Having some trouble with a word problem. A container manufacturer is designing a

Question

Having some trouble with a word problem. A container manufacturer is designing a rectangular box, open at the top and with a square base that is to have a volume of 32ft^3. If the box is to have the least of material used, what must be the dimensions? Thanks in advance.

Explanation / Answer

let side of base be a and height of box be b hence volume becomes (a^2)*b =32 ------(1) least material used => surface area of total box is least ==> base area + area of box on four sides = (a^2) + 4*a*b is least----(2) from (1) b=32/a^2 hence from (2) (a^2) + 4*a*32/a^2 is least ==> (a^2) + 128/a =0 differentiating w.r.t a and equal it to 0 ==> a^3 =64 ==> a=4 ,square side b=height = 32/16 =2 4,2

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