Oil Spill Problem
Question:
The spill wil be classified as a "serious environmental hazard" if cleanup has not begun before the total oil spill reaches 45,000m^2.
What time must the response team get to the spill in order to avoid a "serious environmental hazard"?
The spill wil be classified as a "serious environmental hazard" if cleanup has not begun before the total oil spill reaches 45,000m^2.
What time must the response team get to the spill in order to avoid a "serious environmental hazard"?
Process:
The first thing that I tried to do when I looked at this process was figure out the area of a circle, because the problem was to figure out when the circle reached a total of 45,000 meter squared. From there, I created a table so I could organize my work more efficiently, as shown below. However, as one can see, the time it would reach the serious environmental hazard point, would be in between 12pm and 1pm. I added 13 to the diameter of 12:30 because half of 26 is 13, but when the answer still wasn't exactly 45,000 I decided to go deeper. I then proceeded to figure out how fast the oil would spread per minute, by dividing 26 by 60. 26 represents how many meters it grows per hour, and 60 represents how many minutes there are in every hour. The oil would increase .433 meters per minute, and added that to the diameter for each minute, slowly getting closer to 45,000.
However, when I got to 12:33, I began to lose interest so I decided to get 45,000 and divide that by pi, thus getting me 14,331.2102 rounded to the ten-thousandth place. I did this to undo the process on how to find the area of a circle, and since multiplying pi is the second step, I then proceeded to the first step. I then found the square root of 14,331.2102 and got 119.7130 as the radius, which I then multiplied by two to find the diameter (239.426). To find which minute it was closer to, I began to add .433 to 12:33's diameter until it went above the assigned number, the closest was 12:35. From there, I found the diameter of 12:35 which was 239.165 and I subtracted that by 239.426 to get .261, which I then divided by .007 (the growth rate of the oil per second) to 37 seconds and 2.8 milliseconds. So my final answer was 12:35:37:2.8, as to the exact time the response team should get there at the latest to subdue the problem before it became a serious environmental hazard.
The first thing that I tried to do when I looked at this process was figure out the area of a circle, because the problem was to figure out when the circle reached a total of 45,000 meter squared. From there, I created a table so I could organize my work more efficiently, as shown below. However, as one can see, the time it would reach the serious environmental hazard point, would be in between 12pm and 1pm. I added 13 to the diameter of 12:30 because half of 26 is 13, but when the answer still wasn't exactly 45,000 I decided to go deeper. I then proceeded to figure out how fast the oil would spread per minute, by dividing 26 by 60. 26 represents how many meters it grows per hour, and 60 represents how many minutes there are in every hour. The oil would increase .433 meters per minute, and added that to the diameter for each minute, slowly getting closer to 45,000.
However, when I got to 12:33, I began to lose interest so I decided to get 45,000 and divide that by pi, thus getting me 14,331.2102 rounded to the ten-thousandth place. I did this to undo the process on how to find the area of a circle, and since multiplying pi is the second step, I then proceeded to the first step. I then found the square root of 14,331.2102 and got 119.7130 as the radius, which I then multiplied by two to find the diameter (239.426). To find which minute it was closer to, I began to add .433 to 12:33's diameter until it went above the assigned number, the closest was 12:35. From there, I found the diameter of 12:35 which was 239.165 and I subtracted that by 239.426 to get .261, which I then divided by .007 (the growth rate of the oil per second) to 37 seconds and 2.8 milliseconds. So my final answer was 12:35:37:2.8, as to the exact time the response team should get there at the latest to subdue the problem before it became a serious environmental hazard.
Reflection:
Two processes I did that represent the habits of a mathematician is solving a simpler problem and be systematic. To solve a simpler problem is to start with small portions of the problem and build of from there, and I was able to do this by having little problems I needed to solve to get from one step to the next. For example, I changed the rate of change of the oil from hour to minute the seconds, to get as exact as possible when trying to find the time when the circle reaches 45,000 meters^2. As well as going backwards, because at one point I thought I needed to know when the oil spill started so I began to subtract 26 from 68, to find how wide it was at 5:00, and continued that until there was only 16 left, and I just kept lowering the number until I found a time that it would begin. As to how I used being systematic, I made small changes to look for change when I was trying to find the time, by testing the process by increasing the time by one minute each time, and on a larger scale, I increased the time by an hour to find out the time and how much farther they are from reaching the serious environmental hazard. Another habit of a mathematician that i used, though in my opinion not a lot, was collaborating and listening with your group. I collaborated and shared my answers, choices, and my reasons for doing it. However, I think that I didn't help them learn from this problem, as soon as I learned what I needed to do, I told them the steps and processes while they were still figuring it out, which doesn't help them solve the problem their own way. On the plus side, we all worked together to find the answer, and if one of us got the answer first, we shared it with the others and explained how we got it. We were able to work decently with each other, but I feel that I should let my group mates figure out how to do the problem first instead of giving them a completely different way then what they were originally going to use.
Two processes I did that represent the habits of a mathematician is solving a simpler problem and be systematic. To solve a simpler problem is to start with small portions of the problem and build of from there, and I was able to do this by having little problems I needed to solve to get from one step to the next. For example, I changed the rate of change of the oil from hour to minute the seconds, to get as exact as possible when trying to find the time when the circle reaches 45,000 meters^2. As well as going backwards, because at one point I thought I needed to know when the oil spill started so I began to subtract 26 from 68, to find how wide it was at 5:00, and continued that until there was only 16 left, and I just kept lowering the number until I found a time that it would begin. As to how I used being systematic, I made small changes to look for change when I was trying to find the time, by testing the process by increasing the time by one minute each time, and on a larger scale, I increased the time by an hour to find out the time and how much farther they are from reaching the serious environmental hazard. Another habit of a mathematician that i used, though in my opinion not a lot, was collaborating and listening with your group. I collaborated and shared my answers, choices, and my reasons for doing it. However, I think that I didn't help them learn from this problem, as soon as I learned what I needed to do, I told them the steps and processes while they were still figuring it out, which doesn't help them solve the problem their own way. On the plus side, we all worked together to find the answer, and if one of us got the answer first, we shared it with the others and explained how we got it. We were able to work decently with each other, but I feel that I should let my group mates figure out how to do the problem first instead of giving them a completely different way then what they were originally going to use.