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|00000ab0| 24 41 27 20 3d 20 30 24 | 20 67 69 76 65 73 20 24 |$A' = 0$| gives $|
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|00000bd0| 20 73 65 65 20 74 68 61 | 74 20 24 78 20 3d 20 50 | see tha|t $x = P|
|00000be0| 2f 34 24 0a 69 73 20 74 | 68 65 20 76 61 6c 75 65 |/4$.is t|he value|
|00000bf0| 20 74 68 61 74 20 67 69 | 76 65 73 20 74 68 65 20 | that gi|ves the |
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|00000c10| 6e 73 77 65 72 20 74 68 | 65 20 71 75 65 73 74 69 |nswer th|e questi|
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|00000c30| 70 72 6f 62 6c 65 6d 5d | 20 20 59 65 73 2c 20 64 |problem]|  Yes, d|
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|00000c50| 74 65 72 20 68 61 76 69 | 6e 67 20 64 6f 6e 65 20 |ter havi|ng done |
|00000c60| 61 6c 6c 20 74 68 65 20 | 68 61 72 64 20 73 74 75 |all the |hard stu|
|00000c70| 66 66 21 20 20 54 68 65 | 20 6d 61 78 69 6d 75 6d |ff!  The| maximum|
|00000c80| 20 61 72 65 61 20 69 73 | 20 5c 66 62 6f 78 7b 20 | area is| \fbox{ |
|00000c90| 24 41 20 3d 20 50 5e 32 | 2f 31 36 20 3d 20 32 35 |$A = P^2|/16 = 25|
|00000ca0| 30 30 24 20 73 71 66 74 | 2e 7d 0a 5c 69 74 65 6d |00$ sqft|.}.\item|
|00000cb0| 5b 50 6c 61 75 73 69 62 | 69 6c 69 74 79 5d 20 20 |[Plausib|ility]  |
|00000cc0| 54 68 61 74 20 24 78 24 | 20 69 73 20 31 2f 34 20 |That $x$| is 1/4 |
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|00000d00| 73 74 20 6f 66 20 75 73 | 20 64 69 64 6e 27 74 20 |st of us| didn't |
|00000d10| 6e 65 65 64 20 63 61 6c | 63 75 6c 75 73 20 74 6f |need cal|culus to|
|00000d20| 20 66 69 6e 64 20 69 74 | 20 6f 75 74 2e 20 20 54 | find it| out.  T|
|00000d30| 68 61 74 20 77 65 20 63 | 61 6e 20 65 6e 63 6c 6f |hat we c|an enclo|
|00000d40| 73 65 20 61 6e 20 61 72 | 65 61 20 6f 66 20 20 32 |se an ar|ea of  2|
|00000d50| 35 30 30 20 66 74 24 5e | 32 24 20 77 69 74 68 20 |500 ft$^|2$ with |
|00000d60| 32 30 30 20 66 74 20 6f | 66 20 66 65 6e 63 65 20 |200 ft o|f fence |
|00000d70| 73 65 65 6d 73 20 72 65 | 61 73 6f 6e 61 62 6c 65 |seems re|asonable|
|00000d80| 20 74 6f 6f 2e 20 20 57 | 65 27 64 20 62 65 20 73 | too.  W|e'd be s|
|00000d90| 75 73 70 69 63 69 6f 75 | 73 20 6f 66 20 61 6e 20 |uspiciou|s of an |
|00000da0| 61 6e 73 77 65 72 20 6c | 69 6b 65 20 34 20 66 74 |answer l|ike 4 ft|
|00000db0| 24 5e 32 24 20 6f 72 20 | 6c 69 6b 65 20 34 30 30 |$^2$ or |like 400|
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|00001b80| 61 72 65 73 20 61 72 65 | 20 75 73 65 64 20 74 6f |ares are| used to|
|00001b90| 20 6d 61 6b 65 20 74 77 | 6f 20 63 75 62 65 73 2e | make tw|o cubes.|
|00001ba0| 20 20 48 6f 77 20 73 68 | 6f 75 6c 64 20 74 68 69 |  How sh|ould thi|
|00001bb0| 73 20 62 65 20 64 6f 6e | 65 20 74 6f 20 6d 61 78 |s be don|e to max|
|00001bc0| 69 6d 69 7a 65 20 74 68 | 65 20 74 6f 74 61 6c 20 |imize th|e total |
|00001bd0| 76 6f 6c 75 6d 65 20 6f | 66 20 61 6c 6c 20 66 69 |volume o|f all fi|
|00001be0| 76 65 20 62 6f 78 65 73 | 3f 20 20 45 76 61 6c 75 |ve boxes|?  Evalu|
|00001bf0| 61 74 65 20 79 6f 75 72 | 20 61 6e 73 77 65 72 20 |ate your| answer |
|00001c00| 66 6f 72 20 74 68 65 20 | 6e 75 6d 65 72 69 63 61 |for the |numerica|
|00001c10| 6c 20 76 61 6c 75 65 20 | 24 73 20 3d 20 31 24 2e |l value |$s = 1$.|
|00001c20| 20 20 53 75 67 67 65 73 | 74 69 6f 6e 3a 20 75 73 |  Sugges|tion: us|
|00001c30| 65 20 5c 54 74 7b 50 6c | 6f 74 7d 20 74 6f 20 63 |e \Tt{Pl|ot} to c|
|00001c40| 6f 6e 66 69 72 6d 20 79 | 6f 75 72 20 6e 75 6d 65 |onfirm y|our nume|
|00001c50| 72 69 63 61 6c 20 61 6e | 73 77 65 72 2e 0a 0a 25 |rical an|swer...%|
|00001c60| 20 4c 6f 67 69 73 74 69 | 63 20 67 72 6f 77 74 68 | Logisti|c growth|
|00001c70| 0a 5c 65 6e 64 7b 65 6e | 75 6d 65 72 61 74 65 7d |.\end{en|umerate}|
|00001c80| 0a 5c 65 6e 64 7b 64 6f | 63 75 6d 65 6e 74 7d 0a |.\end{do|cument}.|
+--------+-------------------------+-------------------------+--------+--------+