Problem 35 - Single Variable Calculus

Question

A division of Ditton Industries manufactures the "Spacemaker" model microwave oven. Suppose that the daily total cost (in dollars) of manufacturing \(x\) microwave ovens is $$C(x)=0.0002 x^{3}-0.06 x^{2}+120 x+6000$$ What is the marginal cost when \(x=200\) ? Compare the result with the actual cost incurred by the company in manufacturing the 201 st oven.

Step-by-Step Solution

Answer

The marginal cost when x = 200 is $24, while the actual cost of manufacturing the 201st oven is $23.602. Although the marginal cost is slightly higher than the actual cost, the values are very close, indicating that the marginal cost function provides a good approximation of the actual cost for manufacturing an additional oven.

Step 1: Find the derivative of the total cost function C(x) with respect to x

To find the marginal cost function, we need to differentiate the total cost function C(x) with respect to x. Given total cost function: \(C(x) = 0.0002x^3 - 0.06x^2 + 120x + 6000\) Differentiating the total cost function with respect to x: \(C'(x) = \frac{d}{dx}(0.0002x^3 - 0.06x^2 + 120x + 6000)\)

Step 2: Apply the power rule to differentiate C(x)

Differentiate each term with respect to x using the power rule (for any term ax^n, the derivative with respect to x is n*ax^(n-1)): \(C'(x) = 3(0.0002)x^2 - 2(0.06)x + 120\)

Step 3: Simplify the derivative

Simplify the expression for the marginal cost function C'(x): \(C'(x) = 0.0006x^2 - 0.12x + 120\) This is the marginal cost function.

Step 4: Evaluate the marginal cost function at x = 200

Plug in x = 200 into the marginal cost function: \(C'(200) = 0.0006(200)^2 - 0.12(200) + 120\) Calculate the value: \(C'(200) = 24\) The marginal cost when x = 200 is $24.

Step 5: Calculate the actual cost of manufacturing the 201st oven

To find the actual cost of manufacturing the 201st oven, we can find the difference between the total cost of manufacturing 201 ovens and the total cost of manufacturing 200 ovens. Actual cost of the 201st oven = C(201) - C(200) Plug in x = 201 and x = 200 into the total cost function: Actual cost of the 201st oven = \((0.0002(201)^3 - 0.06(201)^2 + 120(201) + 6000) - (0.0002(200)^3 - 0.06(200)^2 + 120(200) + 6000)\) Calculate the value: Actual cost of the 201st oven = $23.602

Step 6: Compare the marginal cost with the actual cost

The marginal cost of manufacturing when x = 200 is \(24, while the actual cost of manufacturing the 201st oven is \)23.602. The marginal cost is slightly higher than the actual cost, but they are very close in value. This indicates that the marginal cost function provides a good approximation of the actual cost for manufacturing an additional oven.

Key Concepts

Cost Function Derivation

Understanding the derivation of cost functions is crucial for businesses and economists alike as it helps in predicting expenses and setting pricing strategies. In our example, Ditton Industries provides the total cost function for manufacturing 'Spacemaker' microwave ovens as
\[C(x) = 0.0002x^3 - 0.06x^2 + 120x + 6000.\]
To analyze cost behavior, we target the marginal cost which is derived by finding the rate of change of the total cost with respect to the quantity produced, that is,
\[C'(x).\]
Derivation involves using calculus to obtain the function that reports the cost addition of producing one more unit, hence obtaining the marginal cost.

Power Rule Differentiation

Power rule differentiation is a fundamental technique in calculus used to find the derivative of functions with power terms. When dealing with a function like
\[ax^n,\]
the power rule allows us to quickly find its derivative, which is
\[n \times ax^{n-1}.\]
Applying this to each term in the total cost function of Ditton Industries, we differentiate
\[0.0002x^3, -0.06x^2,\] and \[120x\]
separately. The derivative or marginal cost function then simplifies to
\[C'(x) = 0.0006x^2 - 0.12x + 120,\]
which shows how costs change with each additional unit produced.

Actual Cost Calculation

Sometimes, it's essential to consider the actual cost incurred for producing an additional unit to compare it against the estimated marginal cost. The actual cost is determined by computing the difference in total cost for producing 'n+1' units and 'n' units. For our exercise, the actual cost of the 201st oven is acquired by evaluating the total cost function at \(x = 201\) and \(x = 200\), and then finding the difference. This provides the real expense Ditton Industries faces for producing that one extra oven after 200 ovens. Although close to the marginal cost, this actual cost figure serves as a precise check against the estimated marginal expense.

Total Cost Function

The total cost function, \(C(x)\), is the sum of all costs associated with producing 'x' units of a product. In the case of Ditton Industries, the total cost includes a cubic term, a quadratic term, a linear term, and a constant term, each representing different components of cost behavior. The cubic term can stand for a volume discount or increased efficiency at scale, while the quadratic and linear terms can represent variable costs such as labor and materials. The constant term generally represents fixed costs, which are present regardless of the manufacturing volume. Analyzing the total cost function helps businesses discern trade-offs between production levels and costs. Its close cousin, the marginal cost function \(C'(x)\), is an application of the derivative of this function and is critical for making decisions on how much quantity to supply.