A rectangular storage container with an open top is to have a volume of 10 m3. T

Question

A rectangular storage container with an open top is to have a volume of 10 m3. 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.)

Explanation / Answer

Since l = 2w, the volume is: V = h*2w*w = 2hw^2 = 10 => hw^2 = 5 => h = 5/w^2 Cost = C = 20(2w)(w) + 2(12)(h)(2w) + 2(12)(h)(w) => C = 40w^2 + 48hw + 24 hw = 40w^2 + 72hw = 40w^2 + 72(5/w^2)w = 40w^2 + 360/w C' = 80w - 360/w^2 => 0 = 80w - 360/w^2 => 360/w^2 = 80w => 360/80 = w^3 => 4.5 = w^3 => w = 1.65 So the cost for the cheapest container is: C = 40(1.65^2) + 360/1.65 = $327.08

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