Problem 59 - Applied Calculus

Question

The cost (in dollars) of producing \(q\) items is given by \(C(q)=0.08 q^{3}+75 q+1000\) (a) Find the marginal cost function. (b) Find \(C(50)\) and \(C^{\prime}(50)\). Give units with your answers and explain what each is telling you about costs of production.

Step-by-Step Solution

Answer

The marginal cost function is \(C'(q) = 0.24q^2 + 75\). \(C(50) = 5750\) dollars and \(C'(50) = 675\) dollars per item.

Step 1: Understand the Cost Function

The cost function given is \(C(q) = 0.08q^3 + 75q + 1000\), which represents the total cost in dollars to produce \(q\) items. Our goal is to find the marginal cost function, which is the derivative of this cost function with respect to \(q\).

Step 2: Differentiate to Find Marginal Cost

To find the marginal cost, we differentiate \(C(q)\) with respect to \(q\). The derivative \(C'(q)\) gives the marginal cost function.Differentiate term by term:- The derivative of \(0.08q^3\) is \(0.24q^2\).- The derivative of \(75q\) is \(75\).- The derivative of \(1000\) (a constant) is \(0\).Thus, \(C'(q) = 0.24q^2 + 75\).

Step 3: Calculate C(50)

Substitute \(q = 50\) into the original cost function:\[C(50) = 0.08(50)^3 + 75(50) + 1000\]Calculate:- \(0.08 \times 125000 = 1000\)- \(75 \times 50 = 3750\)- Therefore, \(C(50) = 1000 + 3750 + 1000 = 5750\)\(C(50)\) tells us that the cost to produce 50 items is \$5750.

Step 4: Calculate C'(50)

Substitute \(q = 50\) into the marginal cost function:\[C'(50) = 0.24(50)^2 + 75\]Calculate:- \(0.24 \times 2500 = 600\)- Therefore, \(C'(50) = 600 + 75 = 675\)\(C'(50)\) tells us that the marginal cost of producing the 51st item is \$675.

Key Concepts

Cost Function

A cost function is a mathematical expression that represents the total cost of production for producing a certain number of items, often denoted as \(q\). In the given exercise, the cost function is \(C(q) = 0.08q^3 + 75q + 1000\). This function includes several components:

• A variable cost, \(0.08q^3\), which may represent the cost that changes with the production level.
• A linear cost term, \(75q\), which increases proportionally with the quantity produced.
• A fixed cost, represented by \(1000\), which is constant regardless of the production quantity.

Understanding the breakdown of these components helps in analyzing how costs change with different levels of production.

Derivative

The derivative is a fundamental concept in calculus used to determine the rate of change of a function with respect to a variable. In the context of cost functions, the derivative is key to finding the marginal cost. For the function \(C(q) = 0.08q^3 + 75q + 1000\), the derivative is taken with respect to \(q\), the quantity of items. By differentiating each term, we find:

• The derivative of \(0.08q^3\) is \(0.24q^2\).
• The derivative of \(75q\) is \(75\).
• The derivative of the constant \(1000\) is \(0\).

Thus, the marginal cost function, the derivative \(C'(q)\), is \(0.24q^2 + 75\). This function provides important insights into how the cost changes with the production of one additional item.

Production Cost

Production cost refers to the total expense incurred in manufacturing a specific number of products. For this exercise, when \(q = 50\), we substitute into the cost function to find the total cost at this production level: \(C(50) = 0.08(50)^3 + 75(50) + 1000\).Performing the calculations:

• \(0.08 \times 125000 = 1000\)
• \(75 \times 50 = 3750\)
• Adding these to the fixed cost gives \(C(50) = 5750\)

This means the total production cost for 50 items is \$5750. Understanding production cost helps businesses in price setting and cost management strategies.

Marginal Analysis

Marginal analysis is a technique used in economics and finance to analyze small changes in production and their impact on costs and revenues. The marginal cost is a measure derived from this analysis. For \(q = 50\), when we look at the marginal cost function \(C'(q) = 0.24q^2 + 75\), and substitute \(q = 50\), we get:\(C'(50) = 0.24(50)^2 + 75\).After calculation:

• \(0.24 \times 2500 = 600\)
• Adding the constant gives \(C'(50) = 675\)

Thus, the marginal cost of producing the 51st item is \$675. Marginal analysis is crucial for decision-making in production as it tells us if producing one more unit is profitable.