Average and marginal profit Let C(x) represent the cost of pro...

Question

Average and marginal profit Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The profit P(x) of selling x items is P(x) = xp(c) - C(x) (revenue minus costs). The average profit per item when x items are sold is P(x)/x and the marginal profit is dP/dx. The marginal profit approximates the profit obtained by selling one more item, given that x items have already been sold. Consider the following cost functions C and price functions p.
a. Find the profit function P.
C(x) = − 0.04x²+100x+800, p(x)=200, a=1000

a. Find the profit function P.

C(x) = − 0.04x²+100x+800, p(x)=200, a=1000

Accepted Answer

Hello, today we're going to be solving the following problem. We are told that the cost function c and the selling price of each article p are given below. We want to determine the profit function p using the 2 given functions. So, capital P is representing the cost of selling articles. We're going to allow the variable x to equal the amount of articles that are being sold. So for now, let's go ahead and set p equal to an unknown amount of articles which is x. Now, we don't know how many articles we are going to sell, but we do know that the price for each article is given to us as 300. So what this means is that for every article that we sell, we're going to multiply the price to the unknown amount of articles. So we can have x multiplied by the price function, which is little p of x. For every article that we sell, we are making that amount of money. Then we have to consider the cost function. We are paying a certain amount to create the articles. So, from the amount of articles that are being sold, we're going to have to subtract the cost of each article which is given to us as the c function. So in other words, we can allow the capital P function to equal to x multiplied by the p function minus the c function. So now what we want to do is we want to plug in the following functions and simplify the function to get our profit function. So P is equal to x multiplied by the p function, which is given to us as 300, minus the cost function which is given to us as negative 0.05x2 plus 300x plus 700. Now what we want to do is we want to simplify the function. X multiplied by 300 will give us the value of 300x, and what we're going to do is we're going to distribute -one into the c function. Doing so will leave us with positive 0.05x2-300x-700. Now what we want to do is we want to combine the like terms. 300x-300x will cancel each other out. No other terms can be combined with each other, so the profit function, p of x, is equal to 0.05x2 minus 7 100, and this is going to be the simplest form of our profit function. And with that being said, the answer to this problem is going to be d. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

Key Concepts

Profit Function

Average Profit

Marginal Profit

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