Answer to Question #185839 in Calculus for kaye

Question

An open box is to be made with a capacity of 36 000 cu cm. Find the least material

used in making the box if the length is twice its width.

Answer 1

Answer:-

$l=length\\width = w\\h=height$

given l = 2w

$Volume(V) = l\times w \times h$

$V = 2w\times w \times h \\ V = 2w^2\times h \\36000=2w^2\times h\\ now \space for \space the \space material \\ Area = lw + 2wh + 2lh \\ = 2w^2 + {36000\over w} + {72000\over w} \\ =2w^2+{108000\over w }\\ now\\ by\space calculus \\ \frac{d(area)}{dw} = 4w - {108000\over w^2}\\ now\\ \frac{d(area)}{dw}=0\\ clearly \ w = 30 cm \\ and \ now \ l=2w \ so \ , \ l=60 cm \ h = 20cm\\ Area = lw + 2wh + 2lh \\ \boxed{Area = 5400 \ sq \ cm}answer$