Problem 9-88
United Nations to the Rescue
Problem 9-88
As a representative of your country, you have been sent the following letter and given an important task:
Dear Representative to the United Nations:
A critical matter has come to the attention of the United Nations. In the past, when a catastrophe stuck a part of the world, the U.N. gathered supplies to give to people in need. Unfortunately, because the U.N. had to collect supplies from each country at the time of the catastrophe, it was always quite a few days before the supplies could be sent to the areas that needed them the most.
A recommendation has come before the U.N. to create a supply of food and medicine packages for future emergencies. Each food package will be able to feed several hundred people, while each medicine package will supply one first-aid station. I am asking each country to donate the same number of packages so each country shares the burden equally.
I am asking each country to determine how many food and medicine packages they are able to give. You will present your findings at today's United Nations meeting. Please be certain to use the information that your country's Budget Committee has prepared to help you decide how many packages you can afford.
Best of luck, and may our efforts make our world a better place!
Sincerely,
Antonio Guterres (Portugal) The Secretary General of the United Nations
Task: To communicate your country's budget constraints, write an inequality expressing how many food and medicine packages your country is able to give. Let x equal the number of food packages and y equal the number of medicine packages.
Graph the solution region representing the number of medicine and food packets that can be donated by your country. Be prepared to share your graph with the other countries of the United Nations.
Individual Problems:
- Country A: In order to solve for Country A we firstly needed to analyze the given problem. Once reading the given problem, we can deduce that X will be represented as the food packages and Y will be presented as medicine packages. In the problem it states that the cost will be up to but no greater than $300,000, indicating our symbol will be y ≤. Once indicating the symbol, we can then analyze that food and medicine packages can be represented by the given numbers in the problem, where food packages cost $900 and medicine packages cost $600. This being said we can write our inequality as 300,000 ≤ 900x + 600y, which we can then simplify by isolating the Y and moving the 300,000 value to the right, therefore giving us the answer of y ≤ -3/2x + 500.
- Country B: Then, in order to solve for country B, we took the same procedures as Country A. We firstly analyzed the problem and saw that the country was constrained to spending no more than $600,000 giving us the inequality of ≤. Consecutively, we then deduced that each food package costs $500 and each medicine package costs $2000. This being said, we can format the inequality as 500x + 2000y ≤ 600,000. In which by simplifying and isolating the Y, we come to the conclusion of y ≤ -1/4x + 300.
- Country C: After solving for Country B, we then continued on to Country C. We analyzed that instead of having a budget, Country C has a constraint. They need to spend more than $540,000 which indicates that our variable is >. We then analyzed as to how each medicine package costs $3,000, and each food package costs $2,000. With this being said, we can format the inequality as 540,000 > 2,000x + 3,000y, which we can then simplify by isolating the Y variable and moving the 540,000 to the right by subtracting it, therefore leaving us with a slope and y-intercept. Therefore giving us the inequality of y > -2/3x +180.
- Country D: After solving Country C, we then continued on to Country D. In order to approach this problem we firstly deduced which was our starting value in order to determine which inequality symbol to use. We analyzed that the country needed to spend more than $900,000, therefore our inequality symbol would be >. This being said, the country pays $5,000 for food packages and $2,000 for medicine packages. Therefore we can format our inequality as 900,000 > 5,000x + 2,000y, thereby simplifying into the inequality of y > -5/2x + 450, which we determined by isolating the Y variable.
- Country E: Finally, in order to solve for Country E, we took different measures. Due to the fact that they only need food packages, our line will be vertical parallel to the Y-axis and crossing through the X-axis. Therefore analyzing that the problem says that the country needs to donate fewer than 250 food packages to feed their own people, we are able to format our inequality as x < 250.
- Special Assignment: As our last problem to solve for, we took the same measures as Country E. We analyzed that since they had no values for the medicine and food packages, that it would be either a straight or vertical line. Although, in this case it would also be a vertical line such as Country E due to the fact that the problem discusses food packages. Once we overviewed the entirety of the problem, we could see that 100 food packages or more needed to be donated therefore the inequality can be presented as x ≥ 100.
Work done:
How we solved this situation.
- For this problem we could not solve it how we usually would. For starters, each country received their own small sheet of paper. On that paper was a situation for a country. My study team and I received the paper for Country B. My team looked at the sheet and they went straight to solving it. My team said it was fairly easy, the answer they reason came to be was Y is less than or equal to -1/4X + 300. The steps they took to solving it were firstly analyzing the given problem and constraints stated. Once they did this, they then analyzed the problem for X and Y values, meaning the value price of the medicine and food packages. They also looked for the starting values of each problem. Once setting up the inequalities using the format of y=mx+b, we could then simplify the inequality and find our answer.
- We had the equation for Country B but we needed the mathematical equations for the other 4 problems and the mathematical sentence for the "Special" situation. Some members of my team volunteered to venture around the classroom to ask for the equation for the other countries and the Special Assignment. We did this due to the fact that each group in our classroom was assigned a different Country to solve for, therefore once we received the inequalities for the rest of the countries we then made sure to double check their solutions and assure that they were all correct.
- From there all we had to do was place the equations on the graph using rulers, pencils, pens etc. In order to do this, we simply analyzed the previously found equations. Due to the fact that most of the inequalities had y-intercepts and slopes, we firstly plotted the y-intercept. Once this had been done, we then used the slope (rise/run) in order to find the next point, using the numerator as our rise (moving up or down) and the denominator (moving right or left) as our run. Once all the equations were placed we shaded either above or below the line whichever corresponded with the equation/inequality symbol. We also drew either a dashed line or a solid line if it corresponded correctly. Once everything was correct we discovered the common area. We then only shaded in the common area so it could look neater and easier to comprehend and/or understand.
- Once all of that falls into play and you can confirm it is correct, you can discover just how many food and medicine packs each country is able to donate. The way one is able to determine this is by simply analyzing the hexagon shaded region. Any coordinate found within in this region can be a possible solution depending on the given problem and constraints. Aside from this, we are also able to find answers with more ease given the shaded region.
Conclusion
To figure out how much food and medicine packets our country can donate we created an equation. From that point we were able to discover how many medicine packets each country can donate from their equations. We were even able to discover the amount the Special area can donate. Going in order from country A to Special these were the following equations, Country A: Y is less than or equal to -3/2X+500, Country B: Y is less than or equal to -1/4X+300, Country C: Y is greater than -2/3X + 180, Country D: Y is greater than -5/2X+450, Country E: X is less than 250, Special: X is greater than or equal to 100.