Question

suppose a company produces toasters. the total profit the company makes by selling $x$ toasters is given by $p(x)=30x - 0.3x^2 - 200$. the derivative $p(x)$ represents the marginal profit of producing $x$ toasters. which of the following does the marginal profit estimate?
○ the price the company should charge per toaster to maximize their profits
○ the number of toasters the company needs to sell to maximize their profits
○ the number of toasters the company needs to sell to make an additional dollar
○ the additional profit the company makes by selling an additional toaster

Explanation:

Marginal profit, defined as the derivative of the total profit function $P'(x)$, is a core concept that estimates the change in total profit resulting from selling one additional unit of a product. Analyzing the options:

1. The price to maximize profit is related to revenue/cost optimization, not marginal profit's definition.
2. The quantity to maximize profit is found by setting $P'(x)=0$, which uses marginal profit but is not what it estimates.
3. The number of units for an extra dollar is the inverse of marginal profit, not its estimate.
4. This matches the standard definition of marginal profit: the additional profit from one more unit sold.

Answer:

The additional profit the company makes by selling an additional toaster