A box of constant volume `c` is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom. What are the most economic dimensions?

Explanation

Let the height, breadth and length be `h, x, 2 x`

V=`x times 2 x times h`

arrow c=`2 x^2 h`

area of bottom `=` area of top `=2 x^2`

area of sides `=2 xh+2 xh+xh+xh=6 xh`

If `r` be the cost of material for bottom, then for top and sides is `3 r`

`therefore E=r(2 x^2)+3 r(2 x^2+6 x h)`

E=r`(8 x^2+18 x h)`

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