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Math Help - Word problems #2

  1. #1
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    Word problems #2

    I am aware it is late so i will keep this short as possible:
    1.According to postal regulations, the girth plus the length of a parcel sent by fourth-class mail may not exceed 108 in. What is the largest possible volume of a rectangular parcel with two square sides that can be sen by fourth-class mail? The answer says 11664in^3 = 6.75 ft^3.

    2.A poster is to contain 108 cm^2 of printed matter with margins of 6cm each at top and bottom and 2 cm on the sides. What is the minimum cost of the of the poster if it is to be made of material costing 20 cents/cm^2? ans. is $60
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  2. #2
    Senior Member tukeywilliams's Avatar
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    1. So  V = x^{2}y which we want to maximize. Then  4x+y \leq 108 . Note that girth is  2(h+w) or in this case  2(x+x) = 4x . To get the maximum volume we set  4x+y = 108 . Then  y = 108 - 4x and  0 \leq x \leq 27 .

    Then  V = x^2(108-4x) .
     V'(x) = 216x-12x^2 .

     0 = 216x-12x^2 = 12x(18-x) .

    Critical values are  x = 0, \ x = 18 . Checking the endpoints  0,27 (volume is 0) we conclude that the maximum volume occurs when  x = 18 .

    So  V_{\text{max}}  = 11,644 \ \text{in}^3 .

    2. So we want to minimize  C = 20(x-4)(y-12) . Since  xy = 108 ,  y = \frac{108}{x} . Then  C(x) = 20(x-4)\left(\frac{108}{x}-12 \right) . So  C'(x) = \frac{2160}{x} - 240 - \frac{2160(x-4)}{x^2} . Critical values are  6,-6 . We get $60 when we plug in  C(-6) .
    Last edited by tukeywilliams; July 31st 2007 at 12:51 AM.
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