Problem 36 - Single Variable Calculus

Question

Marginal Average Cost of Producing Television Sets The Advance Visual Systems Corporation manufactures a 19 -inch LCD HDTV. The weekly total cost incurred by the company in manufacturing \(x\) sets is $$C(x)=0.000002 x^{3}-0.02 x^{2}+120 x+70,000$$ dollars. a. Find the average cost function \(\bar{C}(x)\) and the marginal average cost function \(C^{\prime}(x)\). b. Compute \(\bar{C}^{\prime}(5000)\) and \(\bar{C}^{\prime}(10,000)\), and interpret your results.

Step-by-Step Solution

Answer

The average cost function is given by \(\bar{C}(x) = 0.000002 x^{2} - 0.02x + 120 + \frac{70,000}{x}\) and the marginal average cost function is \(C'(x) = 0.000004x - 0.02 - \frac{70,000}{x^2}\). At 5,000 sets, the marginal average cost is -0.00088, meaning the average cost decreases with each additional unit produced. At 10,000 sets, the marginal average cost is 0.0022, indicating that the average cost per unit increases with each additional unit produced.

Step 1: Find the average cost function \(\bar{C}(x)\)

To find the average cost function, we need to divide the total cost function \(C(x)\) by the number of sets produced \(x\). Thus, \[\bar{C}(x) = \frac{C(x)}{x}\]

Step 2: Calculate \(\bar{C}(x)\) using the given total cost function

Now we substitute the given total cost function \(C(x)\) into the formula for average cost function: \[\bar{C}(x) = \frac{0.000002 x^{3} - 0.02x^{2} + 120x + 70,000}{x}\] To simplify, divide each term by \(x\): \[\bar{C}(x) = 0.000002 x^{2} - 0.02x + 120 + \frac{70,000}{x}\]

Step 3: Find the marginal average cost function \(C'(x)\)

To find the marginal average cost function, we need to differentiate the average cost function with respect to \(x\). Thus, \[C'(x) = \frac{d(\bar{C}(x))}{dx}\]

Step 4: Calculate \(C'(x)\) using the derived average cost function

Differentiate each term in the average cost function with respect to \(x\): \[C'(x) = 2\times0.000002 x - 0.02 + \frac{-1\times70,000}{x^2} = 0.000004x - 0.02 - \frac{70,000}{x^2}\]

Step 5: Calculate \(\bar{C}'(5000)\) and \(\bar{C}'(10,000)\)

Now we need to compute the values of the marginal average cost function at \(x=5000\) and \(x=10000\): \[\bar{C}'(5000) = 0.000004(5000) - 0.02 - \frac{70,000}{(5000)^2} = -0.00088\] \[\bar{C}'(10,000) = 0.000004(10,000) - 0.02 - \frac{70,000}{(10,000)^2} = 0.0022\]

Step 6: Interpret the results

When producing 5,000 sets, the marginal average cost is negative (-0.00088), meaning that the average cost will decrease for each additional unit produced. On the other hand, when producing 10,000 sets, the marginal average cost is positive (0.0022), implying that the average cost per unit will increase for each additional unit produced.

Key Concepts

Understanding the Average Cost Function

The average cost function plays a critical role in production and economics—it represents the total cost per unit of goods produced. In practical terms, when a company like Advance Visual Systems manufactures an LCD HDTV, their total costs include fixed costs (like building rent and machinery) and variable costs that change with the number of units produced, such as materials and labor.

To calculate the average cost function \(\bar{C}(x)\), we start by taking the total cost \(C(x)\), which includes both fixed and variable costs, and then divide it by the number of goods produced \(x\). It gives us a clear picture of how cost-per-unit changes with production scale. This insight is crucial for businesses to identify the most cost-efficient level of production.

It's also worth noting that the average cost curve can typically exhibit a 'U-shape' in the short run. Initially, costs may decrease as production increases due to the spreading of fixed costs over a larger number of goods. However, after reaching a certain production level, costs can start rising again due to factors such as increased labor costs or inefficiencies at higher output levels.

Marginal Cost Calculation

Grasping Marginal Cost

Let's dive into the concept of marginal cost. This is the cost of producing one additional unit. It's pivotal for businesses to know when the production of the next unit would make or break their bank. In mathematical terms, we compute marginal cost by taking the first derivative of the total cost function \(C(x)\) with respect to \(x\), the number of units produced.

This derivative represents the rate at which costs are changing with respect to production. This information is vital for making decisions—whether to ramp up production or slow it down. If the marginal cost is lower than the selling price, it might be profitable to produce more. However, if the marginal cost is higher, it means that producing more could lead to losses.

Analyzing Results

When we apply this to our example from Advance Visual Systems and calculate the marginal average cost for different production levels, we are essentially looking for the point where producing the next set of televisions either becomes more expensive (indicating we should not increase production further) or less costly (indicating we could benefit from producing more).

Cost Function Differentiation

The process of differentiation is at the heart of understanding and manipulating cost functions. It's like taking a microscope to the cost curve and seeing how it changes, incrementally, as production levels adjust.

Differentiation allows us to calculate the marginal average cost function \(C'(x)\) by finding the derivative of the average cost function \(\bar{C}(x)\), which we initially obtained by dividing the total cost by the number of units. By differentiating, we can find the exact rate at which the average cost per unit changes with each additional unit produced.

Following the differentiation of the given cost function in our problem, we found that the expression for the marginal average cost includes both decreasing and increasing components. The intricate balance of these parts tells us that the cost dynamics are not linear and that for different quantities of production, the unit cost behaves differently, with a critical point where costs shift from decreasing to increasing—a vital detail for any business in making production decisions.