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

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 =

we get

C( L, W ) = 1896 ( ) + 6 LW

so

minimum cost will be when the two partial derivatives is 0

so

= 6W - = 0

so  

= 6L - = 0

L =

so

by solving above equation we get

L = W = = 6.810 cm

and

H =

H =  10.22 cm

i think it's 35 meters

the total distance that martina travels that day is

(distance from home to mom's office) plus (distance from mom's office to school)

plus (distance from school to library) plus (distance from library to friend's house)

plus (distance from friend's house to martina's house).

(we don't know what any of these distance actually is.)