## The college dilemma

For our client, my group had to solve a problem having to do with college admission of in-state and out of state students. There were different constraints that we had to follow. What we had to do for our client was to see what was the optimal amount of students in and out of state to admit to minimize spending.

To make the best recommendation, we started out by reading the problem. This is so we can decipher exactly what the problem wants us to solve, and so that we can extract the important parts for the equations from the word problem. This problem was about a college that wanted to minimize the amount of spending based off of how many in and out of state students they admitted. Each out of state student cost $6000 a year and each in-state student costs $7200. Before we worry about that, we have to work on finding the feasible region. A feasible region shows all the possible answers to a problem that follow certain constraints. Based off this, we have to find the constraints of the problem. To find a constraint, you take clues from the text and turn them into inequalities. For example, the problem states

"The college president wants this class to contribute at least $2,500,000 to the school after graduation. In the past, Big State U has received an average of $8,000 from each in-state student. It has received about $2,000 from each out-of-state student."

Before we start, I am going to assign in-state students with the variable "x" and the out-of-state students with the variable "y". By looking at the problem, we can conclude that the amount that the students contribute in total has to be more than or equal to 2,500,000. Since each student in-state will give 8,000 and each out-of-state student will give 2,000, we know that

8,000x+2,000y≥2,500,000

We can use the rest of the text to find two other restraints for the problem. These two are:

y≥x

and

100x+200y≤85,000

We graph those constraints in a graphing program called Geogabra. In Geogabra, we turn these inequalities into equations. That way, it graphs the line that divides the are that fits the constraint and the area that doesn't. For example, 100x+200y≤85,000 would turn into 100x+200y=85,000. After graphing all three constraints, we find the intersection points of each inequality. Using these points, we create a polygon that shows the feasible region. Any interger value in the area will fit all constraints. This is the graph we made.

To make the best recommendation, we started out by reading the problem. This is so we can decipher exactly what the problem wants us to solve, and so that we can extract the important parts for the equations from the word problem. This problem was about a college that wanted to minimize the amount of spending based off of how many in and out of state students they admitted. Each out of state student cost $6000 a year and each in-state student costs $7200. Before we worry about that, we have to work on finding the feasible region. A feasible region shows all the possible answers to a problem that follow certain constraints. Based off this, we have to find the constraints of the problem. To find a constraint, you take clues from the text and turn them into inequalities. For example, the problem states

"The college president wants this class to contribute at least $2,500,000 to the school after graduation. In the past, Big State U has received an average of $8,000 from each in-state student. It has received about $2,000 from each out-of-state student."

Before we start, I am going to assign in-state students with the variable "x" and the out-of-state students with the variable "y". By looking at the problem, we can conclude that the amount that the students contribute in total has to be more than or equal to 2,500,000. Since each student in-state will give 8,000 and each out-of-state student will give 2,000, we know that

8,000x+2,000y≥2,500,000

We can use the rest of the text to find two other restraints for the problem. These two are:

y≥x

and

100x+200y≤85,000

We graph those constraints in a graphing program called Geogabra. In Geogabra, we turn these inequalities into equations. That way, it graphs the line that divides the are that fits the constraint and the area that doesn't. For example, 100x+200y≤85,000 would turn into 100x+200y=85,000. After graphing all three constraints, we find the intersection points of each inequality. Using these points, we create a polygon that shows the feasible region. Any interger value in the area will fit all constraints. This is the graph we made.

After making the feasible region, I looked at the vertices of the lines that we made. In this case, there are three vertices. Using geogabra, you can easily find the values of these three points by using the intersection tool. Using this tool, I found the intersection points.(as seen on the graph) As you know from the problem, we are trying to

x=250, and y=250,

x spends $7200

y spends $6000

250(6000)=1,500,000

250(7200)= 1,800,000

1,500,000+1,800,000= total cost of $3,300,000

We know the point 283,283 (one of the other vertices, won't work because both x and y values are more than 250. There is no way that that could equal less than the current low of $3,300,000

Lastly I tried the 3rd point which is 235,307

For that, I got a total cost of $3,534,000

Based off of this information, I got the recommendation that they should admit 250 in-state students and 250 out-of-state students

**minimize**the cost of the colleges. We plug in the values for x and y and compute the cost. For example, if:x=250, and y=250,

x spends $7200

y spends $6000

250(6000)=1,500,000

250(7200)= 1,800,000

1,500,000+1,800,000= total cost of $3,300,000

We know the point 283,283 (one of the other vertices, won't work because both x and y values are more than 250. There is no way that that could equal less than the current low of $3,300,000

Lastly I tried the 3rd point which is 235,307

For that, I got a total cost of $3,534,000

Based off of this information, I got the recommendation that they should admit 250 in-state students and 250 out-of-state students

When completing the problem and presenting it to the client, we had different roles we had to fill. The roles in our group were:

Geogabra Expert: Frank

Documentarian: Jocelyn

Facilitator: Liana

Skeptic: Sina

Spokesperson: Sina

Geogabra Expert: In charge of making the graph on Geogabra. This role was vital because without a trustworthy Geogabra expert, your whole client engagement could be messed up. I did a very good job on this part because I got all the work done fast with no problems

Documentarian: This role had the job of writing everything down. Our documenter did an amazing job of writing down all the constraints that we found in the beginning so I could use them in my Geogabra graph

Facilitator: Our group worked very efficiently together with no faults. I really think we are so smart working together, and it shows in our work. We don't get distracted and we just power through the assignments. Liana did not need to handle us getting off topic

Spokesperson: Sina explained everything to the client very well and the client understood all that we were saying. They did not have any questions about the topic and were happy with what we gave them

Skeptic: Sina told me to try the point 235, 307 even though I was 90% sure that 250, 250 was the answer. He made sure that I knew exactly the answer with no doubts

Geogabra Expert: Frank

Documentarian: Jocelyn

Facilitator: Liana

Skeptic: Sina

Spokesperson: Sina

Geogabra Expert: In charge of making the graph on Geogabra. This role was vital because without a trustworthy Geogabra expert, your whole client engagement could be messed up. I did a very good job on this part because I got all the work done fast with no problems

Documentarian: This role had the job of writing everything down. Our documenter did an amazing job of writing down all the constraints that we found in the beginning so I could use them in my Geogabra graph

Facilitator: Our group worked very efficiently together with no faults. I really think we are so smart working together, and it shows in our work. We don't get distracted and we just power through the assignments. Liana did not need to handle us getting off topic

Spokesperson: Sina explained everything to the client very well and the client understood all that we were saying. They did not have any questions about the topic and were happy with what we gave them

Skeptic: Sina told me to try the point 235, 307 even though I was 90% sure that 250, 250 was the answer. He made sure that I knew exactly the answer with no doubts

Celebrations for the Group

Frank(myself): I did an amazing job during this client engagement, I really did! I lead my group and made us stay on task and helped us work fast but effectively. I used the job of collaborating and listening because all of us really worked together to get this job done. Not just me. If anyone needed help I was willing to collaborate with them.

Sina: Sina spoke up and took a big role during this project. Taking on two jobs, he did both to the best of his ability. He did the job of starting small to help him explain what we were doing to the clients. He started at the basic information and built up from there.

Jocelyn: Jocelyn was a joy to work with and kept on track the whole time. She wrote down everything anyone that was solving the problem was saying so that we could reference back to it incase that we needed it. She used the Habit of a Mathematician staying organized so that we could easily get any information we wanted.

Liana: Liana fell behind in the beginning, but she was not afraid to ask for help to catch up. She was no burden on the group and we were glad to have her on the team!

Our group as stated before worked really great together and I could not see a single mistake we made along the way. The only challenge we had was catching Liana back up to us when she left/wasn't paying attention for some reason. After she caught up the first time though, she was following along perfectly. A success we had was successfully finding the restraints fast. This was a success because once you have the restraints, the rest come easy.

The biggest strength I had at my role of Geogabra Expert was to be able to label all the lines and make a key of some sort. This for one makes the graph look a lot nicer and for two, makes the graph easier to read.

Frank(myself): I did an amazing job during this client engagement, I really did! I lead my group and made us stay on task and helped us work fast but effectively. I used the job of collaborating and listening because all of us really worked together to get this job done. Not just me. If anyone needed help I was willing to collaborate with them.

Sina: Sina spoke up and took a big role during this project. Taking on two jobs, he did both to the best of his ability. He did the job of starting small to help him explain what we were doing to the clients. He started at the basic information and built up from there.

Jocelyn: Jocelyn was a joy to work with and kept on track the whole time. She wrote down everything anyone that was solving the problem was saying so that we could reference back to it incase that we needed it. She used the Habit of a Mathematician staying organized so that we could easily get any information we wanted.

Liana: Liana fell behind in the beginning, but she was not afraid to ask for help to catch up. She was no burden on the group and we were glad to have her on the team!

Our group as stated before worked really great together and I could not see a single mistake we made along the way. The only challenge we had was catching Liana back up to us when she left/wasn't paying attention for some reason. After she caught up the first time though, she was following along perfectly. A success we had was successfully finding the restraints fast. This was a success because once you have the restraints, the rest come easy.

The biggest strength I had at my role of Geogabra Expert was to be able to label all the lines and make a key of some sort. This for one makes the graph look a lot nicer and for two, makes the graph easier to read.