2000 H.S. State Mathematics Contest Results

29 April 2000

Carpet is typically manufactured 12' wide and rolled bottom side out onto a cardboard tube with a 4", 6", or 8" outside diameter. These rolls are then sent to a warehouse. A typical warehouse will have hundreds of these rolls in stock. When a retailer makes a sale they will order the piece from the warehouse. Consequently the warehouse is filled with partially used rolls. Most carpet comes with a stiff back and in thickness between 3/8" and 3/4". (Commercial carpet is about 3/8", Berber is about 1/2", Shag is about 5/8" and Plush is about 3/4".)

You are to:

• Develop a mathematical relationship between the appropriate dimensions of the roll and the length of carpet on the roll.

• Devise a system or tool that can be used to easily estimate the amount of carpet left on a roll.

Some suggested variables might be R for the radius of the cardboard tube, T for the thickness of the carpet and A for the amount the radius of the roll is increased with carpet wrapped around the tube.

The Scoring:

You will be scored on a four-point scale for each bulleted area:

Understands and applies mathematical concepts and procedures

Algebraic Sense and Operations

• Develop an appropriate mathematical relationship between the dimensions of the roll of carpet and the length of carpet on the roll

Measurement

• Show correct use of related measurements

Uses Problem Solving

• Use appropriate strategies, procedures, and processes in constructing a solution. Appropriate formulas and algebra techniques are used

• Show consistent and correct use of applicable information and operations. Calculations are error free and appropriate.

Uses Mathematical Reasoning

• Support assumptions, generalizations, and results by showing the important work with supportive explanations of the reasoning

Communicates the Results

• Presents all information in a clear, effective, and organized manner

Makes Connections

• System or tool is easy and efficient to use for estimating the length of carpet on a roll

Details for the Scorers

The Scoring:

You will score teams papers on a four-point scale for each bulleted area:

4 - meets or exceeds all relevant criteria

3 - meets most relevant criteria

2 - meets some relevant criteria

1 - meets few relevant criteria

0 - not scoreable: off topic, no attempt, can't be read, etc.

Understands and applies mathematical concepts and procedures

Algebraic Sense and Operations

• Develop an appropriate mathematical relationship between the dimensions of the roll of carpet and the length of carpet on the roll

A correct mathematical relationship is established between the amount of carpet on a roll and the dimensions of the roll

Measurement

• Show correct use of related measurements

Units are correctly interpreted, used, and noted throughout calculations

Uses Problem Solving

• Use appropriate strategies, procedures, and processes in constructing a solution. Appropriate formulas and algebra techniques are used

• Show consistent and correct use of applicable information and operations. Calculations are error free and appropriate. Calculations are error free and proper and appropriate algebra techniques are used

Uses Mathematical Reasoning

• Support assumptions, generalizations, and results by showing the important work with supportive explanations of the reasoning

Evidence that mathematical reasoning supports: the assumptions, (nap of the carpet flexes and the backing is consistent in dimension so that the location of the backing is used as the radius), generalizations, (each wrap of the carpet is approximately a circle with a radius of the circle being increased by the thickness of the carpet), and results.

Communicates the Results

• Presents all information in a clear, effective, and organized manner

It is easy to follow the thinking, the progression of thought, the interpretations, and the results.

Makes Connections

• System or tool is easy and efficient to use for estimating the length of carpet on a roll

The system or tool devised is easy to operate, easy to learn, convenient, cost effective and unlikely to produce errors

Total Points

Solutions:

Each wrap of the carpet will essentially be a circle with at radius increased by the thickness of the carpet. Let R be the radius of the cardboard tube measured to the outside of the tube. Let the thickness of the carpet be T and the amount the radius is increased because of carpet on the roll be A. Since the carpet is wrapped with the backside out and the backside is the stiff side the first wrap will have a radius of R+T, the second will be R+2T, the third will be R+3T and so on. R and T would be measured in inches.

For example, if the radius was 4 inches, and the thickness was 1/2 inch then the first wrap would be 2(pi)(4+0.5)/12 feet and the second would be 2(pi)(4+2(0.5))/12 feet and so on.

One could count the wraps and get the lineal footage of the carpet by doing:

Lineal Footage = (2(pi)((R+T)+(R+2T)+(R+3T)+...+(R+nT)))/12 where n was the number of wraps.

This formula then would simplify to:

Lineal Footage = (pi)nR/6 + (pi)T(1+2+3+...+n)/6

Since 1+2+3+...+n = (1+n)n/2 then

Lineal Footage would be (pi)nR/6 + (pi)Tn(1+n)/12

However since A/T is n, it would be easier to measure A and use the following formula:

Lineal Footage = (pi)AR/(6T) + (pi)A(1+A/T)/12 which could be written several ways.

This would mean that a roll on a 4 inch diameter cardboard tube with a thickness of 3/8 inches and A=10 inches would be approximately:

Lineal Footage = (pi)(10)(2)/(6(3/8)) + (pi)(10)(1+(10)/(3/8))/12 = 100 ft or 33 lineal yards or 133 square yards.

Easy methods might include:

A programmed calculator that asks for the tube size, thickness and amount on the tube (A) and returns the value calculated by the formula above

Three measuring rods for each tube size and each rod has four sides each side calibrated for nap thickness say of 3/8, 1/2, 5/8, and 3/4 inch naps. It could possibly have eight sides and break the nap down to 1/16 inch increments. The rod would be held up to the roll of carpet to measure A and the length of the carpet could be read directly from the rod that is calibrated according to the formula.

A table of values for each tube thickness:

	4" tube				6" tube				8" tube
A=	0.375	0.500	0.625	0.750	0.375	0.500	0.625	0.750	0.375	0.500	0.625	0.750
0.5	2	1	1	1	2	2	1	1	3	2	2	2
1	4	3	2	2	5	4	3	3	7	5	4	3
1.5	6	5	4	3	8	6	5	4	10	8	6	5
2	9	7	6	5	12	9	7	6	14	11	9	8
2.5	12	9	7	6	15	12	10	8	19	14	12	10
3	15	12	10	8	20	15	12	10	24	18	15	12
3.5	19	15	12	10	24	18	15	13	29	22	18	15
4	23	18	14	12	29	22	18	15	35	26	21	18
4.5	28	21	17	15	34	26	21	18	40	31	25	21
5	33	25	20	17	40	30	24	20	47	35	29	24
5.5	38	29	23	20	46	35	28	24	53	40	33	27
6	43	33	27	23	52	39	32	27	60	46	37	31
6.5	49	37	30	26	58	44	36	30	67	51	41	35
7	56	42	34	29	65	49	40	34	75	57	46	38
7.5	62	47	38	32	73	55	44	37	83	63	51	43
8	69	52	42	36	80	61	49	41	91	69	56	47
8.5	76	58	47	39	88	67	54	45	100	76	61	51
9	84	64	51	43	97	73	59	49	109	82	66	56
9.5	92	70	56	47	105	80	64	54	118	89	72	60
10	100	76	61	51	114	86	70	58	128	97	78	65
10.5	109	82	66	56	124	93	75	63	138	104	84	71
11	118	89	72	60	133	101	81	68	149	112	90	76
11.5	127	96	78	65	143	108	87	73	159	120	97	81
12	137	104	84	70	154	116	94	79	171	129	104	87
12.5	147	111	90	75	165	124	100	84	182	137	111	93
13	158	119	96	81	176	133	107	90	194	146	118	99
13.5	168	127	102	86	187	141	114	95	206	155	125	105
14	180	136	109	92	199	150	121	101	219	165	133	111