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24 September, 06:25

A rectangular box with a volume of 960 ftcubed3 is to be constructed with a square base and top. The cost per square foot for the bottom is 15cents¢ , for the top is 10cents¢ , and for the sides is 1.5cents¢. What dimensions will minimize the cost?

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  1. 24 September, 06:27
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    Yas

    Square base and top, means L=W

    so

    V=HL²

    V=960

    bottom cost=15L²

    top cost=10L²

    sides=2H (L+W) 1.5 or 3H (2L) or 6HL

    so

    total cost=15L²+10L²+6HL or

    TC=25L²+6HL

    eliminate height

    we know

    960=HL²

    divide both sides by L²

    960/L²=H

    sub that for H

    TC=25L²+6 (960/L²) L

    TC=25L²+5760/L

    find where TC is minimum

    take the derivitive

    50L-5760/L²

    or

    (10) (5L³-576) / (L²)

    set numerator to zero

    5L³-576=0

    solve

    L=0.8∛225

    H=960/L²

    H = (20/3) ∛225

    if we were to sub, we would find that minimum cost is 720 (∛15) or aprox 1775.67 cents or $17.76

    dimentions

    L=0.8∛225 aprox 4.86576 ft

    W=0.8∛225 aprox 4.86576 ft

    H = (20/3) ∛225 aprox 40.548 ft

    min cost is $17.76
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