Thinking out of the Box... Problem It s a richer problem than we ever imagined


 Barnard Lewis
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1 From the Mthemtics Techer, Vol. 95, No. 8, pges Wlter Dodge (not pictured) nd Steve Viktor Thinking out of the Bo... Problem It s richer problem thn we ever imgined The bo problem hs been stndrd optimiztion eercise in lmost every clculus tetbook since Leibniz nd Newton invented clculus. With the cpbility of technology in the form of grphing clcultors, this eercise hs recently become stndrd fre erlier in the mthemtics curriculum. We even find it in middle school curricul s nice hndson eercise in dt nlysis. With some vritions in the numericl dimensions of the pper, the problem is similr to the following: Given rectngulr sheet of pper 8.5 inches 11 inches, form bo by cutting congruent squres from ech corner, folding up the sides, nd tping them to form bo without top. To mke bo with mimum cpcity, how lrge should the squre cutouts from the corners of the originl pper be? See figure 1. rough determintion of the cutout size tht results in bo with mimum cpcity. In course prior to clculus, students might be sked to write the function of tht describes the volume of ny bo in which the length of the side of the squre cutout is denoted by. This function is s follows: v() = (8.5 2)(11 2) Depending on the course nd on the technology vilble, students cn gther dt from this function or grph it over the intervl [0, 4.25] nd thus determine the bo of mimum volume, tht is, the bsolute mimum point of the dt or of the grph over the given intervl. A grph of this function, drwn with TI83 grphing clcultor, is given in figure 2. Grphing clcultors hve llowed the bo problem to become stndrd fre erlier in the curriculum 8.5" 11" Fig. 1 Fig. 2 In beginning clculus course, students could use symbolic mnipultor or tke the derivtive of the volume function by hnd, set it equl to 0, solve, nd thus determine the vlue tht gives the bo of mimum volume: In middle school setting, groups of students re often given sheets of pper nd sked to cut uniform squres from the corners, cutting differentsized squres for ech sheet. They then fold up the sides to mke vriety of boes of different sizes nd fill these boes with something, such s popcorn, nd mesure or count the mount needed to fill the boes. In this mnner, students obtin Wlter Dodge, tught mthemtics t New Trier High School in Winnetk, IL He is currently mthemtics contest writer for the North Suburbn Mthemtics Legue of the Chicgo Are. Steve Viktor, is the mthemtics deprtment chir t New Trier High School, Winnetk, IL He is frequent speker nd is especilly interested in the Advnced Plcement clculus progrm. 568 MATHEMATICS TEACHER Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3  Techers Teching with
2 or v() = (8.5 2)(11 2) v() = v'() = = Discrding the solution tht is not in the prcticl intervl [0, 4.25] yields n pproimte solution of = inches. We cn verify this vlue yielding the bsolute mimum by testing the endpoints where the volume is zero nd the volume t = tht is positive. The previously described eperiences usully comprise the totl eposure tht student, or for tht mtter, techer, hs with this problem tht my lso led to interesting eplortions for lgebr nd geometry students. This rticle revels further questions tht cn be investigted from this simply stted problem. We ssigned some of these questions to our AP clculus students s projects to complete outside of clss. We hve hd fun developing some of the lter questions on our own nd pln to use them with students in the future. Question 1 If we lwys strt with squre sheet of pper, does common reltionship eist between the length of the side of this squre nd the length of the side of the smller squres tht re cut out from ech corner? We sked students to eperiment with severl squre sheets of pper of different sizes, gther dt, try to find generl reltionship, nd then prove tht generl reltionship. In the interests of spce, we give only the generl solution for squre sheet of pper tht mesures units by units, s shown in figure 3. The following work could lso be done using symbol mnipultor. Let v() = ( 2) 2, where is in the intervl [0, /2]. Then v() = , v'() = , 0 = We see tht it fctors, so nd 0 = (6 )(2 ) =. 6 The other solution obviously yields minimum volume. The solution mkes students relize tht is the vrible for differentition nd tht, lthough vrible, is constnt with respect to the differentition, tht is, it is one of those very useful fied but still vrible vribles. This concept is lso precursor of multivrite clculus. In ddition, we obtin very simple generl result, which sys tht to find the bo of mimum volume strting with ny squre sheet of pper, we simply mke the squre cutouts t ech corner with side length tht is onesith tht of the side length of the originl squre. Question 2 If we strt with squre but think dynmiclly of incresing one side of tht squre to form lrger nd lrger rectngulr sheets of pper while still keeping the djcent side of fied length, how does the side of the squre cut out from the corners of this pper to form the bo of mimum volume vry s this dynmic side becomes lrger nd lrger? For emple, we consider sheets of pper of the following sizes: 6 inches 6 inches, 6 inches The concept is precursor of multivrite clculus Fig. 3 Photogrph by Richrd Nelson; ll rights reserved Vol. 95, No. 8 November Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3  Techers Teching with
3 This question is much tougher but lso much more rewrding for the persevering student 8 inches, 6 inches 10 inches, 6 inches 12 inches, nd so on. We know tht for the 6 inch 6 inch squre, we cut out 1 inch 1 inch squres. Is the length of the cutout for the 6 inch 8 inch squre more thn 1 inch, less thn 1 inch, or still 1 inch? Wht hppens to the cutout length s the vrible side of the rectngle gets longer nd longer? Does limiting vlue eist? If so, wht is it? We sk our students to eperiment by solving severl concrete emples nd obtining pttern, then generlizing, nd finlly mybe even proving their generliztions. This question is much tougher thn the first one but lso much more rewrding for the persevering student. In this rticle, we offer only flvor of the totl eperience. We ssume tht the originl squre sheet of pper is units units nd tht the side denoted by b is the one tht is incresing in size. We net wnt to find the vlue of for ny vlues of nd b tht yield the bo of mimum volume, s shown in figure 4. b This eqution solves ny bo problem, given the dimensions of the originl sheet of pper, nd b. When b =, it gives, s it should, the solution found in question 1. To get n ide of the solution to the queries given in question 2, we sked our students to fi = 6 nd then consider s function of only b. The result is (b) = (6 + b) 36` `6b`+`b[. We net use grphing clcultor to mke grph of s function of b. On TI83 clcultor, Y1 ssumes the role of, nd X ssumes the role of b. Therefore, the is represents the length of the rectngle whose djcent side is 6, nd the yis represents the cutout size for the corner squres tht yields the bo with mimum volume: y 1 = (6 + ) 36` `6`+`[. with window of : [0, 100] nd y: [0, 3]. See figure 5. Fig. 5 Fig. 4 Agin, symbolic mnipultor cn be used to do the following clcultions. Let v() = ( 2)(b 2), where is in the intervl [0, /2]. Then v() = 4 3 (2 + 2b) 2 + b, v'() = ( + b) + b, 0 = ( + b) + b. Solving by using the qudrtic formul yields We notice tht s the side b increses beyond 6, the cutout size, for the bo with mimum volume lso increses; but limiting vlue, tht is, horizontl symptote for the grph, does seem to eist. Using the tble feture of the TI83 in Ask mode (through TBLSET) nd trying higher nd higher vlues for b, we find tht the limit for b seems to be 1.5 units. See figure 6. We did sk students to try couple of other fied vlues for so tht they might see generl pttern. If students do so, they see tht the cutout = ( + b) ± [` `b`+`b[. For ny vlue of b >, the solution using the positive root is greter thn or equl to /2, so it is not the miml solution tht we desire. Hence, the miml solution is given by = ( + b) [` `b`+`b[. Fig MATHEMATICS TEACHER Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3  Techers Teching with
4 vlue lwys seems to pproch /4, where is the dimension of the fied side of the sheet of pper. A nice symbolicmnipultor eercise for students using TI89, TI92, or the like is to hve it ctully evlute the limit. In this sitution, technology mkes this limit ccessible to gret number of students who would not be ble to do the limit clcultions by hnd: lim = lim ( + b) [` `b`+`b[. b b Some more mthemticlly ble students cn evlute this limit nlyticlly by rtionlizing the numertor s follows: lim ( + b) [` `b`+`b[ b = lim ( + b) [` `b`+`b[ ( + b)+ [` `b`+`b[ b ( + b)+ [` `b`+`b[ 3b = lim b 6( + b + [` `b`+`b[ ) = lim b b b[ b = 4 We know tht for ny size b sheet of pper, where b, the cutout size,, for the length of the side of the squre cut from ech corner to yield the bo of mimum volume lwys stisfies the inequlity <. 6 4 The squre sheet of pper uses the smllest cutout size; nd the more elongted the pper is, the closer the cutout size should be to /4. Question 3 For ny rectngulr sheet of pper tht mesures b, does reltionship eist between the lterl re nd the re of the bse for the bo of miml volume found by cutting congruent squres of side length from the corners of the pper? If so, wht is this reltionship? This question ws one tht we hd not immeditely considered. Only lter did we begin to pursue the reltionship between the lterl re nd the re of the bse of the bo with miml volume. This question becomes fundmentl in the rest of our work. Agin, students should eperiment before seeking the forml result; however, this result is quite esy in its generl form. We ssume tht the originl sheet of pper is b with b, nd the cutoutsqure side length is gin denoted by. Where is in the intervl [0, /2], v() = ( 2)(b 2). Rther thn epnding the term on the right out to obtin polynomil, we cn tke the derivtive in this form using the product rule. One of us hd done so initilly nd noted tht the first term ws the re of the bse of the bo nd wondered whether the remining term hd ny physicl significnce: v'() = ( 2)(b 2) + [( 2)( 2) + (b 2)( 2)] Rewriting this result in slightly different form yields the following: v'() = ( 2)(b 2) [2( 2) + 2(b 2)] 0 = ( 2)(b 2) [2( 2) + 2(b 2)] We notice tht ( 2)(b 2) is the re of the bse of our bo nd tht [2( 2) + 2(b 2)] is the lterl re of the bo. Hence, when we hve the bo of miml volume, the re of the bse minus the lterl re equls 0. Therefore, the bo of miml volume is lwys the bo tht hs the property tht the lterl re is the sme s the re of the bse. This result gives us n esy wy to verify whether ny open bo previously constructed from rectngulr sheet of pper is indeed bo with miml volume. We simply mesure the length, width, nd height of the bo nd then clculte the bse re nd the lterl re. If the two results re equl, the bo is the bo with miml volume; otherwise, it is not. If we hd relied only on symbolic mnipultor, we might not hve been ble to see this reltionship. A symbolic mnipultor gives only the symbolic form tht hs been progrmmed into it. A different form often gives one better insight into generliztion. In this sitution, writing the eqution in our specil symbolic form enbled us to clerly see the reltionship. So fr, we hve just been using rectngulr sheets of pper nd hve been cutting squres out t ech corner to form bo. No reson dicttes tht the piece of pper must be rectngulr. Question 4 ) If the piece of pper tht we strt with is n equilterl tringle, how do we cut out the corners so tht we cn then fold up the sides nd hve bo tht hs n equilterl tringle for bse? b) Once we hve solved prt (), wht is the reltionship between the side of the originl equilterl tringle nd the height,, of the lterl sides of the bo formed in prt () tht gives the bo of mimum volume? We no longer cut squres with side length of out of the corners. Insted, we drw in the ngle Technology mkes this limit ccessible to gret number of students Vol. 95, No. 8 November Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3  Techers Teching with
5 bisectors of the three ngles nd mrk off the sme distnce on ech one. We then connect the three endpoints of these nglebisector segments. We cn see the bse of our solid in figure 7. Finlly, we drw in the si perpendiculr segments from these points to the originl three sides of the pper. The length of these perpendiculr segments is denoted by. Hence, becomes the height of our bo. Thus, for the equilterl tringle, we cut out congruent kites from ech corner. Ech kite hs two opposite right ngles nd the 60degree ngle from the originl equilterl tringle pper. to follow ny simple pttern. We lso tried to obtin the result for regulr hegonl sheet of pper nd the generl regulr ngon sheet of pper. We used the formul A = (1/2)p for the re of the bse of the bo when writing the volume function. During these clcultions, we finlly rrived t better wy to look t our results. We decided to compre the height of the bo of mimum volume with the pothem of the regulr polygonl sheet of pper, not with the side, s we hd been doing. Becuse the pothem is hlf the side of the squre, the mimum bo occurs when the cutout size is onethird of the pothem, or = s 6 = 2 `6` 3 s 3 =, 3 where is the pothem of the squre nd s is the side. We net look t the equilterl tringle, s shown in figure 8. Fig. 7 Compre the height with the pothem of the regulr polygonl sheet of pper To nswer prt (b) of question 4, we need to write formul for the volume in terms of the originl side, s, of the equilterl tringulr pper nd. We cn immeditely write it in generl, but students should do few concrete emples first. We use the fct tht the re of n equilterl tringle is given by Then A = (side)2 3. ````4```` v() = (s 2 3) 2 3, `4` where is in the intervl 0, s 3. `6` We net tke the derivtive, set it equl to 0, nd solve for. Using symbolic mnipultor yields = s 3/18 or = s 3/6. Becuse the ltter result obviously gives minimum volume, our mimum occurs when = s 3/18. This result is not s stisfying s we hd hoped. So fr, we hve worked with two regulr polygons, the squre nd the equilterl tringle. For the squre, the result ws = s/6. We were hoping for some simple reltionship tht would give us the result quickly for ll regulr polygons. The work becme quite tedious, nd the result did not seem 3 Fig. 8 From the digrm, we see tht s = 2 3, so tht = s 3 `18` = ```18``` =. 3 For both the squre nd the equilterl tringle, the height of the bo of mimum volume is onethird the length of the pothem of the originl sheet of pper. Question 5 For ny regulr ngon sheet of pper, if congruent kites re cut from the corners nd then the sides re folded up to form bo with similr ngon for bse, wht is the reltionship between the height 572 MATHEMATICS TEACHER Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3  Techers Teching with
6 of the bo nd the pothem of the originl sheet of pper for the bo of mimum volume? We hve n ide tht the nswer might be = /3, where is the height of the bo nd is the pothem of the originl ngon sheet of pper. To prove this result, perhps we should write the re of the ngon s function of the pothem,, of the originl sheet of pper. For the squre. v() = (2 2) 2 = 4( ) 2. We hve lredy shown tht the mimum occurs when = /3. Thus, if we tke the derivtive of v() nd set it equl to 0, our result will be = /3. For the equilterl tringle. We know tht s = 2 3, so tht v() = ( ) 2 3 `4` = 3 3( ) 2. We notice tht this formul differs only by constnt fctor from tht of the volume of the squre, so the derivtive hs the sme roots. Hence, we gin see tht = /3 is the correct solution. The generl regulr ngon. From figure 9 nd using the fct tht the re of ny regulr ngon is given by A = (1/2)p, where is the pothem nd p is the perimeter, we know tht the volume of the bo is v() = 1 ( )n 2 tn π 2 tn π ; 2 n n v() = n tn π ( )2. n We notice tht this result is just constnt times the formul for the volume of the squre; hence, gin the bo with mimum volume occurs when, the height of the bo, is chosen such tht = /3, where is the pothem of the originl sheet of pper. When we hve the generl regulr ngon volume formul, students cn go bck nd try vlues of n = 3 nd n = 4 nd verify tht these vlues re the ect constnt fctors tht we determined erlier when we did these problems seprtely. Question 6 Wht is the reltionship between the lterl re nd the re of the bse of the bo of mimum volume constructed from regulr ngon sheet of pper? This reltionship is reltively esy one to determine from the fct tht the mimum volume bo hs = /3. Then the re of the bse is 1 ( )p = 1 p = 1 2 p 2 `3` = p 3 = p, which is the lterl re. Hence, the bo with mimum volume mde from ny regulr ngon shped sheet of pper hs lterl re equl to the re of its bse. SUMMARY This clssic problem hs much more to offer thn wht ppers in most tetbooks. Along the wy in our questioning, we used gret del of high school mthemtics, nd we hve discovered interesting geometric nd lgebric reltionships. Finlly, we hope tht when reders see clssic problem, they will think beyond tht problem nd try to find interesting mthemticl generliztions lurking in the bckground. FOR DISCUSSION WITH STUDENTS AND COLLEAGUES We pose further questions tht re relted to the content of this rticle. We hve eplored nswers to the first three of these questions nd would be interested in seeing whether reders gree with our results nd seeing how they obtined their results. Etension 1 For ny given sheet of pper tht mesures b, where b, if we lwys mke the length,, of the side of the cutout squre, so tht = /5, how fr Tn n n Tn n Fig. 9 A section of regulr ngon The formul differs only by constnt fctor from tht of the squre, so the derivtive hs the sme roots Vol. 95, No. 8 November Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3  Techers Teching with
7 will we be from the bo with mimum volume? From prcticl stndpoint, for ll typicl boes tht would normlly be mnufctured, we re relly sking whether we could tell the production stff to lwys cut the squres t the corner of length = /5, where is the shortest dimension of the originl sheet of pper, nd not be too fr from the bo with mimum volume. Etension 2 For the regulr ngon sitution, wht hppens in the limit s the number of sides of the regulr ngon pproches infinity? Wht reltionships do we obtin for cylinder? Etension 3 We hve seen tht with regulr ngon pper nd with rectngulr pper, the bo with miml volume occurs when the lterl re of the bo nd the re of the bse of the bo re equl. Do we continue to obtin this result if we re given ny conve polygon s the originl sheet of pper? The reder should either furnish proof or counteremple. Etension 4 Wht other etensions of this problem should students consider? Etension 5 Wht re pproprite uses of computer lgebr systems nd other technology in these etensions? Wht cn students lern bout pproprite technology use from these bo problems? Etension 6 The techer cn choose nother clssic problem from the mthemtics curriculum. Wht etension questions could be used with tht problem? How might students respond to these questions? Wht mthemtics would they lern or use in the solutions? BIBLIOGRAPHY Dodge, Wlter, Kthleen Goto, nd Philip Mllinson. Soundoff! I Would Consider the Following to Be Proof.... Mthemtics Techer 91 (November 1998): Dossey, John, et l. Focus on Advnced Algebr. Reding, Mss.: AddisonWesley Publishing Compny, Finney, Ross L., George B. Thoms Jr., Frnklin D. Demn, nd Bert K. Wits. Clculus: Grphicl, Numericl, Algebric. Reding, Mss.: Addison Wesley Publishing Compny, Stewrt, Jmes. Clculus. Pcific Grove, Clif.: Brooks/Cole Publishing Compny, MATHEMATICS TEACHER Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3  Techers Teching with
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