Solutions - Quilting: The Mariner's Compass - March 11-15, 1996
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Correct solutions were submitted by:
Akiba Hebrew Academy, Merion, Pennsylvania Daniel Gabriel, David Love, Grade 11 Episcopal School of Acadiana, Cade, Louisiana Nathan Jacobson, Grade 6 Lincoln-Sudbury High School, Sudbury, Massachusetts David Trail, Grade 9 Martin County High School, Stuart, Florida Danielle P. Cegelis and Todd Spahn, Grades 12 & 10 Mt. St. Joseph's Academy, Flourtown, Pennsylvania Claire Bonner and Stephanie Miesnik, Grade 10 Murray Junior High School, Ridgecrest, California Thomas S. Kuo, Grade 9
From: Nathan Jacobson Grade: 6th Grade School: Episcopal School of Acadiana Answer: How many layers? 7 layers How many triangles? 255 triangles How much cloth would we need for the triangles? 3 yards From: Pacetkecil@aol.com Date: Tue, 12 Mar 1996 21:06:04 -0500 To: annie@mathforum.org Subject: Geometry Forum Dear Annie, My school is located in Cade Louisiana. Here is how I got my answer: Layer | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ------------------------------------------------------------ Diameter | 11" | 23" | 35" | 47" | 59" | 71" | 83" | of circle | | | | | | | | ------------------------------------------------------------ # of Tri. | 8 | 18 | 27 | 36 | 46 | 55 | 65 | Layers:7 layers Triangles:8+18+27+36+46+55+65=255 triangles How much cloth for the triangles: 3 yards Nathan Jacobson [Nathan then corrected his solution] Date: Wed, 13 Mar 1996 21:44:12 -0500 Dear Annie, I realize what I did wrong with my chart and found out that there are 190 triangles and 6 layers. Now I need only 2.25 yards for the triangles. Here is how I figured it out. /\/\/\/\/\/\/\/\/\ /\/\/\/\/\/\/\/\/\ /\/\/\/\/\/\/\/\/\ Each row has 18 triangles (9 in one direction /\ and 9 in the other \/) and is 7" high. (190/18) X (7/36 yards) = 2.05 -- Buy 2.25 yards. Nathan ************************************************* David Trail, Grade 9, Lincoln-Sudbury High School Number of Layers = 11" + 12" * n * 90" n * 6 Layers = 6 Number of Triangles = (*d)/4 Layer 1 = (** 11)/4 = 8 Layer 2 = (** 23)/4 = 18 Layer 3 = (** 35)/4 = 27 Layer 4 = (**47)/4 = 36 Layer 5 = (** 59)/4 = 46 Layer 6 = (** 71)/4 = 55 8 + 18 + 27 + 36 + 46 + 55 = 190 190 triangles Length of Fabric (1/2*4.6)7 2.3*7=16.1 190 * 16.1 = 3059 44x = 3059 x = 69.5 69.5/36 * 2 yrds. Amount of Plain White fabric *(5.5*5.5) + 3059 3059 90 8100 + 95 *90 -3154 3154 8100 4946 4946 Square Inches of White fabric ************************************************* Claire Bonner Stephanie Miesnik Mt. St. Joseph's Academy Grade 10 If the original circle were 11 inches, the next larger circle would have to be 23 inches. It would have to be a foot larger because the triangles go on all sides of the circle. That means it would be six inches on every side. The amount of diameter increase would be constant because the height of the triangles is constant. The other diameters would be 35, 47, 59, and 71 inches. To figure out how many triangles surrounded each circle, we used the base of 4 inches and the circumference of each circle. We would find the circumference and then divide by the base of 4. We realized that we must round down to make sure there would be enough room for a full triangle. The total number of circles was 190. There were 8 triangles around the 11 in. circle, 18 triangles around the 23in. circle, 36 triangles around the 35in. circle, 46 triangles around the 47in circle, and finally 55 triangles around the 71in circle. The reason the largest diameter is only 71in. is the next larger circle would have been 83in. If an 83in. circle were surrounded by triangles, the width would go over the 90 inch limit. To find out how much fabric to order, we calculated the the area of one triangle then multiplied it by 190, the number of triangles. The area of all the triangles was 3059 inches squared.With the width of the fabric being 44 in., We knew to divide 3059 by 44. We then converted our calculation to yards. The length of the fabric should be 2 yds. This amount will allow for a little extra fabric just in case. ************************************************* Daniel Gabriel 11th Grade Akiba Hebrew Academy The first thing which we do in solving this problem is find out how many rows of triangles there are. All of the triangles are in a circular shape, with a circle surrounding each row of triangles (i.e. a circle can be drawn around the outermost point of all triangles), and a circle can be drawn to represent each row of triangles. In other words, think about how tall each triangle is, and how far apart the circles, therefore are. Since each row of triangles is 6 inches high, You know that each circle (representing the outside of a layer) is 6 inches from the previous circle. This means that as you add more layers to the quilt, and therefore more rows of triangles, you have more and more concentric circles, each 6 inches apart. The maximum number of rows is reached when the radius of the largest of the concentric circles in a the point where if another row was added, the diameter would be greater than 90. To solve the problem, I drew the original circle, with a radius of 5.5" (11"/2). Around it, I drew a circle (representing the end of row 1 or triangles). This circle, therefore had a radius of 6+5.5, because it was one row of triangles (or 6") away from the first circle. 6 inches from this circle, I drew another concentric circle, whose radius was now 12+5.5. Eventually, by repeating this process, I got up to a circle with a radius of 47.5 inches, which, when doubled gave the circle a diameter greater than 90". From here, I worked backward, subtracting 6", and one layer of triangles ( the outermost circle). I then had the radius of the largest concentric circle equal to 41.5" this is maximum radius which, increasing at increments of 6", will allow the circle to have a diameter less than or equal to 90". I now knew that 41.5" from the center of the original circle (diameter 11") was the end of the outermost layer of triangles. I subtracted the radius of the innermost circle from the radius of the outermost circle (41.5-5.5) to tell me the distance from the beginning of the first layer of triangles to the end of the last layer. I found this distance to be 36", which when divided by 6", the height of each layer independently, gave me a total of 6 layers of triangles. Now on to part two, finding how many triangles there are total... For this part of the problem, when I say circle, I mean the circle on which the bases of a row of triangles rest, not the circle around their tips, like I was talking about before. You know that the innermost circle has a radius of 5.5" (as its diameter is given). You also know that outside of it, there are five more circles, each with a radius 6" bigger than the preceding circle. So, using C=2(pi)r you can find the circumference of each of the circles, and figure out how many triangles can rest on each circle and then add the number of triangles from each circle to get the total number of triangles on all 6 circles. You find the circumference of the innermost triangle (C=2(pi)5.5= 34.558") You then divide this number by 4, because each base is 4" long. This leaves you with 8.64, but since only whole triangles count, you have 8 whole triangles on the innermost circle. You repeat this process with all 5 remaining circles: add 6 to the radius of the preceding circle, multiply by 2(pi), divide by 4, and round down to the whole number. Once you do this for all of the rows, you find the following: innermost row (r=5.5): 8 triangles row 2 (r=11.5): 18 triangles row 3 (r=17.5): 27 triangles row 4 (r=23.5): 36 triangles row 5 (r=29.5): 46 triangles row 6 (r=35.5): 55 triangles You now add this number of triangles together to find the total number. The total number of triangles is 190. The problem is not over yet - now, you have to find out how much fabric you need to cut out these triangles. Each triangle is given new dimensions, including the seams. Triangles are now 4.6" X 7" high. You know that a piece of fabric is 44" wide, so to find how many triangles can be made from one piece of fabric going across, divide 44 by 4.6, and again round down to the whole number. This tells you that you can have a maximum of 9 triangles going across each row of fabric. Since there are 190 triangles, and 9 per row, you can divide to find out that there must be 21 rows (again rounding down to the nearest whole number). Since each row must be the height of a triangle, or 7", you can multiply to find out that you need 147" of fabric. Since 1 yard equals 36", divide again to find the number of yards needed. You need 4.083 yards of fabric. Since the problem tells us to round up to the nearest quarter yard, you round this to 4.25 yards of fabric. At 11:34 PM 3/15/96 -0500, you wrote: >This is where I think you went wrong. Remember that we are talking >triangles here, not rectangles. Does that change your answer any? > > -Annie Whoops, sorry about that. You know that you can have triangles pointing downward as well. This means that you can acrually have 18 triangles going across each row of fabric. You then divide 190 by 18 to tell you you need 11 rows, rounding up to the whole numebr. Multiple this by 7", the height for each row to give you 70" of fabric. divide this by 36" and round up to find you need 2 yards of fabric. I hope this is better. Please let me know. Thanks. Daniel Daniel A. Gabriel ************************************************* David Love Akiba Hebrew Academy 11th grade The problem first asks you to find how many layers there are, and how many triangles, for a 90 inch quilt with a center circle 11" diameter, and isosceles triangles with 4"bases, and are 6" high. To start this problem, I divided the 90 inches in half, because was only working in one direction, or half the quilt. The first layer has a radius of 5.5", the next one's radius is 11.5, and they keep increasing by 6 because they are 6 inches high. The last layer of triangles occur at 35.5", because at the next highest one 41.5, 6 inch triangles would not completely fit into the quilt. Starting at 5.5, and going to 35.5, there are 6 layers of triangles. The next step is figuring out the total number of triangles. Because the circumference of a circle is given by 2xr, and you know the radius of the initial circle is 5.5". You know the circumference is 2*x*5.5, or approximately 34.55. Since the bases are 4", you can have 8 complete triangles for the first layer. Repeting this process for the next layer with a 11.5 radius, you get 18 triangles, then 27,36,46,and 55 triangles for the following layers. This adds to a total of 190 triangles. The next part asks you how much fabric that is, given there is a 1/4" seam, making the triangles approximately 4.6 inches at the base, and 7 inches tall. You can find the area of each triangle by A=(bh)/2. Incorporating the new dimensions you get the area to be 16.1 square inches for triangle. Because there are 190 triangles, there is a total of 3059 square inches. Since you know the width of the fabric is 44", you can divide that into the total square inches to get the length in inches. It comes out to be 69.52272 inches. The problem asks for the amount in yards, so you divide by 36 getting 1.93, which rounds up to 2 yards of fabric, as your final answer. ************************************************* Danielle P. Cegelis, grade 12 Todd Spahn, grade 10 Martin County High School Stuart, Florida ANSWERS: 1) There are 6 layers of triangles on the quilt. 2) There are 190 triangles on the quilt. 3) We will need 2.75 yards of fabric in order to complete all of the triangles. EXPLANATION: 1) It was given that the diameter of the original circle was 11 inches and that each triangle would add 6 inches onto the radius of that circle for each layer that was added, thus I calculated how many layers there could be without the diameter of the layers exceeding 90 inches. 11(the original diameter)+[12(the length added per layer)*6(the number of layers added)]=83 inches. 2) I figured the cicumference of the circle and the divided that by 4 (the length of the base of each triangle). The original value reduced to 8 triangles, then I did the same for the next circle (with diameter=23 inches) and the number of triangles reduced to 18 and so on... 3) The last portion of the problem was a bit more difficult unless I am just confused about what you meant. I took the question as meaning you wanted to know how much fabric you would need to make all 190 complete triangles (not needing to sew scraps of cloth together to make another triangle). So I calculated that you would only be able to use 36.8 inches of the 44 inch wide fabric because if you set the bases of 8 of the triangles together the running length would be 36.8 inches. By using 7 inches of the fabric you could cut out 15 complete triangles. To cut out all 190 complete triangles at the rate of 15 triangles per one 44"by7" of cloth you would need to measure out 13 pieces 44"by7" . If you add all of those pieces together you have a running length of 91 inches which, when divided by 36 to calculate yardage, equals a little over 2.52 yards, you must round up to the next quarter yard in order to fit all of the triangles. Sorry my explanations are so wordy. Thank you, and have a nice day. ************************************************* From: Thomas S. Kuo School: Murray Junior High School, Ridgecrest, California Grade: 7th 1. 6 layers make the quilt under 90". I got this by making a table as shown in the following: # layer | Diameter of circle | Cross Section | # of triangle | (in inches) | (includes circle | | | and triangles) | ---------------------------------------------------------------- 1 | 11 | 23 | 11*pi/4 ~ 8 2 | 23 | 35 | 23*pi/4 ~ 18 3 | 35 | 47 | 47*pi/4 ~ 27 4 | 47 | 59 | 59*pi/4 ~ 36 5 | 59 | 71 | 71*pi/4 ~ 46 6 | 71 | 83 | 83*pi/4 ~ 55 7 | 83 | 95 | out of range 90" 2. From the above table, I got the total number of triangles 190. 3. A total of two and one quarter yards of fabric is needed for the triangles. * * * * * * * * * * * * * . . . . . . * * * * * * * * E * * * * * * * * *. * * * * * * * * * . * * * * * * * * * . * * * * * * * * * . * * * * * * * * * * * * * * * *. . . . . . * * * * * *. . . A B C D (1) AB = 4.6", ED = 7", and CD = 2.3". Let n be the number of triangles along the bottom (there is another n triangles along the top). Solve equation 4.6 n + 2.3 <= 44 for n n <= 9.06 and n = 9 for an integer. (2) There are therefore 18 triangles in a piece of fabric of size 44" x 7". (3) To determine the length of fabric, I have to divide 190 by 18 which is 10.56 which means I need 11 pieces of fabric of size 44" x 7" to make the 190 triangles. (4) I take the 11 pieces and multiply it by 7" which is 77" in total or 2.14 yards which rounds up to 2 and 1 quarter yards. Extra Four and half square yards white fabric are needed to fill out the 90" square. I got it with: 90 * 90 - 1/2 * 4 * 6 * 190 = 5820 square inches = 4.49 square yards ~ 4 1/2 square yards
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