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Chapter 4: Problem 7

Calculate total cost, disregarding any fixed costs. Total cost from marginal costs. Raggs, Lid., determines that its marginal cost, in dollars per dress, is given by \(C^{\prime}(x)=-\frac{2}{25} x+50,\) for \(x \leq 450\) Find the total cost of producing the first 200 dresses.

Short Answer

Expert verified
The total cost of producing the first 200 dresses is \$8400.

Step by step solution

01

Integrate the Marginal Cost Function

To find the total cost of producing the first 200 dresses, integrate the marginal cost function. The marginal cost function is given by \[ C^{\text{'} }(x) = -\frac{2}{25}x + 50 \] Integrating this function will give us the total cost function.
02

Find the Integral

Compute the indefinite integral of the marginal cost function: \[ \int \left(-\frac{2}{25}x + 50\right) dx \] This results in: \[ C(x) = -\frac{2}{25} \cdot \frac{x^2}{2} + 50x + C_0 \] Simplify the expression: \[ C(x) = -\frac{1}{25}x^2 + 50x + C_0 \]
03

Apply the Initial Condition

Since we are only interested in the total cost of producing the first 200 dresses, disregard the constant of integration, \( C_0 \), as it represents fixed costs: \[ C(200) = -\frac{1}{25}(200)^2 + 50(200) \]
04

Calculate the Total Cost

Now substitute \( x = 200 \) into the total cost function: \[ C(200) = -\frac{1}{25}(40000) + 10000 \] Simplify: \[ C(200) = -1600 + 10000 \] So the total cost is: \[ C(200) = 8400 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Marginal Cost Function
Marginal cost is the additional cost incurred in producing one more unit of a good or service. It is a vital concept in economics because it helps firms decide the optimal level of production. In this exercise, the marginal cost function is given by the equation: This function describes how the cost of producing an additional dress changes as the number of dresses, denoted by 饾懃, increases. In this case, the marginal cost function is given by: 饾應鈥�(饾挋)=鈭捖�5饾挋+饾煋饾煄 where: 鈭捖�5饾挋 represents the variable cost per dress 饾煋饾煄 represents the fixed cost per dress Understanding the marginal cost function is crucial for cost calculation.
Integration
Integration is a core mathematical process used to calculate the
Indefinite Integral
To find the total cost function, we start by calculating the indefinite integral of the marginal cost function. The indefinite integral is represented by the formula: 鈭�(鈭捖�5饾挋+饾煋饾煄)饾拝饾挋 In this case, integrating 鈭捖�5饾挋 gives us the term 鈭捖金潚櫬�25, and integrating 饾煋饾煄 gives us 饾煋饾煄饾挋. Thus the indefinite integral of the marginal cost function becomes: 饾應(饾挋)=鈭捖金潚櫬�25+饾煋饾煄饾挋+饾應鈧� 饾應鈧� is the constant of integration, which represents the initial condition or fixed costs in this context.
Initial Condition
An initial condition is an extra piece of information used to find the exact value of the constant of integration. In cost calculation problems, the initial condition often represents fixed costs. For simplicity, if we disregard any fixed costs in this exercise, the equation simplifies to: 饾應(饾挋)=鈭捖金潚櫬�25+饾煋饾拹饾挋 Our focus here is to calculate the total cost of producing the first 饾煇饾煄饾煄 dresses.
Total Cost Function
The total cost function gives the total cost of producing '饾挋' number of units of a product. For the given exercise, the total cost function was found using the indefinite integral: 饾應(饾挋)=鈭捖金潚櫬�25+饾煋饾拹饾挋 By substituting 饾挋=饾煇饾煄饾煄 into the total cost function, we get: 饾應(饾煇饾煄饾煄)=鈭捖�(饾煇饾煄饾煄虏25)+饾煋饾煄(饾煇饾煄饾煄) 饾應(饾煇饾煄饾煄)=鈭掟潫忦潫旔潫庰潫�+饾煆饾煄饾煄饾煄饾煄 Therefore, the total cost of producing the first 饾煇饾煄饾煄 dresses is 饾煐饾煉饾煄饾煄 dollars.

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