Please solve, to find the dimensions for least materials used.

Question

If an open box has square base and a volume of 256 in 3 and is constructed from a tin sheet. Find the dimenstions of the box using the least amount of material.

Answer 1

256= b x b x h 256=b²h

Surface Area = 4xbxh+ b x b S=4bh+b²

S=4bh+b²-λ[256-b²h]

dS/db= 4h+2b-2bhλ

dS/dh= 4b-b²λ

dS/dλ=256-b²h

Eq 1: b[4h+2b-2bhλ]=0

Eq 2: h[4b-b²λ]=0

Eq 3: λ[256-b²h]=0

Using Eq 2: [4b-b²λ]=0 4b=b²λ λ=4/b

Using Eq 1: [4h+2b-2bhλ]=0 4h+2b-2bh(4/b)=0 4h+2b-8h=0 b=2h

Using Eq 3: [256-b²h]=0 256=(2h)²h 256=4h³ h=4 therefore b=8

SA=4bh+b²

SA=4*8*4+8*8

SA=192