by the rational expression $$ A=\frac{500+6 x+0.01 x^{2}}{x} $$ (b) Complete the table by calculating the average cost per shirt for the given values of \(x .\) $$ \begin{array}{|c|c|}\hline x & {\text { Average cost }} \\ \hline 10 & {} \\\ {20} \\ {50} \\ {100} \\ {200} \\ {500} \\ {500} \\ {1000} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

The average cost per shirt decreases generally as more shirts are produced, exemplifying economies of scale.

Step 1: Understand Total Cost

The given cost function for producing \(x\) shirts is \(C(x) = 500 + 6x + 0.01x^2\). This expression represents the total cost in dollars of manufacturing \(x\) shirts.

Step 2: Define Average Cost

The average cost per shirt, \(A\), is the total cost divided by the number of shirts produced. This is expressed as \(A = \frac{500 + 6x + 0.01x^2}{x}\). This rational expression simplifies the cost to a per-shirt basis.

Step 3: Calculate Average Cost for x=10

Substitute \(x = 10\) into the average cost formula to find \(A\). So, \(A = \frac{500 + 6 \times 10 + 0.01 \times 10^2}{10} = \frac{500 + 60 + 1}{10} = \frac{561}{10} = 56.1\).

Step 4: Calculate Average Cost for x=20

Substitute \(x = 20\) into the formula: \(A = \frac{500 + 6 \times 20 + 0.01 \times 20^2}{20} = \frac{500 + 120 + 4}{20} = \frac{624}{20} = 31.2\).

Step 5: Calculate Average Cost for x=50

Substitute \(x = 50\): \(A = \frac{500 + 6 \times 50 + 0.01 \times 50^2}{50} = \frac{500 + 300 + 25}{50} = \frac{825}{50} = 16.5\).

Step 6: Calculate Average Cost for x=100

Substitute \(x = 100\): \(A = \frac{500 + 6 \times 100 + 0.01 \times 100^2}{100} = \frac{500 + 600 + 100}{100} = \frac{1200}{100} = 12.0\).

Step 7: Calculate Average Cost for x=200

Substitute \(x = 200\): \(A = \frac{500 + 6 \times 200 + 0.01 \times 200^2}{200} = \frac{500 + 1200 + 400}{200} = \frac{2100}{200} = 10.5\).

Step 8: Calculate Average Cost for x=500

Substitute \(x = 500\) (appears twice): \(A = \frac{500 + 6 \times 500 + 0.01 \times 500^2}{500} = \frac{500 + 3000 + 2500}{500} = \frac{6000}{500} = 12.0\).

Step 9: Calculate Average Cost for x=1000

Substitute \(x = 1000\): \(A = \frac{500 + 6 \times 1000 + 0.01 \times 1000^2}{1000} = \frac{500 + 6000 + 10000}{1000} = \frac{16500}{1000} = 16.5\).

Key Concepts

Rational Expressions

Rational expressions are an important tool in algebra, especially when dealing with functions and equations. They are fractions in which both the numerator and the denominator are polynomials. In the context of cost functions, they can help us determine the average cost per item produced. This is because the total cost of production needs to be divided by the number of items, creating a ratio or fraction. The given cost function, which is a polynomial, can be easily turned into a rational expression in this way. Advantages of using rational expressions include:

• Making complex algebraic equations easier to work with.
• Providing useful insights into the relationship between the total and per-unit cost by simplifying expressions.
• Helping visualize how costs change in relation to production levels.

Understanding rational expressions is essential, as they are frequently used in business, economics, and everyday decision-making scenarios.

Cost Functions

Cost functions are mathematical models that express the total cost incurred by a company in producing a specific number of goods. The total cost is generally affected by fixed costs, such as rent or machinery expenses, and variable costs, like raw materials and labor. In the exercise, the total cost function is given by \[C(x) = 500 + 6x + 0.01x^2\], where:

• The constant term \(500\) represents fixed costs.
• The linear term \(6x\) represents costs that vary directly with production.
• The quadratic term \(0.01x^2\) illustrates increasing marginal costs, which occur as production expands.

This polynomial reflects the detailed manner in which manufacturing costs increase with production volume. Businesses need to grasp the nuances of cost functions, as they inform pricing strategies and profitability. By analyzing these equations, businesses can forecast potential expenses and create cost-effective production plans.

Average Cost Calculation

To understand how the average cost is calculated, imagine needing to determine the cost per shirt when you are a clothing manufacturer. Calculating the average involves dividing the total cost by the number of units produced, resulting in a rational expression. In our problem, the average cost per shirt seamlessly derives from the total cost function. It is expressed as \[A = \frac{500 + 6x + 0.01x^2}{x}\].Here's how the average cost calculation works:

• Substitute different values of \(x\) (number of shirts produced) into the formula.
• Simplify the expression to find the average cost for that specific number of shirts.

For example, to find the average cost when producing 10 shirts, you would calculate \[A = \frac{500 + 60 + 1}{10} = 56.1\].Calculating average cost is vital for businesses because it allows insight into cost efficiency and helps identify the production level at which economies of scale are achieved. As production increases, average cost often decreases, highlighting the benefits of producing goods in bulk.