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17 July, 15:54

A box of volume 72 m3 with a square bottom and no top is constructed out of two different materials. the cost of the bottom is $40/m2 and the cost of the sides is $30/m2. find the dimensions of the box that minimize total cost.

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  1. 17 July, 17:32
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    Square bottom, so L=W

    yah

    V=HL²

    surface area (minus the top) is

    SA=2H (2L) + LW

    SA=4HL+L²

    great

    given V=72

    72=HL²

    solve for H

    72 / (L²) = H

    umm

    sorry, I'm all over the place

    cost=30 (4) HL+40L² or

    cost=120HL+40L²

    subsitute for H

    cost=120 (72 / (L²)) L+40L²

    cost=8640/L+40L²

    find the minimum using your calculator or take the derivitive

    min at L=3∛4≈4.7622031559046

    anyway, sub back

    72/L²=H

    72 / ((3∛4) ²) = H

    2∛4=H

    the dimentions of the box are

    height is 2∛4 or about 3.1748m

    length and width are 3∛4 or about 4.7622m
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