Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 
b. Interpret the meaning of your results in part (a).

Verified step by step guidance

1. To interpret the meaning of the results from part (a), we first need to understand the function C(x) = −0.02x² + 400x + 5000. This function represents the cost of producing x lawn mowers.
2. The term −0.02x² indicates that the cost function is quadratic, and the negative coefficient suggests that the cost increases at a decreasing rate as more lawn mowers are produced. This is typical of economies of scale where producing more units can reduce the average cost per unit.
3. The term 400x represents the linear component of the cost function, which implies that there is a direct cost associated with each lawn mower produced. This could include materials, labor, and other variable costs.
4. The constant term 5000 represents fixed costs, which are costs that do not change with the number of lawn mowers produced. These could include rent, salaries of permanent staff, and other overheads.
5. In part (a), you likely found the derivative of C(x) to determine the marginal cost, which is the cost of producing one additional lawn mower. Interpreting this result helps understand how the cost changes with production levels and can inform decisions on optimal production quantities.

Key Concepts

Cost Function

A cost function, such as C(x) = -0.02x² + 400x + 5000, represents the total cost of producing a certain number of goods, in this case, lawn mowers. It typically includes fixed costs and variable costs, where the quadratic term indicates that costs may increase at a decreasing rate after a certain production level.

Derivative and Marginal Cost

The derivative of the cost function, C'(x), provides the marginal cost, which is the cost of producing one additional unit. Understanding this concept is crucial for interpreting how production levels affect costs and for making decisions about scaling production.

Interpretation of Results

Interpreting results involves analyzing the output from the cost function and its derivative to understand the economic implications. This includes assessing how changes in production volume impact overall costs and profitability, which is essential for making informed business decisions.