Question

Design a rectangular milk carton box of width ww, length ll, and height hh which holds 474 cm3474 cm3 of milk. The sides of the box cost 2 cent/cm22 cent/cm2 and the top and bottom cost 3 cent/cm23 cent/cm2. Find the dimensions of the box that minimize the total cost of materials used

Answer 1

Answer:

L = W = 6.810 cm

H =  10.22 cm

Explanation:

given data

volume L W H = 474 cm³

sides of the box cost = 2 cent/cm²

top and bottom cost 3 cent/cm²

to find out

dimensions of the box that minimize the total cost of material use

solution

we know here L W H = 474

here L is length and B is width and H is height

so when we minimize the cost function

C( L, W, H ) = (2) 2 H ( L + W ) + (3) 2 L W

so put here H

substitute H = $\frac{474}{LW}$

we get

C( L, W ) = 1896 ( $\frac{1}{W} + \frac{1}{L}$ ) + 6 LW

so

minimum cost will be when the two partial derivatives is 0

so

$\frac{dC}{dL}$ = 6W - $\frac{1896}{L^2}$ = 0

so  

$\frac{dC}{dW}$ = 6L - $\frac{1896}{W^2}$ = 0

L = $\frac{316}{W^2}$

so

by solving above equation we get

L = W = $({{316})^{1/3}$ = 6.810 cm

and

H = $\frac{474}{6.810^2}$

H =  10.22 cm