SIAM REVIEWVol. 26, No. 2, April, 1984

(C) 1984 Society for Industrial and Applied Mathematics004

CLASSROOM NOTES IN APPLIED MATHEMATICS

EDITED BY MURRAY S. KLAMKIN

This section contains brief notes which are essentially self-contained applications of mathematics thatcan be used in the classroom. New applications are preferred, but exemplary applications not well known orreadily available are accepted.

Both "modern" and "’classical" applications are welcome, especially modern applications to current realworldproblems.

Notes should be submitted to M. S. Klarnkin, Department of Mathematics, University of Alberta,Edmonton, Alberta, Canada T6G 2G 1.

ESTIMATING THE LENGTH OF MATERIAL WRAPPED AROUND ACYLINDRICAL CORE*

FRANK H. MATHIS AND DANNY W. TURNERS"

Abstract. We present three methods for estimating the length of material (e.g., carpet, paper) that iswrapped around a cylindrical core. Relationships among the methods are explored.

Introduction. The problem we wish to address originated from a request by ourcampus Central Receiving Office for "an equation" to compute the number of linear unitsof carpet remaining on many unlabelled rolls that were in storage. Neither unwinding therolls nor weighing them was practical. So a relatively simple procedure, using measure-ments taken from the end of a roll, was desired.

Below we shall develop three methods to approximate the desired length. Eachmethod will calculate the exact length based on specific assumptions about the geometryof the roll. Although the mathematics involved .is elementary, we feel that the results areinteresting and even somewhat surprising.

For all three methods, we assume that the material is wrapped around a cylindricalcore having radius r. Consider one circular end of the core and let point P be at the centerof this circle. Let point Q, on the circumference of the circle, correspond to the internalstarting position of the wrap and let point R, on the inner edge of the material, correspondto the terminal position of the wrap. For computational simplicity we assume that P, Qand R are collinear. Indeed if they are not, our calculations will be off no more than onecircumference of the inner core (i.e., 2rrl), an error which we consider to be acceptable.Fig. illustrates the situation.

We define the outer radius r2 to be the distance from P to R. We denote by n thenumber of wraps the material makes from Q to R and by the thickness of the materialwhich we assume to be constant throughout the wrap.

Method A. In the simplest approach, which requires only elementary geometry, weassume that the material is wrapped in concentric circles for which the difference betweenconsecutive radii is t. We may lhen either sum up the average circumferences of eachlayer or equivalently calculate the average circumference overall and multiply by n. Wethus obtain the first length formula:

L 2r(r +(2i-i=1 1))=nr(r+r2).*Received by the editors August 20, 1982, and in revised form March 3, 1983.’Department of Mathematics, Baylor University, Waco, Texas 76798.

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264 CLASSROOM NOTES

FIG. 1. End view ofmaterial wrapped around a cylindrical core. r is the distancefrom P to R.

This method is simple enough; however, it may be challenged since it does notaccount for the "bump" which occurs when each layer is wrapped above the starting pointQ. The next two methods attempt to take care of this.

Method B. If we assume that the material is flexible enough to allow a tautwrapping, then the geometry of the problem will appear as in Fig. 2. Here we are lookingat the end of the roll again. The circle with center P and radius r represents an end of thecore. Point S is located so that the line through points T and S is tangent to theaforementioned circle at S. Angle QPS is denoted by a and is the same as angle RTV. Thelength of line segment TS is called l. Now it is an easy exercise in trigonometry andgeometry to. write down the total length of material, L2, that is represented in Fig. 2. First,the length corresponding to the circular part of the ith wrap is approximated by

r, +(2i- 1) (27r-a), i=1,2,. .,n.

Note that an average radius is used and that a arccos (r/(rl + t)). The lengthassociated with each of the n rectangles is /t(t + 2rl). The length associated with the

I’

FIG. 2. Detailed end viewfor the geometry ofMethod B (n 3).

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CLASSROOM NOTES 265

ith sector is (2i 1)(t/2)a. Here we use the average radius again. Adding all of our termsand simplifying yields

L_ n[Tr(r, + r2) +/-

where a and have been previously given in terms of rl and t.

Method C. Suppose that the material is not as flexible as the geometry of method Brequires, but we may assume that the material forms a spiral from points Q to R. We maydefine this spiral in polar coordinates by

r rl + 0, O<-_O<=2mr.

Then the length of the spiral is given by the integral

L3 f2. (r, + (t/2r)O)2 + (t/2r) dO.

Using elementary calculus techniques we obtain a closed form for L3. Let g(x)x/x + (t/2r). Then

L3 =7 rzg(r2) r,g(r,) + {ln (r + g(r2)) -In (r, + g(r))}

Comparison of the methods. At this point we might favor method A because of itssimplicity. However, we should ask if the geometries of methods B and C will producelengths substantially different from that calculated by method A. To answer this we willinvestigate the differences in the approximations obtained from the three methods. Themajor tool for this comparison is the following.

THEOREM 1. Let LI, L2, L3 be the approximations given by methods A, B, and C,respectively. Then

(1) L L 2 xf (r r)3/2n-/a + O(n-3/:),3

and

(2) L3- L(r,- r)

ln r)4rn- + O(n-3).

Proof This is simply an application of Taylor’s theorem. We will present the steps toobtain (1). Equation (2) follows in a similar manner.

Since

a arctan + 0r 3

L2 mr(r + r2) + n(l- ar)

Zl + nr, [ ()3 + O(()5)]"But by substituting

1= 2x/-t/l+--- 2xft(l+O())=2r’(r--rl)(l+O(n-’))2rl n

into the above, we obtain (1).

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266 CLASSROOM NOTES

The above theorem implies that all three methods yield essentially the sameapproximation provided n is large relative to (r2- r)/rl. We may now seek conditionswhich will assure that the approximations differ by less than some reasonable error, sayone percent. For simplicity we will obtain bounds using only the leading terms of theright-hand sides of (1) and (2). Although the results will be meaningful only in anasymptotic sense, i.e., as n tends to infinity, they are sufficient to give a representativecomparison. In particular, the next theorem investigates the difference in L2 and L1.

THEOREM 2. Let El (2x//34h)(r2 r)3/2rt-1/2 and suppose that there is apositive number z such that < zEr and n >-_ 31 z. Then El/L is less thanO.O1.

Proof Note that < z-r implies that (r rl)/rln <= z z. Then since r -+- r > r rwe have

El 2 x[ (r rl)3/2 2 x[ [r2 rl]H-3/2 __<

1/2 _1L 3rl r + r 3r tIXrn n

2-z < .01

As an example of the z in the above theorem, consider z 0.1. Then we may expectL and L2 to differ by less than one percent provided the thickness of the material is lessthan one tenth of rl and there are ten or more wraps.

The next theorem presents a similar result for Ll and L3.THEOREM 3. Let E ((r rl)/47r In (rz/rl) t1-1 and suppose that <-_ rl and n > 3.

Then E2/LI is less than 0.01.Proof Since rE > rl > 0, In (r/r) < (rE rl)/rl. Also since rE + r > rE rl and

(r2 rl)/nrl <= 1, we have

E2 (r2- rl)In (r2/rl)L 4nTt"2 (r + r) 47r2rt <

.01

Conclusion. We have presented three methods for calculating approximations forthe length of a material wrapped in a cylindrical roll, each based on different assumptionsconcerning the geometry of the roll. However, in most practical situations we have shownthat all three approximations agree to within an accuracy of roughly one percent.

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