Out of a circular piece of paper, you wish to form a cone cup, so you cut out a circle wedge (with its extreme at the circle center) and join the resulting straight sides, forming a conical cup.

What size should the wedge be, to maximize the capacity of the cone?

(In reply to Extension to this problem... by Alec)

The following program evaluates the total volume numerically:

DEFDBL A-Z
pi = ATN(1) * 4
dr = pi / 180

c = 2 * pi
FOR angle = 0 TO 180 STEP 5
  c1 = c * angle / 360: c2 = c - c1
  r1 = c1 / (2 * pi): r2 = c2 / (2 * pi)
  h1 = SQR(1 - r1 ^ 2): h2 = SQR(1 - r2 ^ 2)
  v1 = pi * h1 * r1 * r1 / 3: v2 = pi * h2 * r2 * r2 / 3
  PRINT USING "### ##.##### ##.##### ##.#####"; angle; v1; v2; v1 + v2
NEXT angle

At this wide separation, we get:

angle   v1      v2      v1+v2
  0  0.00000  0.00000  0.00000
  5  0.00020  0.16913  0.16933
 10  0.00081  0.23168  0.23249
 15  0.00182  0.27473  0.27654
 20  0.00323  0.30700  0.31023
 25  0.00504  0.33203  0.33706
 30  0.00725  0.35167  0.35892
 35  0.00985  0.36709  0.37694
 40  0.01285  0.37906  0.39191
 45  0.01623  0.38815  0.40438
 50  0.02000  0.39479  0.41480
 55  0.02416  0.39931  0.42347
 60  0.02868  0.40199  0.43067
 65  0.03358  0.40303  0.43661
 70  0.03884  0.40265  0.44148
 75  0.04445  0.40098  0.44543
 80  0.05042  0.39817  0.44859
 85  0.05673  0.39435  0.45108
 90  0.06337  0.38962  0.45299
 95  0.07034  0.38407  0.45441
100  0.07762  0.37780  0.45542
105  0.08521  0.37088  0.45609
110  0.09309  0.36338  0.45648
115  0.10126  0.35537  0.45663
120  0.10970  0.34690  0.45661
125  0.11840  0.33804  0.45644
130  0.12734  0.32883  0.45617
135  0.13652  0.31932  0.45584
140  0.14591  0.30956  0.45547
145  0.15550  0.29958  0.45508
150  0.16527  0.28943  0.45470
155  0.17521  0.27914  0.45435
160  0.18530  0.26874  0.45404
165  0.19552  0.25827  0.45379
170  0.20584  0.24776  0.45360
175  0.21625  0.23724  0.45349
180  0.22672  0.22672  0.45345

which checks out in the sense that at a little above 65 degrees, the opposite cone is maximized.  The total of the two cones seems maximized between 115 and 120 degrees, so tightening the increments we get (showing the pertinant part):

116.4  0.10359805  0.35304234  0.45664039
116.5  0.10376572  0.35287480  0.45664052
116.6  0.10393349  0.35270709  0.45664058
116.7  0.10410137  0.35253921  0.45664058
116.8  0.10426936  0.35237115  0.45664051
116.9  0.10443745  0.35220292  0.45664038
117.0  0.10460566  0.35203452  0.45664017

and tightening further:

116.60  0.1039334902  0.3527070940  0.4566405843
116.61  0.1039502735  0.3526903135  0.4566405869
116.62  0.1039670578  0.3526735312  0.4566405889
116.63  0.1039838431  0.3526567471  0.4566405903
116.64  0.1040006295  0.3526399614  0.4566405909
116.65  0.1040174170  0.3526231739  0.4566405909
116.66  0.1040342056  0.3526063846  0.4566405902
116.67  0.1040509953  0.3525895937  0.4566405889
116.68  0.1040677860  0.3525728009  0.4566405869
116.69  0.1040845778  0.3525560065  0.4566405843

and

116.640  0.104000631082  0.352639959835  0.456640590917
116.641  0.104002309784  0.352638281162  0.456640590947
116.642  0.104003988497  0.352636602473  0.456640590970
116.643  0.104005667221  0.352634923766  0.456640590986
116.644  0.104007345955  0.352633245041  0.456640590996
116.645  0.104009024700  0.352631566300  0.456640591000
116.646  0.104010703456  0.352629887541  0.456640590996
116.647  0.104012382222  0.352628208764  0.456640590986
116.648  0.104014060999  0.352626529970  0.456640590969
116.649  0.104015739787  0.352624851159  0.456640590946
116.650  0.104017418585  0.352623172331  0.456640590916

so it seems 116.645 degrees is close enough for practical purposes.

Posted by Charlie on 2005-04-09 16:59:09