Problem 59 - Calculus Early Transcendentals: Pearson New International Edition

Question

The total cost of producing and selling \(n\) units of a certain commodity per week is \(C(n)=1000+n^{2} / 1200\). Find the average cost, \(C(n) / n\), of each unit and the marginal cost at a production level of 800 units per week.

Step-by-Step Solution

Answer

The average cost per unit is \(\frac{1000}{n} + \frac{n}{1200}\) and the marginal cost at 800 units is approximately 1.33.

Step 1: Calculate Average Cost

The average cost per unit is found by dividing the total cost by the number of units, \(n\). Use the formula for the average cost function:\[\text{Average Cost} = \frac{C(n)}{n} = \frac{1000 + \frac{n^2}{1200}}{n}.\]Next, simplify the expression:\[\frac{1000}{n} + \frac{n^2}{1200n} = \frac{1000}{n} + \frac{n}{1200}.\]

Step 2: Simplify Average Cost Expression

Now, simplifying the expression obtained for average cost, we have:\[\text{Average Cost per unit} = \frac{1000}{n} + \frac{n}{1200}.\]This expression gives us the average cost per unit for producing \(n\) units.

Step 3: Find Marginal Cost Function

Marginal cost is derived by differentiating the total cost function, \(C(n)\), with respect to \(n\). We have:\[C(n) = 1000 + \frac{n^2}{1200}.\]Taking the derivative, \[\frac{d}{dn}(1000 + \frac{n^2}{1200}),\] we obtain:\[C'(n) = \frac{d}{dn}(1000) + \frac{d}{dn}(\frac{n^2}{1200}) = 0 + \frac{2n}{1200} = \frac{n}{600}.\]

Step 4: Compute Marginal Cost at 800 Units

Substitute \(n = 800\) into the marginal cost function \(C'(n) = \frac{n}{600}\):\[C'(800) = \frac{800}{600} = \frac{4}{3} \approx 1.33.\]This represents the marginal cost at a production level of 800 units per week.

Key Concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It provides tools to deduce information about the behavior of functions. In the world of economics and business, calculus helps in analyzing costs, revenues, and profits. A key idea in calculus is the derivative, which tells us how a function changes as its input changes. In cost analysis, calculus plays a vital role in determining marginal costs, which are essential for making efficient production decisions. By understanding calculus, businesses can optimize production and pricing strategies, ultimately boosting their profitability.

Average Cost

Average cost is a crucial concept in understanding production economics. It refers to the total cost of production divided by the quantity of goods produced, giving us a per-unit cost. For example, if the total production cost is expressed as a function such as \(C(n) = 1000 + \frac{n^2}{1200}\), where \(n\) is the number of units, the average cost \(C(n)/n\) can be found by dividing the total cost function by \(n\). This simplifies to \(\frac{1000}{n} + \frac{n}{1200}\), showing how variable costs per unit change with the number of units produced. Analyzing average cost helps businesses price their products effectively to cover costs and generate profits.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. In cost functions, the derivative represents the rate of change of total costs, known as the marginal cost.Taking the function \(C(n) = 1000 + \frac{n^2}{1200}\) as an example, the derivative \(C'(n)\) is derived through differentiation. We calculate \(\frac{d}{dn}(1000) + \frac{d}{dn}(\frac{n^2}{1200})\), which simplifies to \(0 + \frac{2n}{1200}\). This further reduces to \(\frac{n}{600}\).The derivative lets us understand how the cost of producing one additional unit changes with production quantities, allowing businesses to make informed cost-management decisions.

Economic Analysis

Economic analysis involves evaluating financial information to make strategic business decisions. One aspect of this is analyzing costs, including average and marginal costs, to understand the impact of production changes on profitability.Using economic analysis, businesses determine how changes in production levels affect costs. Marginal cost, which is found using the derivative of the total cost function, represents the additional cost of producing one more unit. This analysis helps firms decide whether increasing production will be economically beneficial.For instance, calculating the marginal cost at 800 units as \(C'(800) = \frac{4}{3} \approx 1.33\) informs a firm about the cost implications of producing one more unit beyond this level, aiding in production planning and pricing strategies.