Problem 90 - Introductory Algebra for College Students

Question

A company that manufactures small canoes has costs given by the equation $$ C=\frac{20 x+20,000}{x} $$ in which \(x\) is the number of canoes manufactured and \(C\) is the cost to manufacture each canoe. a. Find the cost per canoe when manufacturing 100 canoes. b. Find the cost per canoe when manufacturing \(10,000\) canoes. c. Does the cost per canoe increase or decrease as more canoes are manufactured? Explain why this happens.

Step-by-Step Solution

Answer

The cost per canoe when manufacturing 100 canoes is $220. For 10,000 canoes, the cost per canoe is $22. The cost per canoe decreases as more canoes are manufactured due to economies of scale.

Step 1: Substitute Value of Canoes

For part a, substitute \(x = 100\) into the equation \(C =\frac{20x + 20,000}{x}\) to find the cost per canoe when manufacturing 100 canoes.

Step 2: Calculate Cost Per Canoe for 100 Canoes

After substituting the value, the equation becomes \(C =\frac{20(100) + 20000}{100}\). Calculation gives \(C = 220\), meaning the cost per canoe when manufacturing 100 canoes is $220.

Step 3: Substitute Value of Canoes for 10,000 Canoes

For part b, substitute \(x = 10,000\) into the equation \(C =\frac{20x + 20,000}{x}\) to find the cost per canoe when manufacturing 10,000 canoes.

Step 4: Calculate Cost Per Canoe for 10,000 Canoes

After substituting the value, the equation becomes \(C =\frac{20(10000) + 20000}{10000}\). Upon simplifying, we find that \(C = 22\), implying the cost per canoe when manufacturing 10,000 canoes is $22.

Step 5: Interpret The Cost Change

For part c, it is observed that with the increase in the number of canoes manufactured, the cost per canoe decreases. This happens due to the nature of the formula which reflects the idea of economies of scale - a principle that says that as the volume of production increases, the cost per unit decreases.

Key Concepts

Economies of Scale

Understanding the concept of economies of scale is crucial for businesses and students of economics alike. This idea pertains to the cost advantage that businesses experience when production becomes efficient, as costs can be spread over a larger number of goods.

Economies of scale occur due to various factors. One contributing factor is the ability to purchase raw materials in bulk, which often comes at a discount. Fixed costs, like rent or salaries, also play a role. These costs do not increase with the volume of production, so as more units are produced, the fixed cost per unit drops.

The example of the canoe manufacturing company beautifully illustrates this principle. As the company produces more canoes, the cost of production for each canoe decreases. Not only does this concept help a business in assessing performance but also in strategic planning. Staying competitive requires an understanding and application of economies of scale.

Cost Per Unit

Cost per unit is a fundamental measurement in both accounting and operations management. It tells us how much it costs to produce a single unit of product. This measure is pivotal for pricing, budgeting, and financial forecasting.

In algebra, we often calculate cost per unit using rational expressions, where the cost formula is expressed as a ratio of two polynomial expressions. For instance, as seen in the canoe manufacturer's cost equation \( C=\frac{20x+20,000}{x} \), where \( x \) is the number of canoes produced.

By calculating the cost per unit at different production volumes, as was done for 100 canoes and then 10,000 canoes, a business can determine the most cost-effective scale of operation. When the production volume increases, the cost per unit can significantly drop, which underscores the interplay between cost analysis and economies of scale.

Rational Expressions

Rational expressions are fractions wherein the numerator and the denominator are both polynomials. Seen frequently in algebra, these expressions play an integral role in forming relationships in cost analysis and in various domains of mathematics and real-world applications.

The canoe manufacturer's cost equation is an example of a rational expression used to model the cost structure of production. When dealing with rational expressions for cost analysis, it's important to understand the implications of the variables involved. For example, in \( C=\frac{20x+20,000}{x} \), \( x \) represents the number of canoes made, and it affects the cost \( C \) directly.

Understanding how to manipulate and simplify these expressions is essential for accurately finding costs at different production levels. The reduction in cost per unit as production increases, as seen in the canoe case, is also a testament to the utility of rational expressions in understanding real-world business scenarios.