Problem 55 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

MANUFACTURING COST A manufacturer is capable of producing 5,000 units per day. There is a fixed (overhead) cost of \(\$ 1,500\) per day and a variable cost of \(\$ 2\) per unit produced. a. Express the daily cost \(C(x)\) as a function of the number of units produced, and sketch the graph of \(C(x)\). b. Find the average daily \(\operatorname{cost} A C(x)\). What is the average daily cost of producing 3,000 units per day? c. Is \(C(x)\) continuous? If not, where do its discontinuities occur?

Step-by-Step Solution

Answer

a. \(C(x) = 1500 + 2x\). b. The average cost function is \(AC(x) = \frac{1500}{x} + 2\). The average cost for 3,000 units is \$2.50 per unit. c. \(C(x)\) is continuous everywhere.

Step 1: Define the Cost Function

To express the daily cost, combine the fixed cost and the variable cost. The fixed cost is \$1,500 per day. The variable cost is \$2 per unit. Therefore, the total cost function, \(C(x)\), where \(x\) is the number of units produced per day, is given by: \[C(x) = 1500 + 2x\]

Step 2: Sketch the Graph of \(C(x)\)

Plot the cost function \(C(x) = 1500 + 2x\). The graph is a straight line with a y-intercept of \$1,500 and a slope of 2. For example, at \(x = 0\), \(C(x) = 1500\), and at \(x = 5000\), \(C(x) = 1500 + 2(5000) = 11500\).

Step 3: Calculate the Average Cost Function

The average daily cost, \(AC(x)\), is given by dividing the total cost \(C(x)\) by the number of units produced, \(x\). Therefore, \[AC(x) = \frac{C(x)}{x} = \frac{1500 + 2x}{x} = \frac{1500}{x} + 2\]

Step 4: Calculate the Average Daily Cost for 3,000 Units

To find the average daily cost of producing 3,000 units per day, substitute \(x = 3000\) into the average cost function: \[ AC(3000) = \frac{1500}{3000} + 2 = 0.5 + 2 = 2.5\] Therefore, the average daily cost of producing 3,000 units is \$2.50 per unit.

Step 5: Determine Continuity of \(C(x)\)

The function \(C(x) = 1500 + 2x\) is a linear function. Linear functions are continuous everywhere. Therefore, \(C(x)\) is continuous for all values of \(x\). There are no discontinuities.

Key Concepts

Cost Function

In manufacturing, understanding the cost function is crucial.
A cost function, usually denoted as \(C(x)\), helps in determining the total expense incurred when producing a certain number of units, \(x\).
Factoring in both fixed and variable costs is essential to formulating this function.
Fixed costs are everyday expenses that do not change with the level of production. In our problem, this is \$1500\ per day.
On the other hand, variable costs change with production levels. Here, it is \$2\ per unit produced.
The total cost function hence is: \[C(x) = 1500 + 2x\] The significance of understanding this function is to forecast expenses and manage budgets in a manufacturing setting.

Average Cost Function

The average cost function, \(AC(x)\), tells us the cost per unit produced.
It is derived by dividing the total cost function \(C(x)\) by the number of units, \(x\).
The formula is: \[AC(x) = \frac{C(x)}{x} = \frac{1500 + 2x}{x} = \frac{1500}{x} + 2\] Understanding this concept helps businesses determine the cost-efficiency of their production levels.
For instance, producing 3,000 units has an average cost of \$2.50\ per unit since: \[AC(3000) = \frac{1500}{3000} + 2 = 0.5 + 2 = 2.5\] This average cost function allows companies to decide the most cost-effective production level.

Continuity in Functions

The concept of continuity in functions ensures that the function behaves predictably.
A function is continuous if its graph can be drawn without lifting the pencil.
For \(C(x) = 1500 + 2x\), it's a linear function, which implies it's continuous for all values of \(x\).
Linear functions like this one do not have any breaks, holes, or jumps.
Therefore, there are no points of discontinuity in \(C(x)\).
This predictability is crucial for planning manufacturing processes and expenses as it means that costs increase steadily with production.

Linear Functions

Linear functions represent relationships with a constant rate of change.
The general form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In our cost function \(C(x) = 1500 + 2x\), \(2\) is the slope, indicating the rate at which costs increase per unit produced, and \$1500\ is the y-intercept, or fixed cost.
Linear functions provide simplicity and clarity in interpreting how costs grow with production.
This method is crucial for forming budgets and making production decisions based on expected strategies and cost analysis.