Planning Platforms POW
For this problem of the week, we were introduced to several different variables, people, and questions that we needed to use in order to solve the problem. A city is getting ready for a baton twirling event, in which they will have several platforms in which the baton twirlers will stand on. Kevin is the leader of the twirlers, and has several decisions to make concerning the platforms. He needs to decide on the number of platforms, the height of the first platform, and the difference in height between the platforms. Camila, his assistant, also has a few things to do. She needs to know the height of the tallest platform, and also plans to hang strips of fabric from the front of the platforms, so she needs to know how much material she should buy. Unfortunately for her, she can’t do her job until Kevin does his. In this problem, we take on the role of Camilla’s assistant, and we need to give Camilla the information she needs when Kevin makes up his mind. To do this, we have to create two formulas that will allow us to find the height of the tallest platform as well as one that will tell us the total amount of fabric that Camila needs to buy. The formulas need to be in terms of the number of platforms, the height of the first platform, and the difference in height between adjacent platforms. Even though the problem has a lot of information and seems complicated at first, the only things we have to figure out for this problem are the two formulas.
We started off the problem by simply restating some basic, but important information that we need to get the solution. We had two columns, one for what Kevin needs to know, which consisted of the number of platforms, height of the first platform, and the difference in height between platforms, and Camilla’s, which had the height of the tallest platform and the height increase for each platform. With the basic information that we had, my table and I made a sketch of the potential platforms and their heights. We thought that maybe it had something to do with slope intercept form, or y=mx+b, because it had a constant growth in height no matter what. We set up a basic problem, saying that the first platform was 1 foot, the second platform was 2 feet, and the third platform was three feet. So the total amount of material needed was 6 feet of fabric. Even though this wasn’t really was the problem was asking for, we tried it out anyways. Here is a picture of our sketch.
We started off the problem by simply restating some basic, but important information that we need to get the solution. We had two columns, one for what Kevin needs to know, which consisted of the number of platforms, height of the first platform, and the difference in height between platforms, and Camilla’s, which had the height of the tallest platform and the height increase for each platform. With the basic information that we had, my table and I made a sketch of the potential platforms and their heights. We thought that maybe it had something to do with slope intercept form, or y=mx+b, because it had a constant growth in height no matter what. We set up a basic problem, saying that the first platform was 1 foot, the second platform was 2 feet, and the third platform was three feet. So the total amount of material needed was 6 feet of fabric. Even though this wasn’t really was the problem was asking for, we tried it out anyways. Here is a picture of our sketch.
We then created another rough sketch concerning the second half of the problem, figuring out the total amount of material needed to cover the platforms. We made the first platform 2 ft tall, the second 4 feet tall, the third six feet tall, and the fourth eight feet tall. So the platforms had two feet of space in between them. We made it really simple and added up all the heights to get the total amount of fabric needed. Here is a picture of the sketch.
But the next question was, how could we figure out these two pieces of information without sketching each one out? Lets start with a new scenario. There are a total of 5 platforms. The first platform is 12 feet, the second is 15 feet, the third is 18 feet, the fourth is 21 feet, and the fifth is 24 feet. Each platform is obviously three feet apart in height.
From this example problem, we can pull both solutions using the equations we created. For Kevin’s decision, he has three variables. The number of platforms (n), the difference in height of the platforms (d), and the height of the first platform (h). To find the height of the tallest platform, we came up with the equation T=(n x d) - d + h. After plugging in the numbers, your solution is 24, and if you relate that back to the drawing, it proves it to be correct.
Now to figure out Camillas dilemma, or how much material she needed to cover the platforms. In order to find it, we had to use the solution from the previous equation to find the solution to this one. The equation was (t+h)(.5n), or (height of the tallest + height of the shortest)(.5 x number of platforms). If you plug in the numbers and solve the equation, you get 90 feet. If you add up all of the heights of the platforms from the diagram, it proves to be correct. You can use both of these equations to solve for any amount of platforms, heights, etc.
The two habits of a mathematician that I used while solving this problem were Visualize and Seek Why and Prove. For visualizing, almost every problem or way I tried to solve the problem I drew out the platforms and all their dimensions. This helped me figure out which variables related back to the problem. For Seek Why and Prove, I related a lot back to the original problem, which is usually harder to do, but for this problem it was easier. All in all I think I performed pretty well with this problem in such little time, and I ended up fully understanding the problem and doing a lot of the work on my own. But collaborating with my group also opened my mind to a lot of different ways to look at the problem. I think I deserve an A because of this.
The two habits of a mathematician that I used while solving this problem were Visualize and Seek Why and Prove. For visualizing, almost every problem or way I tried to solve the problem I drew out the platforms and all their dimensions. This helped me figure out which variables related back to the problem. For Seek Why and Prove, I related a lot back to the original problem, which is usually harder to do, but for this problem it was easier. All in all I think I performed pretty well with this problem in such little time, and I ended up fully understanding the problem and doing a lot of the work on my own. But collaborating with my group also opened my mind to a lot of different ways to look at the problem. I think I deserve an A because of this.