P.o.w. 5: Possible Patches
In this problem, Ralph Lauren, is cutting 3 in.-by-5 in. patches from a 17in.-by-22in. rectangular piece of satin. Mr. Lauren refuses to piece together patches made out of the leftover satin. Our job was to create a diagram to find out how many pieces he could cut out of the rectangular piece of satin. After we had done this, we had to apply the same process to different dimensions and record our observations. For the first problem, I managed to fit twenty-three 3 in.- by-5in. patches in the 17 in.-by-22in piece. In the next problem, I managed to fit two 9-by-10 patches in a 17-by-22 piece. The rest of my work is detailed in this table. When Creating these diagrams I tried to fit each piece as tightly as possible, so that’d there’d possibly be more room for a new rectangle. (Clarification: The Regions covered with the X pattern are unused pieces of satin).
It has been very difficult for me to find any sort of pattern or discover any formulas, but here are my observations:
-If the area of the satin piece is divisible by the area of a patch, then the division’s quotient is the maximum amount of patches one can fit in. -If the sum of the width and the length of the patch size are greater than one side of the satin piece, the total amount of patches is equivalent to the amount of times the length of the patch can be multiplied without becoming longer than the length of the piece of satin. |
-If two rectangles are able to have the same area and they have different dimensions, they are able to hold different amounts of equivalent triangles.
-For both the 5-by-12 and 10-by-12 patches within the 17-by-22 pieces of satin, you are able to create a rectangle of unused material in the center. It should be noted that the if you add two adjacent sides within one of the patches the sum would be equal to a side length of the total satin piece. I believe these observations may be the building blocks to finding a formula for the maximum number of patches, but otherwise I have had trouble finding many patterns. |
Reflection
I believe I deserve at least a 9/10, because I found the answers to each problem and I attempted to explore it as far as I could. I put a lot of effort into trying to find a formula, but I was unable to find any pattern. However, I was able to find some recurrences, as seen above. My biggest accomplishments for this P.O.W. would be looking for patterns and staying organized. I looked for patterns by attempting find relationships between the lengths and widths of the smaller and larger rectangles and by looking for similarities in the maximum amount of patches that could be found in each rectangle. This work soon became very challenging once I began including more and more variables, and would’ve definitely become muddled if I hadn’t stayed organized. I stayed organized by making sure to keep all my work together and by recording all my results in a clear table. I made sure to label this table properly to make sure I knew what I was looking at. In the end, I found this problem very, very frustrating, since it didn’t resemble a lot of work I’ve done before in Geometry, Trigonometry, and Calculus and because I was unable to “fit the puzzle pieces together” to find one unifying formula or pattern.