Final answer:
To calculate the maximum profit, differentiate the profit function which is revenue minus expense, set the first derivative equal to zero, confirm it's a maximum with the second derivative, then evaluate the profit function at that price.
Step-by-step explanation:
To find the maximum profit, we must analyze the revenue (R) and expense (E) functions for a given product. The expense function given is E = -1,850p + 800,000, and the revenue function is R = -100p2 + 20,000p. The profit function (P) is defined as the revenue minus the expenses (P = R - E).
Substituting the given R and E functions into the profit formula we get:
P = (-100p2 + 20,000p) - (-1,850p + 800,000)
P = -100p2 + 20,000p + 1,850p - 800,000
P = -100p2 + 21,850p - 800,000
To find the maximum profit, we differentiate the profit function with respect to price (p) and set the first derivative equal to zero. This will give us the price at which the profit is maximized. We then calculate the second derivative to ensure that it is a maximum, and finally, we evaluate the profit function at the found price to determine the maximum profit value.