The Formal Write-Up to the Big Question
1. Self-Assessment
I learned how to collaborate in a group while working on this problem, as well as the different methods to discover the maximum amount of profit. Although I felt like I helped the group eventually come to a decision, I wouldn’t consider myself one of the MAJOR contributors. Because of this, I would grade myself a 15/20. In the beginning, I simply allowed others to discover the solutions and chimed in from time to time. It was near the end of the project where I began to really get into the work. There were a couple Habits of a Mathematician that I used during the course of this problem, the three largest being ‘Look for Patterns,’ ‘Start Small’ and ‘Be Confident, Persistent and Patient.’ While trying to solve the equation, it was important for my group and I to start out using simple numbers and growing from a solid base. Starting from too high of a number plugged into the equation would end up confusing us and ruining our work. In addition to starting small, we would have to find a pattern with the simple numbers that we plugged in first. Using this pattern, we would then be able to plug in larger and larger numbers. The pattern would be useful when we were creating equations to represent our data. Finally, in order to figure out the pattern, we would had to be patient with ourselves and persistent in plugging in numbers to discover if they worked. We had to be confident that the next number would be the answer we were looking for, in order to have to motivation to continue solving the equations.
2. Problem Statement
The problem is a question about maximum profit. Two bakers are able to make two types of cookies: Plain and Iced. They have of 32 pounds of icing, 110 pounds of batter, oven space for 140 cookies and 15 hours of preparation time. Each dozen plain cookie requires 1 pound of dough and 0.1 hour of preparation time. Each dozen iced cookies requires 0.7 pounds of dough, 0.4 pounds of icing and about 0.15 hour of preparation time. The plain cookies sell for $6 a dozen and the iced sell for $7 a dozen. The problem asks us to calculate how many of each type of cookie the bakers need to make in order to make their profit as high as possible, while staying within the constraints.
3. Process Description
In order to solve this problem, my group and I began to work on discovering a balance between all of the constraints. We kept finding numbers that would work for two or three of the constraints, but didn’t work for the rest. When this happened, we would either lower or raise the numbers to try and squeeze a solution out of the problem. Eventually, we came up with a solution that included 24 dozen plain cookies and 24 dozen iced cookies. This only used up 40.8 pounds of our batter and 9.6 pounds of our icing. It would only take 6 hours to prepare the cookies and used up 48 spots in the oven. It worked for all the constraints, but it was not even CLOSE to the maximum profit. We only acquired $36 for the plain cookies and $48 for the iced cookies, leaving us with a total profit of $84.
We took break from the ‘Big Question’ to answer a similar version of the problem. This problem, called ‘A Simpler Cookie’ had only one constraint: The bakers only had 15 hours of preparation time, and the time to make each dozen of cookies was the same. Everything else was endless. We were asked to discover at least five different combinations of plain and iced cookies while staying within the 15 hour constraint. At this time, our teacher supplied us with five different equations in order to solve the problem.
x= plain cookies y= iced cookies
1. (Dough Amount Used) x + 0.7y ≤ 110
2. (Icing Used) 0.4y ≤ 32
3. (Oven Space) x + y ≤ 140
4. (Preparation time) 0.1x + 0.15y ≤ 15
5. (Profit) 1.5x + 2y = Profit
These problems were really meant for the ‘Big Question,’ since they had the constraints not included in ‘A Simpler Cookie.’ However, by using the equation for preparation time, we were able to discover the maximum amount of profit for the second problem: 0.1 (150) + 0.15 (0) ≤ 15. 150 is the number of plain cookies that we decided to make. We did not use any iced cookies, since they required more time and cost more to make. We plugged both numbers into the x and y slots and came out with $225 as our final profit. In the end, after trying out many other solutions, (Including 0.1 (45) + 0.15 (70) ≤ 15 and 0.1 (0) + 0.15 (100) ≤ 15, which gave us profits of $207.50 and $200.) it was decided that 0.1 (150) + 0.15 (0) ≤ 15 would yield the highest profit. After discovering the solution to ‘A Simpler Cookie,’ we transferred our equations to the ‘Big Question.’ This time, we had to discover the maximum profit while remaining within more constraints. We couldn't use the solution we had already found, since it did not fit within the dough limit. However, with a few tweaks to the numbers, we were able to discover what we thought was the maximum profit.
4. Solution
We eventually came up with the solution that fit in all of the constraints and that we believed was the largest profit. This consisted of 30 dozen plain cookies, 80 dozen iced cookies, 86 pounds of batter, 32 pounds of icing, 15 hours and 110 spaces used in the oven. This gave us a final profit $205. We used the different equations and made sure that all the numbers fit in the final solution. However, after the fact, another group came up with a solution that was 50 cents higher. This solution included 33 dozen plain cookies, 78 dozen iced cookies, 87.6 pounds of batter, 31.2 pounds of icing, 15 hours and 111 spaces used in the oven. This gave them a final product of $205.50. After multiple attempts at discovering a higher solution, it was decided by my group that $205.50 was the MAXIMUM amount of profit. Adding any more cookie to either the plain or the iced would drop the profit or raise the numbers over the constraints.
The Plain Cookie Design that I made. (It's a rock)
The Iced Cookie Design that I made. (It's a princess)