Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 
a. Determine the average and marginal costs for x = 3000 lawn mowers.

Verified step by step guidance

1. To find the average cost, use the formula for average cost: \( \text{Average Cost} = \frac{C(x)}{x} \). Substitute \( C(x) = -0.02x^2 + 400x + 5000 \) and \( x = 3000 \) into the formula.
2. Calculate \( C(3000) \) by substituting \( x = 3000 \) into the cost function \( C(x) = -0.02x^2 + 400x + 5000 \).
3. Once you have \( C(3000) \), divide it by 3000 to find the average cost for producing 3000 lawn mowers.
4. To find the marginal cost, first determine the derivative of the cost function \( C(x) \). The derivative \( C'(x) \) represents the marginal cost.
5. Calculate \( C'(3000) \) by substituting \( x = 3000 \) into the derivative \( C'(x) \) to find the marginal cost for producing the 3000th lawn mower.

Key Concepts

Cost Function

A cost function, such as C(x) = -0.02x² + 400x + 5000, represents the total cost of producing x units of a good. It typically includes fixed costs and variable costs, where the quadratic term indicates how costs change with production levels. Understanding the structure of the cost function is essential for calculating average and marginal costs.

Average Cost

The average cost is calculated by dividing the total cost C(x) by the number of units produced, x. It provides insight into the cost per unit at a specific production level. For example, the average cost for producing 3000 lawn mowers can be found by evaluating C(3000) and dividing by 3000, which helps in assessing efficiency and pricing strategies.

Marginal Cost

Marginal cost refers to the additional cost incurred from producing one more unit of a good. It is derived from the derivative of the cost function, C'(x), evaluated at a specific production level. For x = 3000, calculating the marginal cost involves finding C'(3000), which indicates how production decisions impact overall costs and profitability.