Question

Suppose that we're making and selling some widget and the profit (in dollars) when making a widgets is P(x) = 30 x 0.07² - 680. Find the marginal profit formula and the marginal average profit formula. Then determine the marginal profit and the marginal average profit when 54 widgets are made.

The marginal profit function is P'(x) =

Answer 1

The marginal profit function derived from the provided profit function P(x) is P'(x) = 30 - 0.14x. The marginal profit at 54 widgets is $22.44 per widget. The marginal average profit function needs additional steps for simplification and differentiation before substituting the value of 54 widgets.

Step-by-step explanation:

The provided profit function for the widgets is P(x) = 30x - 0.07x² - 680. To find the marginal profit function, we need to calculate the derivative of this function with respect to x, which gives us P'(x) = 30 - 0.14x. The marginal profit at x = 54 is then found by substituting 54 into the marginal profit function, which yields P'(54) = 30 - 0.14(54) = 30 - 7.56 = 22.44 dollars per widget.

To find the marginal average profit, we first need to calculate the average profit function, which is the total profit P(x) divided by the number of widgets x. This gives us the average profit function AP(x) = (30x - 0.07x² - 680)/x. The marginal average profit is the derivative of this average profit function with respect to x. After simplifying and finding the derivative, we can find the marginal average profit for 54 widgets similarly.