How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?

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How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?

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Answer 1 · 2015-03-11T00:40:27

Note that varying the length and width to be other than equal reduces the volume for the same total (length + width); or, stated another way, $w = l$ $w = l$ for any optimal configuration.

Using given information about the Volume, express the height ( $h$ $h$ ) as a function of the width ( $w$ $w$ ).

Write an expression for the Cost in terms of only the width ( $w$ $w$ ).

Take the derivative of the Cost with respect to width and set it to zero to determine critical point(s).

Details:
Volume $= w \times l \times h = w^{2} h = 534$ $= w xx l xx h = w^2 h = 534$
$\to h = \frac{534}{w^{2}}$ $rarr h = (534)/(w^2)$

Cost = (Cost of sides) + (Cost of top and bottom)
$C = (4 \times (4 w \times h)) + (8 \times (2 w^{2}))$ $C= (4 xx (4w xx h)) + ( 8 xx (2 w^2))$

$= (4 \times (4 w \times \frac{534}{w^{2}}) + (8 \times (2 w^{2}))$ $= (4 xx (4w xx (534)/(w^2)) + (8 xx (2 w^2))$

$= 8544 w^{- 1} + 16 w^{2}$ $= 8544 w^(-1) + 16 w^2$

$\frac{d C}{d w} = 0$ $(d C)/(dw) = 0$ for critical points
$- 85434 w^{- 2} + 32 w = 0$ $- 85434 w^(-2) + 32 w = 0$
Assuming $w \neq 0$ $w != 0$ we can multiply by $w^{2}$ $w^2$
and with some simple numeric division:
$- 267 + w^{3} = 0$ $- 267 + w^3 = 0$
and
$w = {(267)}^{\frac{1}{3}}$ $w = (267)^(1/3)$
$= 6.44$ $= 6.44$ (approx.)

$\to l = 6.44$ $rarr l = 6.44$
and $h = 12.88$ $h = 12.88$ (approx.)

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How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?

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