Question

Suppose a large company makes 25,000 gadgets per year in batches of \(x\) items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, it has been determined that the total cost \(C(x)\) of producing 25,000 gadgets in batches of \(x\) items at a time is given by $$C(x)=1,250,000+\frac{125,000,000}{x}+1.5 x.$$ a. Determine the marginal cost and average cost functions. Graph and interpret these functions. b. Determine the average cost and marginal cost when \(x=5000\). c. The meaning of average cost and marginal cost here is different from earlier examples and exercises. Interpret the meaning of your answer in part (b).

Short answer

Also, find the average cost and marginal cost when producing a batch of 5000 gadgets, and interpret the meaning of both in this context. Answer: The marginal cost function is \(C'(x) = -\frac{125,000,000}{x^2}+1.5\) and the average cost function is \(A(x) = \frac{1,250,000}{x}+\frac{125,000,000}{x^2}+1.5\). When producing a batch of 5000 gadgets, the average cost is approximately $345 per gadget, and the marginal cost is approximately $0, indicating no significant change in the total cost by producing an additional gadget in a batch of 5000 items.

Step-by-step solution

Find the marginal cost function

To find the marginal cost function, we'll differentiate the total cost function, C(x), with respect to x: $$C'(x)=\frac{d}{dx}(1,250,000+\frac{125,000,000}{x}+1.5x)$$ Using the power rule and the chain rule, we get: $$C'(x)=0-\frac{125,000,000}{x^2}+1.5$$ So the marginal cost function is: $$C'(x)=-\frac{125,000,000}{x^2}+1.5$$

Find the average cost function

To find the average cost function, we will divide the total cost function C(x) by the number of items produced x: $$A(x)=\frac{C(x)}{x}=\frac{1,250,000+\frac{125,000,000}{x}+1.5x}{x}$$ Simplifying the expression, we get: $$A(x)=\frac{1,250,000}{x}+\frac{125,000,000}{x^2}+1.5$$ So the average cost function is: $$A(x)=\frac{1,250,000}{x}+\frac{125,000,000}{x^2}+1.5$$

Graph the marginal cost and average cost functions

Graph the functions C'(x) and A(x) using any graphing software or calculator. The graph shows how the marginal cost and average cost change as the number of items produced in a batch (x) varies. Observe the trend of these functions to interpret their meanings.

Calculate the average cost and marginal cost when x = 5000

To find the average cost and marginal cost for a batch size of 5000, plug x = 5000 into both functions: Average cost: $$A(5000)=\frac{1,250,000}{5000}+\frac{125,000,000}{5000^2}+1.5\approx 345$$ Marginal cost: $$C'(5000)=-\frac{125,000,000}{5000^2}+1.5\approx 0$$

Interpret the meaning of average cost and marginal cost

The average cost function gives the cost per item when producing gadgets in batches of x items. In this case, when x = 5000, the average cost is approximately $345 per gadget. The marginal cost function represents the change in total cost when producing one more item in a batch. In this case, when x = 5000, the marginal cost is approximately $0, which indicates that there is no significant change in the total cost by producing one more gadget in a batch of 5000 items. In this specific context, the average cost represents the cost of producing each gadget in a batch of x items, while the marginal cost indicates how the cost would change by producing an additional gadget in that batch.

Key concepts

Marginal Cost

The concept of marginal cost is an essential part of calculus and economics, particularly when analyzing production costs in businesses. Marginal cost refers to the additional cost incurred when producing one additional unit of a product. To find this in a given problem, we use the differentiation technique.
To determine the marginal cost from a cost function, we take the first derivative of the total cost function, denoted by \(C(x)\). This results in the marginal cost function \(C'(x)\), which shows the rate at which total cost changes with an increase in production quantity.
In our example, the marginal cost function is calculated by differentiating the cost function \(C(x) = 1,250,000 + \frac{125,000,000}{x} + 1.5x\). Using differentiation rules, particularly the power rule, we arrive at:

• \(C'(x) = -\frac{125,000,000}{x^2} + 1.5\)

With this function, we can predict cost changes for any additional item produced. Understanding marginal cost helps businesses make decisions about increasing or decreasing production.
When we calculate \(C'(5000)\) for \(x = 5000\), we find that it is around $0. This indicates that at this level of production, producing an additional gadget does not significantly change the total cost.

Average Cost

The average cost function allows us to determine the cost associated with producing each unit when a company makes a total number of units. It is calculated by dividing the total cost \(C(x)\) by the number of units \(x\). This yields the average cost function \(A(x)\).
For the given cost function \(C(x)\), the average cost function is found as follows:

• \(A(x) = \frac{1,250,000 + \frac{125,000,000}{x} + 1.5x}{x}\)
• Simplified: \(A(x) = \frac{1,250,000}{x} + \frac{125,000,000}{x^2} + 1.5\)

The average cost function tells us how the cost per item changes depending on the batch size \(x\). This is useful for determining the most cost-efficient production levels and pricing strategies.
When we substitute \(x = 5000\) into the average cost function, we find that \(A(5000) \approx 345\). This indicates that when producing 5000 gadgets in a batch, each gadget costs about $345. As smaller or larger batches are made, the average cost per gadget may vary significantly.

Differentiation

Differentiation is a fundamental technique in calculus used to find the rate at which a quantity changes. It is particularly powerful in economics and business for analyzing cost functions, profit functions, and more.
The process of differentiation involves finding the derivative of a function. The derivative represents how a function's output value changes as the input changes. For a function like the total cost \(C(x)\), the derivative \(C'(x)\) gives us the marginal cost — the rate of change of total cost with respect to the number of units produced.
Differentiation uses several rules like the power rule or chain rule to compute derivatives. For example, differentiating \(\frac{125,000,000}{x}\) involves using the power rule to convert the function into a form that's easier to differentiate.

• Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\)
• Chain Rule: Useful for composed functions to differentiate nested functions.

Understanding differentiation is crucial for effectively utilizing calculus in solving practical business problems, such as optimizing production and minimizing costs. By mastering this technique, companies can make informed decisions regarding production processes based on cost analyses.