The total profit (in dollars) from the sale of x gas grills...

The total profit (in dollars) from the sale of $x$ gas grills is $$ P(x)=20 x-0.02 x^{2}-320 \quad 0 \leq x \leq 1,000 $$ (A) Find the average profit per grill if 40 grills are produced. (B) Find the marginal average profit at a production level of 40 grills and interpret the results. (C) Use the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced.

The total profit (in dollars) from the sale of $x$ gas grills is
$$
P(x)=20 x-0.02 x^{2}-320 \quad 0 \leq x \leq 1,000
$$
(A) Find the average profit per grill if 40 grills are produced.
(B) Find the marginal average profit at a production level of 40 grills and interpret the results.
(C) Use the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced.

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Key Concepts

Average Profit Calculation

This concept involves determining the profit earned per unit by dividing the total profit by the number of units produced. It is a fundamental idea in economics and business, allowing one to assess the efficiency or profitability of producing each unit from a given total profit function.

Marginal Analysis

Marginal analysis uses the derivative of a function to measure the rate at which a quantity changes as its input changes. In economic contexts, taking the derivative of an average profit function with respect to the number of units produced yields the marginal average profit, which indicates the instantaneous rate of change in the average profit per unit when one more unit is produced.

Linear Approximation

Linear approximation relies on the idea that for small changes in the input of a differentiable function, the change in the function’s value can be closely estimated by multiplying the derivative at a point by the change in the input. This concept is crucial in estimating the effect of a slight increase or decrease in production on the average profit per unit.

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