Economics


Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².


a. Find the average cost per machine of producing the first 100 washing machines.

Verified step by step guidance

1. First, understand the given cost function: c(x) = 2000 + 100x - 0.1x², where c(x) represents the total cost of producing x washing machines.
2. To find the average cost per machine for the first 100 machines, we need to calculate the total cost for producing 100 machines, which is c(100).
3. Substitute x = 100 into the cost function: c(100) = 2000 + 100(100) - 0.1(100)².
4. Calculate c(100) to find the total cost of producing 100 machines.
5. Finally, divide the total cost c(100) by 100 to find the average cost per machine: Average cost = c(100) / 100.

Key Concepts

Average Cost

Average cost is calculated by dividing the total cost by the number of units produced. It provides insight into the cost efficiency of production. For the given function c(x), the average cost per machine for producing x washing machines is c(x)/x, which helps determine the cost per unit for a specific production level.

Cost Function

A cost function, such as c(x) = 2000 + 100x − 0.1x², represents the total cost of producing x units. It includes fixed costs, variable costs, and sometimes quadratic terms that account for changes in cost efficiency. Understanding this function is crucial for calculating total and average costs at different production levels.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically in the form ax² + bx + c. In economics, they can model cost functions where the cost changes non-linearly with production levels. The term −0.1x² in the cost function indicates diminishing returns or increasing inefficiencies as production increases, affecting the average cost calculation.