A rectangular storage container with an open top is to have a volume of 10 m^3. The length of this base is twice the width. Material for the base costs $20 per square meter. Material for the sides costs $12 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)
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Let width of base be W then legth = 2W
also V = LWH = 2W²H = 10, so H = 5/W²
cost of base = 20*W*2W = 40W²
cost of sides = 12(2LH + 2WH) = 24H(L+W) = 72WH = 360/W
total cost C = 40W² + 360/W
C' = 80 W - 360/W² = 0 for a maximum
W³ = 4.5
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W = 1.651 M
L = 3.302 M
H = 1.834 M
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also V = LWH = 2W²H = 10, so H = 5/W²
cost of base = 20*W*2W = 40W²
cost of sides = 12(2LH + 2WH) = 24H(L+W) = 72WH = 360/W
total cost C = 40W² + 360/W
C' = 80 W - 360/W² = 0 for a maximum
W³ = 4.5
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W = 1.651 M
L = 3.302 M
H = 1.834 M
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The height of the container is required. Otherwise, there are two unknowns.