Asked by Rahul
A box of constant volume c is to be twice 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 most economic dimensions ?
Answers
Answered by
Steve
w * 2w * h = c
So, h = c/(2w^2)
So if the bottom costs $1/m^2, the total cost is
C(w,h) = w*2w + 3*w*2w + 3*2(wh + 2wh)
= 7w^2 + 18wh
C(w) = 7w^2 + 18w*c/(2w^2)
= 7w^2 + 9c/w
dC/dw = 14w - 9c/w^2
dC/dw=0 when (14w^3-9c) = 0
That is, when w = ∛(9c/14)
and so on to the length and height.
So, h = c/(2w^2)
So if the bottom costs $1/m^2, the total cost is
C(w,h) = w*2w + 3*w*2w + 3*2(wh + 2wh)
= 7w^2 + 18wh
C(w) = 7w^2 + 18w*c/(2w^2)
= 7w^2 + 9c/w
dC/dw = 14w - 9c/w^2
dC/dw=0 when (14w^3-9c) = 0
That is, when w = ∛(9c/14)
and so on to the length and height.
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