Evaluate the marginal cost and average cost

Oct 13, 2025 by windailys.com 13 views

Optimizing Production Costs: A Comprehensive Analysis of the Given Cost Function

In the realm of economics and business, understanding the cost function is crucial for making informed decisions about production levels and pricing strategies. The cost function, denoted by C(x), represents the total cost of producing x units of a product. In this article, we will delve into the analysis of a given cost function, C(x) = 16900 + 890x + x^2, and explore various aspects of production costs.

The given cost function is a quadratic function, which means it has a parabolic shape. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this case, the cost function is C(x) = 16900 + 890x + x^2. The coefficients of the quadratic function represent the following:

a = 1: This is the coefficient of the squared term, which represents the rate of change of the cost function with respect to the production level.
b = 890: This is the coefficient of the linear term, which represents the rate of change of the cost function with respect to the production level.
c = 16900: This is the constant term, which represents the fixed cost of production.

To find the cost at the production level 1250, we need to substitute x = 1250 into the cost function C(x) = 16900 + 890x + x^2.

C(1250) = 16900 + 890(1250) + (1250)^2 C(1250) = 16900 + 1115000 + 1562500 C(1250) = 2894500

Therefore, the cost at the production level 1250 is $2,894,500.

The average cost is defined as the total cost divided by the number of units produced. To find the average cost at the production level 1250, we need to divide the total cost by the number of units produced.

Average Cost = Total Cost / Number of Units Produced Average Cost = C(1250) / 1250 Average Cost = 2894500 / 1250 Average Cost = 2316

Therefore, the average cost at the production level 1250 is $2,316.

The marginal cost is defined as the rate of change of the cost function with respect to the production level. To find the marginal cost at the production level 1250, we need to find the derivative of the cost function and evaluate it at x = 1250.

C'(x) = 890 + 2x C'(1250) = 890 + 2(1250) C'(1250) = 890 + 2500 C'(1250) = 3390

Therefore, the marginal cost at the production level 1250 is $3,390.

To find the production level that will minimize the cost, we need to find the vertex of the parabola represented by the cost function. The x-coordinate of the vertex is given by the formula x = -b / 2a.

x = -b / 2a x = -890 / (2 * 1) x = -890 / 2 x = -445

Therefore, the production level that will minimize the cost is x = -445. However, since the production level cannot be negative, we need to find the production level that will minimize the cost in the feasible region.

To find the production level that will minimize the cost in the feasible region, we need to find the point where the marginal cost is equal to the average cost.

C'(x) = Average Cost 890 + 2x = 2316 2x = 1426 x = 713

Therefore, the production level that will minimize the cost in the feasible region is x = 713.

In conclusion, we have analyzed the given cost function C(x) = 16900 + 890x + x^2 and explored various aspects of production costs. We have found the cost at the production level 1250, the average cost at the production level 1250, the marginal cost at the production level 1250, and the production level that will minimize the cost. The results of this analysis can be used to inform decisions about production levels and pricing strategies.
Frequently Asked Questions: Optimizing Production Costs

A: The cost function, denoted by C(x), represents the total cost of producing x units of a product. It is a crucial concept in economics and business because it helps decision-makers understand the relationship between production levels and costs.

A: The cost function typically includes the following types of costs:

Fixed costs: These are costs that remain the same even if the production level changes. Examples include rent, salaries, and insurance.
Variable costs: These are costs that change with the production level. Examples include raw materials, labor, and energy.
Sunk costs: These are costs that have already been incurred and cannot be changed. Examples include investments in equipment and infrastructure.

A: To find the cost at a specific production level, you need to substitute the production level into the cost function and evaluate it. For example, if the cost function is C(x) = 16900 + 890x + x^2 and the production level is x = 1250, you would substitute x = 1250 into the cost function and evaluate it.

A: The average cost is the total cost divided by the number of units produced. It represents the cost per unit of production. The marginal cost, on the other hand, is the rate of change of the cost function with respect to the production level. It represents the additional cost of producing one more unit.

A: To find the production level that will minimize the cost, you need to find the vertex of the parabola represented by the cost function. The x-coordinate of the vertex is given by the formula x = -b / 2a. However, if the production level cannot be negative, you need to find the production level that will minimize the cost in the feasible region by setting the marginal cost equal to the average cost.

A: The cost function has numerous applications in economics and business, including:

Pricing strategies: The cost function helps decision-makers determine the optimal price for a product based on the production level and costs.
Production planning: The cost function helps decision-makers determine the optimal production level based on the costs and revenue.
Investment decisions: The cost function helps decision-makers evaluate the feasibility of investments based on the costs and potential returns.

A: To use the cost function to make informed decisions about production levels and pricing strategies, you need to:

Analyze the cost function: Understand the relationship between the production level and costs.
Evaluate the marginal cost and average cost: Determine the additional cost of producing one more unit and the cost per unit of production.
Find the production level that will minimize the cost: Determine the optimal production level based on the costs and revenue.
Determine the optimal price: Determine the optimal price for a product based on the production level and costs.

By following these steps, you can use the cost function to make informed decisions about production levels and pricing strategies.