Question

A new refrigerator is being designed. Magnets will be used to hold the door closed, and the rubber seal thickness can be selected to adjust the separation between the magnets (and therefore the force). The total length of rubber seal required to go around the door is 6m, and the cost of rubber seal is given equation 2. The pull force when opening the door should be 12N. Determine which of the magnet combinations should be used, and the combined cost of the magnets and seals per refrigerator.

Table 2. Material costs of neodymium magnets

Magnet size	Cost per unit
10mm	$4.10
12.7mm	$8.15
25mm	$43.20

$Cost/m of rubber seal = 30 T^-0.7                    Equation 2

where T = thickness of each seal in mm (half the separation between the magnets)

25mm		10mm		12.7mm		10-12.7mm
Separation mm	Force N	Separation mm	Force N	Separation mm	Force N	Separation mm	Force N
250	0.014715	45	0.034335	120	0.0036297	130	0.0015696
240	0.015696	42	0.038259	110	0.0066708	120	0.001962
230	0.0192276	40	0.053955	100	0.0093195	110	0.002943
220	0.0207972	38	0.060822	90	0.0135378	100	0.0041202
210	0.0241326	35	0.07848	80	0.018639	90	0.0060822
200	0.02943	32	0.099081	70	0.0319806	80	0.0090252
190	0.03924	30	0.161865	60	0.0526797	70	0.014715
180	0.048069	27	0.199143	50	0.099081	60	0.0249174
170	0.054936	25	0.259965	40	0.197181	50	0.0464994
160	0.065727	23	0.299205	30	0.468918	40	0.092214
150	0.0897615	20	0.44145	25	0.77499	30	0.206991
140	0.112815	17	0.638631	20	1.2753	20	0.555246
130	0.150093	15	1.03005	15	2.06991	15	1.3734
120	0.204048	45	0.034335			10	2.99205
110	0.283509
100	0.385533
90	0.565056
80	0.77499
70	1.13796

Answer 1

Soln.-

Information from question

Cost of 3 different sizes of magnets given. Cost of rubber seal shall be governed by the equation, Cost/m = 30*T**(-0.7). Thickness T is selected as half of the magnet size.

Cost of magnet and corresponding rubber seal calculated as per above equation is tabulated below:

Magnet size	Magnet cost / unit	Rubber seal thickness, T	Cost of rubber seal /m = 30*T**(-0.7)	Cost of 6 m long rubber seal	Lenghth Combinations as given in question to get 6 m and separation force of 12 N
10 mm	$4.10	5 mm	$9.72	$58.34	No combination found
12.7 mm	$8.15	6.35 mm	$8.23	$49.36	120 numbers each of 50 mm length, 120*0.099081 =11.9 N (~ 12 N)
25 mm	$43.20	12.5 mm	$5.12	$30.72	Nocombination found

Hence the possible combination found is 12.7 mm size magnet and 6.35 mm thick rubber seal to get approximately 12 N separation force.

Cost of magnets per refrigerator = $8.15*120 = $978

Cost of rubber seals per refrigerator = $49.36

Hence, combined cost of magnets and rubber seals per refrigerator= $978 + $49.36 = $1027.36