Problem 160 - Calculus Volume 1

Question

The cost function, in dollars, of a company that manufactures food processors is given by \(C(x)=200+\frac{7}{x}+\frac{x^{2}}{7}, \quad\) where \(x\) is the number of food processors manufactured. a. Find the marginal cost function. b. Find the marginal cost of manufacturing 12 food processors. c. Find the actual cost of manufacturing the thirteenth food processor.

Step-by-Step Solution

Answer

a. Marginal cost: \(-\frac{7}{x^2} + \frac{2x}{7}\). b. $3.380. c. $3.527.

Step 1: Differentiate Cost Function

To find the marginal cost function, we need to take the derivative of the cost function \(C(x)\). The given cost function is \(C(x)=200+\frac{7}{x}+\frac{x^{2}}{7}\). Differentiating term by term, we get:\[C'(x) = 0 - \frac{7}{x^2} + \frac{2x}{7}.\]Thus, the marginal cost function is \(C'(x) = -\frac{7}{x^2} + \frac{2x}{7}\).

Step 2: Calculate Marginal Cost at x=12

To find the marginal cost of manufacturing 12 food processors, substitute \(x=12\) into the marginal cost function \(C'(x)\).\[C'(12) = -\frac{7}{12^2} + \frac{2 \times 12}{7} = -\frac{7}{144} + \frac{24}{7}.\]Calculating these values gives us:\[C'(12) = -\frac{7}{144} + \frac{24}{7} \approx -0.0486 + 3.4286 \approx 3.380.\]

Step 3: Calculate Cost for 13th Processor

To find the actual cost of manufacturing the thirteenth food processor, calculate the difference in cost of manufacturing 13 and 12 processors using the original cost function.First, find \(C(13)\):\[C(13) = 200 + \frac{7}{13} + \frac{13^2}{7} = 200 + \frac{7}{13} + \frac{169}{7}.\]Calculating gives us:\[C(13) \approx 200 + 0.5385 + 24.1429 \approx 224.6814.\]Next, find \(C(12)\):\[C(12) = 200 + \frac{7}{12} + \frac{12^2}{7} = 200 + \frac{7}{12} + \frac{144}{7}.\]Calculating gives us:\[C(12) \approx 200 + 0.5833 + 20.5714 \approx 221.1547.\]Then, the actual cost of the thirteenth processor is:\[C(13) - C(12) \approx 224.6814 - 221.1547 \approx 3.5267.\]

Key Concepts

Cost Function Differentiation

To understand how to find the marginal cost function, we first need to differentiate the cost function. A cost function is a mathematical formula used to determine the total cost incurred by a company to produce a specific number of products. It usually depends on the number of products made, represented by a variable like \( x \). In this case, our cost function is given by:
\[ C(x) = 200 + \frac{7}{x} + \frac{x^2}{7} \]
By differentiating this cost function, we can find the marginal cost function, which tells us the rate at which the total cost changes as the production level \( x \) changes. Differentiation here involves applying the basic derivative rules to each term:

• The constant \(200\) becomes zero because constants have no rate of change.
• The term \( \frac{7}{x} \) is differentiated using the rule for derivatives of fractions, giving us \( -\frac{7}{x^2} \).
• The term \( \frac{x^2}{7} \) is differentiated as \( \frac{2x}{7} \) using the power rule.

So, the derivative or the marginal cost function, \( C'(x) \), is:
\[ C'(x) = -\frac{7}{x^2} + \frac{2x}{7} \]This function \(C'(x)\) provides insight into how costs change with each additional unit produced.

Calculating Marginal Cost

The marginal cost of producing one more unit, in this case, a food processor, can be found by evaluating the marginal cost function at a specific \( x \) value. Here, we are interested in finding this cost when 12 food processors are manufactured.
To do this, we substitute \( x = 12 \) into our marginal cost function:
\[ C'(12) = -\frac{7}{12^2} + \frac{2 \times 12}{7} \]
Calculating each term separately:

• \(-\frac{7}{144} \approx -0.0486\)
• \(\frac{24}{7} = 3.4286\)

Adding them together gives:
\[ C'(12) \approx -0.0486 + 3.4286 \approx 3.380 \]
This value, \(3.380\), represents the estimated cost change when producing the 12th food processor. The marginal cost function is crucial as it helps businesses make decisions about producing additional items based on how costs will increase.

Actual Cost Calculation

Calculating the actual cost of producing an additional item involves more than just looking at the marginal cost. Here, we determine the exact cost increase when manufacturing one more item.
In this scenario, we are interested in the cost to make the 13th food processor. To find this, we calculate the difference between the total cost of producing 13 processors and the cost for 12.
First, compute \( C(13) \):
\[ C(13) = 200 + \frac{7}{13} + \frac{169}{7} \]

• \(\frac{7}{13} \approx 0.5385\)
• \(\frac{169}{7} \approx 24.1429\)

Thus, \( C(13) \approx 224.6814 \).
Next, compute \( C(12) \):
\[ C(12) = 200 + \frac{7}{12} + \frac{144}{7} \]

• \(\frac{7}{12} \approx 0.5833\)
• \(\frac{144}{7} \approx 20.5714\)

Thus, \( C(12) \approx 221.1547 \).
Subtract \( C(12) \) from \( C(13) \) to find the actual cost of the 13th processor:
\[ C(13) - C(12) \approx 224.6814 - 221.1547 = 3.5267 \]
This result, \(3.5267\), reflects the real cost added by manufacturing the 13th processor. Actual cost calculations provide precise financial details that cannot be captured by marginal cost alone, hence proving crucial for budgeting and financial planning.