Problem 8 - Algebra A Combined Function

Question

For a certain model of fax machine, the manufacturing cost \(C\) per machine is given by the equation $$ C=\frac{250 x+10,000}{x} $$ where \(x\) is the number of fax machines manufactured and \(\operatorname{cost} C\) is in dollars per machine. a. Find the cost per fax machine when manufacturing 100 fax machines. b. Find the cost per fax machine when manufacturing 1000 fax machines. c. Does the cost per machine decrease or increase when more machines are manufactured? Explain why this is so.

Step-by-Step Solution

Answer

a. $350 b. $260 c. Cost decreases as more machines are manufactured.

Step 1: Understand the Cost Function

The cost function given is \( C = \frac{250x + 10,000}{x} \), where \( C \) is the cost per fax machine and \( x \) is the number of machines manufactured. This function will help us find the cost for different values of \( x \).

Step 2: Substitute x = 100 into the Cost Function

Substitute \( x = 100 \) into the equation: \[ C = \frac{250 \times 100 + 10,000}{100} \]. Calculate the numerator: \( 250 \times 100 = 25,000 \). Add the fixed cost: \( 25,000 + 10,000 = 35,000 \). So now the equation is \( C = \frac{35,000}{100} \).

Step 3: Calculate Cost for 100 Machines

Divide the total cost by 100: \( \frac{35,000}{100} = 350 \). Thus, the cost per fax machine when manufacturing 100 machines is \( 350 \) dollars.

Step 4: Substitute x = 1000 into the Cost Function

Substitute \( x = 1000 \) into the equation: \[ C = \frac{250 \times 1000 + 10,000}{1000} \]. Calculate the numerator: \( 250 \times 1000 = 250,000 \). Add the fixed cost: \( 250,000 + 10,000 = 260,000 \). So now the equation is \( C = \frac{260,000}{1000} \).

Step 5: Calculate Cost for 1000 Machines

Divide the total cost by 1000: \( \frac{260,000}{1000} = 260 \). Thus, the cost per fax machine when manufacturing 1000 machines is \( 260 \) dollars.

Step 6: Analyze the Cost Behavior

Compare the costs for 100 machines and 1000 machines. The cost decreased from \( 350 \) dollars to \( 260 \) dollars. This indicates that the cost per machine decreases as the number of machines manufactured increases. This is because the fixed cost of \( 10,000 \) is spread over more units, reducing the average cost per unit.

Key Concepts

Manufacturing Cost

Manufacturing cost refers to the total expense incurred in producing a good. This cost typically consists of variable and fixed components. In the given problem, the manufacturing cost function is expressed as:\[ C = \frac{250x + 10,000}{x} \]Here, \( 250x \) represents the variable cost, which changes with the number of machines manufactured. The \( 10,000 \) is the fixed cost, which doesn't change regardless of how many machines are made.

• Variable Cost: This is the cost that varies directly with the level of production. For every fax machine produced, the cost is \( 250 \) dollars.
• Fixed Cost: This remains constant and is not affected by the production level. In this case, it is \( 10,000 \) dollars spread across all fax machines manufactured.

Understanding these components is crucial as it helps in determining the cost per unit for different levels of production. By calculating how these costs spread over larger quantities, manufacturers can evaluate their pricing strategies more effectively.

Economies of Scale

Economies of scale occur when the cost per unit decreases as the volume of production increases. This is a significant concept for manufacturers aiming to reduce production costs.In the exercise, as the number of fax machines manufactured increases from 100 to 1000, the cost per machine drops from \( 350 \) dollars to \( 260 \) dollars. This reduction is a classic example of economies of scale. Let's understand why:

• The fixed cost of \( 10,000 \) is distributed over an increasing number of units, reducing each unit's share of the cost.
• Efficiency improves as production ramps up, which could lower the variable costs further.

By producing more, manufacturers can achieve lower average costs, giving them a competitive advantage. Recognizing this pattern is valuable for business planners who aim to optimize resource utilization and cost management.

Variable Costs

Variable costs are those expenses that change in proportion to the production volume. An increase in production results in higher variable costs and vice versa.In the problem, each fax machine has an associated variable cost of \( 250 \) dollars. This cost scales directly with the number of machines:

• If 100 machines are manufactured, the total variable cost is \( 250 \times 100 = 25,000 \) dollars.
• If 1000 machines are manufactured, the total variable cost becomes \( 250 \times 1000 = 250,000 \) dollars.

Variable costs allow flexibility in manufacturing, as they adjust with production levels, unlike fixed costs. Understanding the behavior of variable costs assists businesses in forecasting expenses and setting appropriate sales prices to ensure profitability. In this exercise, observing how variable costs interplay with fixed costs helps in understanding broader financial implications on manufacturing.