A Math Forum Project

Geometry Forum - Problem of the Week Solutions - Carefully Cutting Cloth, Jan. 16- 20, 1995 Annie says: This problem generated some great solutions! I asked Joe Ganley, the original poster of the problem, what inspired the question, and he said: It's actually very close to what one might guess. My wife has a piece of material of the specified size, and wants to make a tree skirt (like for a Christmas tree) that is as big as possible. -- Joe Several issues came up with this problem. First, and one that several people stumbled over is, "What's a practical answer to the problem?" Sure, you could make a circle of diameter 45", and you wouldn't have to cut anything. Or you could figure out the total area of fabric available, and figure out what diameter circle has that area, and while those are "correct", they aren't really good answers to the question. So what makes a good answer to the question? Sean Nichols and I went around on this for a while, and I have included all three of his messages on this thread. However, several folks came up with good answer that are certainly practical. David Barber wins the prize, if there is one, by making the biggest circle without using "more" seams. When you are solving a problem like this, that doesn't have purely mathematical answers, there is often more than one way to solve it, and while a couple of the answers might be more efficient or a little bit better, some of the other ones will do just fine (as seen here). All the answers lacked one little "practical" insight, though. When you cut up fabric and sew it back together, you lose some area as the seams have to overlap, and take up fabric. That's the ultimate in practical, and one that you would have to consider if the finished piece had to conform to some minimum requirement (like making a bag to cover a certain box). There was one other problem that some people had a problem with. While they figured that a good way to start would be to make the given rectangle into a square, they tried to do so by finding the total perimeter of the rectangle and dividing by four, thinking this would give them the edgelength of the new square. That's all well and good if you're talking about fences and similar objects, where perimeter is the only variable, but conserving perimeter doesn't conserve area, and we needed to deal with area in this problem. David Barber David Barber, Grade 8 student at Upper Canada College Preparatory School in Toronto, Ontario, Canada Cut the 116 by 45 piece in half. Then cut the one half into four pieces, two 58 x 8 pieces and two 58 x 14.5 pieces. Re-sew these pieces so that the 58 by 14.5 pieces lie along the 58 inch side and the 58 by 8 pieces are centred on the 45 inch side. A circle with diameter 74 can be drawn in the middle of this shape. Diagram 58 ************************************ 8 * * * * ****** ****** * 8 * * * * * * * * * * 58 * * * * * * * * * * * * * * * ****** ******* * * 8 * * *********************************** The new piece is 74 from top to bottom and the two 58x8 pieces start 8 inches down from the top on both sides. The circle has a diameter of 74 inches. Sent by Doug McDougall Head of Mathematics, Upper Canada College Preparatory School Uyen Nguyen Solution for POW for Jan.16-20: Submitted by Uyen Nguyen, Steel Valley High School, Pa. A circle with the largest diameter possible would fit into a rectangle closest to a square. I had to find the values of the dimensions closest to each other that would still give me the same area. I found the square root of 5220 which was approximately 72.24956747. Therefore, one of the dimensions was around 72 and the other very close to it. Through trial and error I found the other dimension to be 72.5. 72 x 72.5 = 5220, the same as the first rectangle. Now I had to rearrange the first rectangle to come close to this shape. 1. Cut 43.5 in. form the 116 to get a piece 72.5 in. by 45 in. (Piece A), and a piece 43.5 in. by 45 in. (Piece B) 2. Cut rectangle B into 4 pieces that fit dimensions of 27 in. by 72.5 in. that would be added to A. 4 pieces are: C = 27 x 43.5 D = 29 x 18 E = 14.5 x 9 F = 14.5 x 9. 3. Attach rectangle C,D,E,F to rectangle A to form a large rectangle, 72.5 in. by 72 in. 72.5 ----------------------------------- | | E | F | | C | D | |-----------------------------------| | | | | | | | | | | | | | A | 72 in. | | | | | | | | | | | | | | ----------------------------------- Sean Nichols Hello. This is Sean Nichols, at Burnaby South Secondary School, in Burnaby, BC, Canada. Following is my solution to the Problem of the Week for January 16-20: The problem, as stated, contains in fact two distinct problems. It is asking (1) To cut the cloth such that the resulting circle is as large as possible, (2) To make the fewest number of cuts possible. Cutting the given cloth to these two specifications will not necessarily give the same circle: Let's take the last case first. The smallest number of cuts to be made is, simply enough, 0. If we make no cuts to the cloth, the maximum diameter of any circle contained in the cloth would be equal to the smallest side, or 45". (Area = pi*r^2 = about 1590sq.in) However, if we want to make the circle any larger, we must cut more and more rectangles out of the cloth, and so the second condition (to cut as few rectangles as possible) is no longer applicable: The larger the circle, the more rectangles. Since no matter how many times we cut rectangles off, the area will remain the same, then the upper limit of the area of any circle contained in the cloth should be the same as the area of the original rectangle, or 45x116 = 5220sq.in. - It should be possible to make a square out of the rectangle, again having an area of 5220, and therefore a side of about 72.2". Then, theoretically, small rectangles could be cut out of the corners, and attached to the sides, and so on, yielding a figure that resembles a circle. Of course, since we are cutting out rectangles, not a smooth line, the resulting shape will not be _exactly_ a circle, but will be pretty close, especially as the cuts made get smaller and smaller. So, The maximum _limit_ for the area of the circle is 5220, and since pi*r^2 = 5220, then r = about 40.76, and the limit for the diameter is 81.52490677...". Thank You - Sean Nichols [Annie pesters Sean to think about a "practical" answer to the question.] Annie, I'm not sure I understand exactly what you're trying to say... The two objectives seem mutually exclusive - it seems to be a matter of priorities - are we trying to get the largest possible circle, or the fewest possible cuts? Both cannot be achieved at the same time. As I have already stated, The more cuts we make, the larger the circle, and vice versa (The fewer cuts, the smaller the circle) - to a certain point. From your reply, it seems to me as if we are asking primarily for the largest possible circle, which would involve a nearly infinite number of cuts (!!). As you pointed out, this is undesirable, as too many tiny rectangles are impossible to sew together again, and get a circle out of. However, how many is too many? How small is too small? At what point do we say "Enough"? I agree, if this point can be established, we have some boundaries within which the problem can be solved, and have a reference point from which can be determined the smallest number of cuts. As the problem stands, however, it is impossible to solve it to any degree of satisfaction. Thanks - Sean Nichols [Annie writes back to address Sean's concerns] Annie, The exact answer to this problem depends (as you pointed out), on the exact purpose this cloth circle is to serve. If we had the original rectangle, we could cut a circle with a diameter of 45" (smallest side of rectangle). from this rectangle could be constructed a square, which is the quadratic allowing us the largest diameter of any circle. The area of this square would necessarily be the same as that of the rectangle (5220sq.in), so the length of one side is the square root of that, or about 72.2". So, we cut 72.2" off the long side, giving us a remainder of 43.8". Our cloth is now 45" x 72.2" To get the short side up to 72.2", we have to lengthen the side by (72.2 - 45 = 27.2"). Our "remainder" cloth is 45"x43.8" We need to make this into 72.2"x27.2", so off the short side (43.8") we can cut a length of 27.2". This will give us a piece of cloth 27.2x45", which can be tacked onto the length-deficient side of our object square. We now have a 27.2"x72.2" square which we want to fill with a 16.6"x45" piece if cloth. A third cut will give us a 27.2"x16.6" piece, which can be used to fill the gap in the object. We make a fourth cut, to give us a piece of cloth 10.6"x17.8", and so on... We could theoretically go on forever, and never _quite_ achieve a perfect square (or maybe we could, but I haven't followed it through that far...) But, since the diameter of the circle will be 72.2" (the length of one of the squares), and therefore the radius 36.1" So as long as the distance from the center of the square to the inside corner of the missing "gap" in the cloth is larger than 36.1", there is no problem. In this case, after 4 cuts, it is 36.06" - close enough, I figure. If not, 2 more cut - and - pastes will give us a gap small enough to provide a large enough space for the radius. If we wanted to carry this further, claiming that we needed a larger circle, we could cut small pieces off the corners, as I claimed in my earlier solutions, and paste these onto the centers of the sides. This would require 2 more cuts for each further iteration (1 per corner), and so on. Thank You, - Sean Nichols. Kristina Almquist Kristina Almquist 9th grade I first multiplied 116 x 45 and got the area of 5220 in ^2. I figured the largest diameter would have to se in a square and a square is the easiest made with 4 congruent squares. I divided 5220 by 4 = 1305...so each smaller square would have to have 1305 in ^2= 36 in on a side. This larger square made up 4 squares (36 x 36) will have an area of 5184 in^2 and a diameter of 72. To cut out these pieces start in the top left corner and count 36 in. down and 36 in. over and 3 times going lengthwise for the 3 large squares (36 x 36). Brelow the place the 3 squares were in 9 inch strip. Count over 36 inches on this, 3 times going lengthwise. Continue the line of the bottom of the boxes over, So you have a strip 36 x 8 and a piece 9 x 8. Since 5184 is 36 in. less than 5220 we need to cut off 1 36 piece of the long strip. Cut the 9 x 8 into 4 strips of 9 x2. Sew the three pieces together then lie the widest of the strips next to each other then the 7 x 36...then the four 9 x 2's will go end to end to fill the rest of the space. Candice Wiegand Candice Wiegand 9th grade To solve this, first start with the strip. if we can't start working with 116" measurement for a diameter, we'll work up from 45". Make that into a 45" x 45" square. We still have 71" x 45" of cloth left. If we cut the remaining pieces into 17.75 " x 45" strips and attach to the previously made square, the new area will be approximately 3936 in^2. That would have used up all the cloth with only 5 cuts and resulting in a circle with a diameter of approximately 62.75 inches. Previous page || Next problem || Previous problem || Table of Contents || Forum Home Page