Cookie Writting Assignment

Cookie Writing Assignment

For the cookie optimization problem, I will be focusing on five main points in determining the optimal value to sell and the max profit. The five points I will focus on involve the cost function, revenue function, profit maximization, determining the best price, and dive into some potential variables that could be introduced to the equation.

For determining my cost as a function of demand, first I had to find the relevant information in the given problem. I determined that the materials and the price to make each cookie are what constitutes as cost. Using these numbers, I made an equation as follows: C(x) = 200+0.25x. I know that my cost for cookies will include my material costs and the price to make each cookie, as it was given as $200 for materials as a flat rate, plus $0.25/cookie (x) because it was $4 per 16 cookies. (4/16).

For determining my revenue as a function of demand I need demand and price (x, p). Because I do not know what demand is I show that with the variable x. The price was given to us as an equation of demand (p = 1.25-0.0025x). In order to make my revenue function I will need to multiply demand by the price-demand equation, or distribute an x to the price-demand equation. After multiplying my demand by my price, I got the revenue equation: R(x) = 1.25x-0.0025x^2.

Next step in the optimization problem is determining my profit as a function of demand. I know that profit in any occasion is measured by subtracting my cost from my revenue, so in this case I will subtract my cost function from my revenue function [(200+0.25x)- (1.25x-0.0025x^2)]. After subtracting my cost from my revenue, I have my profit as a function of demand: P(x) = -0.0025x^2+x-200.

The next step is to determine the price at which I should sell my cookies at. In order to do this, I took the derivative of my profit function, leaving me with the equation P^1(x) = -0.0005x+1. Then I set the equation equal to zero in order to determine the quantity I should be producing, which is 200 cookies. Next, I take the second derivative P^11(x) = -.0005 and plug my quantity (200) into it: P^11(200) = -.0005<0. This tells me that the profit function is concave down at x=200. The last step is go back to my price demand function and plug in the amount of cookies I will be producing in order to determine