Math – Greenbacks or Greenspace

Greenbacks or Greenspace is a math problem that I’ve worked on since 11th grade. In this math problem I had to try and solve how much land we can split between different companies.
Nature or Business

Problem Statement:
In the problem it is asking what is the best way to divide the land and what will be the cheapest way to do it. There are three different areas where the land has been given to the city of San Francisco. George Lucas was kind enough to give 300 acres, soon right after that the Navy give 100-acre of the shipyard at Hunter’s Point. Then right afterwards the AT&T Park gives 150 acres to the city as well. There are two groups of people out there who want to buy the lands. There is the business group and the nature group. The business group feels that they should cover the central land (Hunter’s Point and AT&T Park) while the nature should cover the other land. But the nature group feels like they should cover the central land so that people can have easy access to nature. The cost prices are:
Lucas’s presidio (L) Nature= $50 Business= $500, Hunter’s Point (S) Nature= $200 Business= $2000, AT&T Park (P) Nature= $100 business= $1000. Overall you need to find out which plan will be the most cost effective and the best way to use the land.
The group came down to these agreements with each other about the land:
1. At most, 200 acres of Hunter’s Point Shipyard and the Parking Lots could go for nature.
2. The amount of Shipyard for nature and Presidio. Land used for business together had to equal exactly 100 acres.
3. As stated above, at least 3,000 acres would go for business.
The main math we will be using to solve this will be the matrices. Here will be my plan for how I should solve it.
My plan to solve this is at first list all the constraints equations and inequalities. From there I will find all the combinations and eliminate the ones that do not work with the constraints. I would double-check each combination that do not work and explain why. Once I am done with listing the right combinations I will use those combinations and plug them into matrices. The next step I would plug it into the matrices into the calculator I will multiply the matrices together and get the answer. From that answer I will then compare it to the equation that are listed. Then once I find the answer I will check them with the constraints to make sure they are able to fit in well with the equation, if it doesn’t I will cross it out and mark it wrong. As for the one that are right I will mark them as well with a different marking. Then I will keep on checking each combination until I am done with the list. Once I am done I will go back to check it and see which one is rights. The one that are right I will find the costing price by multiplying them by the land coast.
Here is my outline step to make it clearer:
1. List all the constraints equations and inequalities.
2. Turn the inequalities into equations
3. Find the combinations
4. Eliminate combinations that does not work
5. Double-check the combinations
6. Plug the combinations in as matrices
7. Plug in the matrices into the calculator
8. Write down the answer
9. Check the answer to the constraints
10. Mark it if the answer fits in the constraints is right or wrong and explain why.

The Math Work:
Step 1: List the Constraint:
1) LN+LB=300
2) SN+SB=100
3) PN+PB=150
4) SN+LB=100
5) SN+PN≤200
6) LB+SB+PB≥300
7) LN≥0
LB≥0
9) SN≥0
10) SB≥0
11) PN≥0
12) PB≥0

The reason: The reason of why I list the constraint is because it will be easy for me and for other people to know what I am looking for. What equations I am working with. Plus you need to test these constraints out to see which one will be the best to work out with.

Step 2: Make a list of constraints that will not work together:
7 and 8: Putting these two together violates constraint # 1
9 and 10: Putting these two together violates constraint # 2
11 and 12: Putting these two together violates constraint # 3
8 and 9: Putting these two together violates constraint # 4
9 and 11: Putting these two together violates constraint # 5
5 and 9: Putting these two together violates constraint # 3 & 5
1 and 7: Putting these two together violates constraint # 1 & 4
5 and 11: Putting these two together violates constraint # 3& 5
6 and 10: Putting these two together violates constraint # 2, 3 & 4
7 and 9: Putting these two together violates constraint # 1, 2, & 6
6 and 7: Putting these two together violates constraint # 1, 2, 3, & 4
The reason: The reason of why I tested which constraints work is because it better to know which one doesn’t work so you wouldn’t have to sit there and do bunch of math work and not needing to test over 1000 combinations there might be.
Step 3: Put the constraints into combinations that will work:
1,2,3,4,5,6 1,2,3,4,6,8 1,2,3,4,8,10 1,2,3,4,9,12 1,2,3,4,10,11 1,2,3,4,10,12
1,2,3,4,5,8 1,2,3,4,6,9 1,2,3,4,8,11
1,2,3,4,5,10 1,2,3,4,6,11 1,2,3,4,8,12
1,2,3,4,5,12 1,2,3,4,6,12
The reason: It is very important to know what combinations you are working with because these will be the one that you will be testing. It is also nice to have it down in a list so you would know which one you have done and which one you have not done yet when you are testing them.

Step 4 part 1: Testing the combinations in matrices:
The History of Matrices:
The main number rule of matrices is [x]=[A]-1[B]. Therefore for the next few problems you will see it set up as that way. But you will see it set up as [A]-1[B]=[X], then the other around. But no matter what it will still stay the same. The reason it is step up like this is to help people to solve equations a lot more easily way and faster then doing everything by the hand.
Why always use combo 1,2,3,4?:
The reason of why we use it is because 1,2,3,4 is equations not inequalities therefore it ill always need to be use.

Why use Matrices? :
The reason of why you need to use this to solve the problem is because you have numbers in here that are missing. Therefore you are unable to solve the problem. Most of the time when people look at a problem and there is missing numbers the first thing comes to mind is solving the equations and being into an equation form. Well Matrices are equations but are able to be plug into the calculator. Also because most of the work that are being done in the calculator the answer should come out right, as long it is always double-check.

Below is how you will take the equations and set them up in matrices:
1) Combination: 1, 2, 3,4,5,6
Ln Lb Sn Sb Pn Pb
[ 1 1 0 0 0 0 ] [ LN] [ 300]
[ 0 0 1 1 0 0 ] [ LB ] [100 ]
[ 0 0 0 0 1 1 ] X [SN] = [ 150 ]
[ 0 1 1 0 0 0 ] [ SB ] [ 100 ]
[ 0 0 1 0 1 0 ] [ PN ] [ 200 ]
[ 0 1 0 1 0 1 ] [ PB ] [ 300 ]
* The above math is how you should set it up at first from taking it out of the equation format.
* The below math will show you how to set it up to solve. After this one I will no longer show the set up from the equation to format because once you get the hang of it you no longer need that to help you and you can always look back at the first as to where go where.
Below is the set of how you should multiply the matrices:
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [ 300 ] [50 LN]
[0 0 1 1 0 0] [100 ] [250 LB]
[0 0 0 0 1 1 ] X [150] = [-150 SN] ← Those are the answer to the
[0 1 1 0 0 0 ] [100] [250 SB] combinations.
[0 0 1 0 1 0 ] [200] [350 PN]
[0 1 0 1 0 1 ] [300] [-200 PB]
[A]-1 [B] [X]

The Reason: This combination does not work because it violates constraint number 6. Also the fact that the answers have negatives numbers in them as well. This shouldn’t be there.

Step 4 part 2: Testing the combinations in matrices:

2) Combination: 1, 2, 3,4,5,8
Ln Lb Sn Sb Pn Pb
[ 1 1 0 0 0 0 ] [ 300 ] [ 300]
[0 0 1 1 0 0 ] [100 ] [ 0]
[0 0 0 0 1 1 ] X [150 ] [100 ]
[0 1 1 0 0 0 ] [100 ] = [ 0 ]
[0 0 1 0 1 0 ] [200] [100]
[0 1 0 0 0 0] [ 0 ] [ 50]
The Reason: Yet again this combination does not work because it violates constraint number 6.

3) Combination: 1, 2, 3, 4, 5, 10
Ln Lb Sn Sb Pn Pb
[ 1 1 0 0 0 0 ] [ 300 ] [300]
[0 0 1 1 0 0 ] [ 100 ] [0]
[0 0 0 0 1 1 ] X [150 ] [100]
[0 1 1 0 0 0 ] [100 ] = [0 ]
[0 0 1 0 1 0 ] [200] [100]
[0 0 0 1 0 0] [ 0 ] [50]
The Reason: Yet again this combination does not work because it violates constraint number 6.

4) Combination: 1, 2, 3, 4, 5, 12
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300 ] [ 250]
[0 0 1 1 0 0 ] [100 ] [ 50]
[0 0 0 0 1 1 ] X [150 ] [50 ]
[0 1 1 0 0 0 ] [100 ] = [50]
[0 0 1 0 1 0 ] [200] [150]
[0 0 0 0 0 1] [ 0 ] [0]
The Reason: Yet again this combination does not work because it violates constraint number 6.

5) Combination: 1, 2, 3,4,6,8
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300] [300]
[0 0 1 1 0 0 ] [100] [0]
[0 0 0 0 1 1 ] X [150] [100 ]
[0 1 1 0 0 0 ] [100 ] = [0]
[0 1 0 1 0 1 ] [300 ] [-150]
[0 1 0 0 0 0] [0 ] [ 300]
The Reason: The reason of why this one does not work is because it has a negative number in the answer.

6) Combination: 1, 2, 3,4,6,9
Ln Lb Sn Sb Pn Pb
[ 1 1 0 0 0 0 ] [ 300 ] [ 200]
[ 0 0 1 1 0 0 ] [100 ] [ 100]
[ 0 0 0 0 1 1 ] X [150 ] [ 0 ]
[0 1 1 0 0 0 ] [100 ] = [100]
[0 1 0 1 0 1 ] [300] [ 50]
[0 0 1 0 0 0 ] [ 0 ] [100]
The Reason: The reason of why this does work is because it does not have a negative number in the answer and it does not violate a constraint.

Step 4 part 3: Testing the combinations in matrices:
7) Combination: 1, 2, 3, 4,6,11
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [ 300 ] [ 225]
[0 0 1 1 0 0 ] [100 ] [ 75]
[0 0 0 0 1 1 ] X [150 ] [ 25]
[0 1 1 0 0 0 ] [100 ] = [75]
[0 1 0 1 0 1 ] [300] [ 0]
[0 0 0 0 1 0 ] [ 0 ] [150]
The Reason: The reason of why this does work is because it does not have a negative number in the answer and it does not violate a constraint.

Combination: 1, 2, 3, 4,6,12
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300] [200]
[0 0 1 1 0 0 ] [100 ] [100]
[0 0 0 0 1 1 ] X [150] [ 0 ]
[0 1 1 0 0 0 ] [100 ] = [100]
[0 1 0 1 0 1 ] [300 ] [150]
[0 0 0 0 0 1 ] [0 ] [0]
The Reason: Yet again this combination does not work because it violates constraint number 6.

9) Combination: 1, 2, 3, 4,8,10
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0] [300]
[0 0 1 1 0 0] [100 ]
[0 0 0 0 1 1 ] X [150 ]
[0 1 1 0 0 0 ] [100 ] =
[0 1 0 0 0 0 ] [ 0 ]
[0 0 0 1 0 0 ] [ 0 ]
The Reason: This does not work at all. When I was plugging this into the graphing calculator it came out as an error. I went over and over it again and it came out the same thing. So I asked John for help and he told it came out the same way as he did. It turn out it was the fact that the line could not intersects at all.

10) Combination: 1, 2, 3, 4,8,11
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300] [300]
[0 0 1 1 0 0 ] [100] [ 0]
[0 0 0 0 1 1 ] X [150] [100]
[0 1 1 0 0 0 ] [100] = [ 0]
[0 1 0 0 0 0 ] [0 ] [0]
[0 0 0 0 1 1 ] [ 0] [150]
The Reason: Yet again this combination does not work because it violates constraint number 6.
Step 4 part 4: Testing the combinations in matrices:

11) Combination: 1, 2, 3, 4,8,12
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300 ] [300]
[0 0 1 1 0 0 ] [100 ] [ 0]
[0 0 0 0 1 1 ] X [150 ] [100]
[0 1 1 0 0 0 ] [100 ] = [ 0]
[0 1 0 0 0 0 ] [ 0 ] [150]
[0 0 0 0 0 1 ] [ 0 ] [ 0]
The Reason: Yet again this combination does not work because it violates constraint number 6.

12) Combination: 1, 2, 3, 4,9,12
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300] [200]
[0 0 1 1 0 0 ] [100] [100]
[0 0 0 0 1 1 ] X [150] [ 0 ]
[0 1 1 0 0 0 ] [100] = [100 ]
[0 0 1 0 0 0 ] [ 0] [150]
[0 0 0 0 0 1] [ 0] [ 0]
The Reason: Yet again this combination does not work because it violates constraint number 6.

13) Combination: 1, 2, 3, 4, 10, 11
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300 ] [300]
[0 0 1 1 0 0 ] [100 ] [0]
[0 0 0 0 1 1 ] X [150 ] [100 ]
[0 1 1 0 0 0 ] [100 ] = [ 0 ]
[0 0 0 1 0 0 ] [ 0 ] [ 0]
[0 0 0 0 1 0 ] [ 0] [150]
The Reason: Yet again this combination does not work because it violates constraint number 6.

14) Combination: 1, 2, 3, 4, 10, 12
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300 ] [300]
[0 0 1 1 0 0 ] [100 ] [0 ]
[0 0 0 0 1 1 ] X [150 ] [100 ]
[0 1 1 0 0 0 ] [100 ] = [0 ]
[0 0 0 1 0 0 ] [ 0 ] [150]
[0 0 0 0 0 1] [ 0 ] [0]
The Reason: Yet again this combination does not work because it violates constraint number 6.

Step 5: Looking for the right answer:

6) Combination: 1, 2, 3,4,6,9
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300 ] [200]
[0 0 1 1 0 0 ] [100 ] [100]
[0 0 0 0 1 1 ] X [150] [ 0 ]
[0 1 1 0 0 0 ] [100] = [100]
[0 1 0 1 0 1] [300] [ 50]
[0 0 1 0 0 0 ] [ 0 ] [100]
The Reason: The reason of why this does work is because it does not have a negative number in the answer and it does not violate a constraint.
And
7) Combination: 1, 2, 3, 4,6,11
Ln Lb Sn Sb Pn Pb
[1 1 0 0 0 0 ] [300 ] [225]
[0 0 1 1 0 0 ] [100 ] [75]
[0 0 0 0 1 1 ] X [150 ] [25]
[0 1 1 0 0 0 ] [100 ] = [75]
[0 1 0 1 0 1] [300] [ 0]
[0 0 0 0 1 0] [ 0] [150]
The Reason: The reason of why this does work is because it does not have a negative number in the answer and it does not violate a constraint.

Step 6: Once you find the right answer, use it to solve the cost:

For problem six we will take the answers that we got and multiply them by their own cost. Here is the list of the answer just to refuse your memory.
6) Ln=200 7) Ln=225
Lb=100 Lb=75
Sn=0 Sn=25
Sb=100 Sb=75
Pn=50 Pn=0
Pb=100 Pb=150
From here we take our answer and put them into the equation from multiplying from their cost of the land and adding them up.

Here is how we do number 6:
(200 x 50)+(100 x 500)+(100 x 2000)+(50 x 100)+(100 x 1000)= $ 365,000
* Notice how I did not multiply PN because of the fact that it is 0 therefore the answer will just come out as 0 and will not affect the cost.
1, 2, 3, 4, 6, 9
LN – 200 $50 x 200 = $10,000
LB – 100 $500 x 100 = $50,000
SN – 0
SB – 100 $2,000 x 100 = $200,000
PN – 50 $100 x 50 = $5,000
PB – 100 $1,000 x 100 = $100,000
Total = $365,000

Here is how we do number 7:
(225 x 50)+(75 x 500)+(25 x 200)+(75 x 2000)+(150 x 1000)= $ 353,750
* Notice how I did not multiply PN because of the fact that it is 0 therefore the answer will just come out as 0 and will not affect the cost.
1, 2, 3, 4, 6, 11
LN – 225 $50 x 225 = $11,250
B
LB – 75 $500 x 75 = $37,500
SN – 150 $200 x 25 = $5,000
SB – 75 $2,000 x 75 = $150,000
PN – 0
PB – 150 $1,000 x 150 = $150,000
Total = $353,750
Answer to the problem:
The best way to go about on this problem is to go with the combination 1, 2, 3, 4,6,11. This come down that the LN should have 225, LB should have 75, SN should have 25, SB should have 75, PN should have nothing, and PB should have 150. The cost of this will come down to $ 353,750. This is the cheapest and the best way to go about this.
I know this is the best way to go because the only way for you to solve the
equations are by doing them in matrices. Doing this you will be able to solve the missing numbers that you will need to find the cost. It is also a faster way for you to solve the problem because you are not doing it by solving the X,Y, and Z by it self. It is also less likely for the answers to be wrong because all of the real math work is being done on the calculator. So every time I enter it into the calculator I always double checking the set up to make sure I was right. So I know that my answer are right because I spend a about good 3 hrs rechecking everything in the calculator and rechecking if it works or not. In way of looking at it, you look at the numbers and you notice that there are some numbers at there that is a tad bit higher, so you could guess the ones that are low numbers will be the cheapest way to go about it.
Even though this is the best way to solve the money issue and how the city should use the land I believe that the Business should get most of the land because of the fact that more people go to the malls more. The city already lack with big malls that it will be best to add more big malls to it. The only malls we have in the city right now is the one in downtown and the one in the sunset. Overall there aren’t any other malls in the city. You would have to drive out of the city to go to another mall or take Bart. If we can not use the lands to build another mall in the city we could at lest use it to build independent food stores. There are very few areas in the city that also lack of independent food place, the only area that have the most independent food stores is the Sunset area. Beside the Sunset area there really isn’t another area out there. This is the reason of why I think we should give business more land. It will help our people and also help build the San Francisco.
Here I will reshow the answer that I got:
(225 x 50)+(75 x 500)+(25 x 200)+(75 x 2000)+(150 x 1000)= $ 353,750
* Notice how I did not multiply PN because of the fact that it is 0 therefore the answer will just come out as 0 and will not affect the cost.
1, 2, 3, 4, 6, 11
LN – 225 $50 x 225 = $11,250
B
LB – 75 $500 x 75 = $37,500
SN – 150 $200 x 25 = $5,000
SB – 75 $2,000 x 75 = $150,000
PN – 0
PB – 150 $1,000 x 150 = $150,000
Total = $353,750

Reflection:
At first when I looked at the problem the only thing that came to my mind was to use matrices. Because I saw we had to solve the equations and inequalities I knew it was going to take me a really long time just by doing it by hand. Then right after that I remember the whole matrices system and how it was made to solve for those. So I told my group about it and we decide to check it out as that. So right after that we took out the notes from the previous day and started to look back at it and putting it into the system.
So then right after that was done, I went back to the start and starting to work on my problem statement. I knew what steps I should get take and I knew what I should do. I also look back at my notes to make sure I was doing and planning my steps right. Once I was done I stay with my plan and didn’t change it at all. I made sure I was doing the same steps in my real math work. But every time I was thinking I might have done something wrong I went back and double-check my work. Then I wrote it down the answer onto my paper and kept on doing it like that. I knew I was going need to type this so I skip the whole set up write up because it will just case me to waste time. But I made sure the first two problems I wrote down the set up and explain of what was going on. The reason why I did was because I knew I might forget my steps or I might put it away and not look at for another week which will make me forget of what I needed to do. Then after that I wrote down the whole answer. I also made sure I label almost everything because I was scared that I might miss read something. I already know how I work and something told me I might miss read something or miss placed something. I also always double-check if everything was working well with the constraints as well. I did this because it helped me feel that I wasn’t doing anything wrong. Once I found out which constraints work I put them into the equation and started to do the everyday kind of math to solve for the cost value. Then once I was done I double-checked my math and talked to other few people about the answer and then made sue what the person did the same steps as me as well. Once I was fully sure with myself I started to type up the paper all in one night. Because I know if I didn’t get it done within the week I will get it due before the due date and that will make a lot of bad. I also worked on the paper before school and during class time. Making sure I get the big pieces done.
If Mayor Newsom changed anything in the plan(showed in 2nd page) or the math then there the cost price and the land will be so mess up to the point it might come out costing more then before. That or he will be lost if with the changes and everything might be turn out wrong. Also with more piss off people. But if Mayor Newsom doesn’t go with the plan then you might no be able to find the cheapest thing. Even if you change one part of the matrices like for an example 1,2,3,4,6,9 ROW 1 COLUM 1: 1 into 2. Then you will get way off numbers that will bring up the prices up. WAY UP and it wont solve the problem.
Another way of using this problem is if you wanted to make a sort amount of ice creams and candy, of which one will be the best to sell on different days. You could also use this into how many house should go in which part of the city. Depending which section of the city and the cost also the number house that is needed. This is another way for you to use matrices into our daily life. There are many different ways of doing this and many different ways of to solve problems.