E-OP04 · Different materials and a height limit
A closed box has a square base of side x cm, height h cm and volume \(128\ \mathrm{cm^3}\). Base material costs 3 units per \(\displaystyle \mathrm{cm}^{2}\); the top and four side faces cost 1 unit per \(\displaystyle \mathrm{cm}^{2}\).
Express the total material cost C in terms of x only.
Find the dimensions and cost of the least expensive box without a height restriction. Justify the minimum.
Find the cheapest permitted box when the height must not exceed 2 cm.
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