Problem 9-69
United Nations Constraints
Problem 9-69
The United Nations asked every nation to write constraints that best approximate it's country's shape (the U.N. thinks this will help find each country's area). Honduras sent in it's constraints, but some of the information is unreadable. With your team, determine the missing parts of the inequalities and rewrite them on your paper.
Honduras Constraints:
Honduras is bordered by Guatemala, Nicaragua, and El Salvador. On the north, the country shares a vast stretch of coast with the Caribbean Sea. On the south, it shares a small stretch with the Pacific Ocean. Several islands are also found on the country's coast. Honduras is best known for the production and export of agricultural products (coffee, tropical fruits and sugar cane) as well as for its textile industry (maquila). Remittances sent by Hondurans abroad represent 20% of the country's GDP. This Central American country has a 3,000-year-old Mayan culture, beautiful beaches, and delicious cuisine. Honduras' tropical weather enriches its lush flora and fauna. Slightly larger than the state of Tennessee, Honduras is mountainous and the only Central American country without active volcanoes.
How we solved this
- In order to solve for our first line, we mainly focused on finding the y-intercept and determining how the slope affected the line. With this being said, we then we were able to deduce that from our y-intercept being 3 and our slope was X or 1, we determine this line.
- Due to the fact that this line did not have a y-intercept, we focused on the slope. In order to solve for the y-intercept we merely used the origin as our set point and followed the rule of the slope which was a rise of 1 and a run of 2, therefore also following whether it was a negative or positive line. In which we determined the y-intercept to be -1.
- Due to the fact that this line had no slope it was fairly easy to identify it since it would be a horizontal line going through the y-axis.
- In order to identify this line, we then continued these steps of locating the y-intercept, in which this case was four, and followed through what the slope indicated. Therefore we climbed a rise of 2 and had a run of 3, going in a negative direction to find this line.
- We then followed the same process as stated in step 4, concluding as to how we found the given lines, due to their slopes and y-intercepts.
Note: In order to verify our findings, and deduce as to which symbols fit with which line, we merely analyzed the shading of the line and whether the lines were dotted or full. Due to the fact that we had all full lines, we knew we wouldn't be using symbols such as (>,<), therefore based on upward and downward shading we configured which symbols corresponded to each line. When shading was below the given line, ≤, was used. Meanwhile if the line had shading above it, ≥
, would be used.
Individual Lines and Explanations:
- Line 1: In order to find and match this line we firstly saw that it had a y-intercept of 3 in which was plotted above the x-axis an going upwards, indicating a positive slope. Consequently we then analyzed that the slope rose 1 unit (up) and ran 1 unit (right), giving us a 1/1 slope which can also be presented as X. Therefore indicating that our line equation is y ≤ x+3. We can also confirm this since the inequality symbol is ≤ meaning the shading is under the line as given above.
- Line 2: In order to find this line, we found the y-intercept of -1. In which we consequently found its next point. By doing this, we were able to confirm that it's rise was 1 (up) and run was 2 (right), therefore giving us the slope of 1/2 since slope can also be presented as rise/run. Once finding the slope and y-intercept we then were able to write out the equation once analyzing that the shading was above the line, leaving us with y ≥ 1/2x-1.
- Line 3: Due to the fact that this line was a horizontal line going through the y-axis, we were able to deduce that this line would not have a slope and would simply only have a y-intercept. With this being said, we analyzed that the line intersected the y-axis at positive 2, giving us the y-intercept and allowing us to conclude that the shading is below the line. Therefore revealing the inequality to be y ≤ 2.
- Line 4: Analyzing that this line is going down, we can firstly conclude that this will have a negative slope. With this being said, we consequently found the y-intercept, which was positive 4 due to the fact that this is where the line intersected the y-axis. Once finding this point, we then continued to find the continuing coordinate to find our slope, in which we found our rise to go down 2 units and run 3 units to the right, giving us a slope of -2/3. Knowing this and that the shading is below the line, we can conclude that the inequality is given by y ≤ -2/3x+4.
- Line 5: Through the process of elimination we have been able to match all the lines, but nonetheless we will check our findings. In order to do this for the last line, we firstly found the y-intercept which was -1, such as line 2. Once finding our y-intercept we then followed the slope to find our next point. By doing this, we had a rise of 2 units (up) and run of 3 units (left), thereafter giving us a slope of -2/3. By then analyzing that the shading is upwards, we can conclude that the equation for this inequality is y ≥ -2/3x-1.