SOLUTION: A box manufacturer is to produce a closed box of volume 4500 cubic inches, where the base is a rectangle having length three times its width. What are the dimensions of the box con

Algebra ->  Finance -> SOLUTION: A box manufacturer is to produce a closed box of volume 4500 cubic inches, where the base is a rectangle having length three times its width. What are the dimensions of the box con      Log On


   

Question 1078617: A box manufacturer is to produce a closed box of volume 4500 cubic inches, where the base is a rectangle having length three times its width. What are the dimensions of the box constructed with the least amount of material?
To achieve a minimum surface area of ______________square inches, the box should be __________inches wide, _________inches long and _____________inches high.
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39615) About Me  (Show Source):
You can put this solution on YOUR website!
Bottom or top dimensions,
length 3w, width w
h, for height

Given volume 4500,
3w%2Aw%2Ah=4500
3w%5E2h=4500
h%2Aw%5E2=1500


Total surface area
3w%5E2%2B3w%5E2%2B3wh%2B3wh%2Bwh%2Bwh
6w%5E2%2B8wh

Substitute for h.
total surface area:
6w%5E2%2B7500%2Fw------MISTAKEN STEP
-
should be
6w%5E2%2B8w%281500%2Fw%5E2%29
6w%5E2%2B%288%2A1500%29%2Fw
6w%5E2%2B12000%2Fw------correct surface area formula
-

(removed solution steps which followed)

Answer by ikleyn(52772) About Me  (Show Source):
You can put this solution on YOUR website!
.
The solution by "josgarithmetic" is WRONG.
Below find the correct solution.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Bottom or top dimensions, 
length 3w, width w

h, for height


Given volume 4500,
3w%2Aw%2Ah=4500
3w%5E2h=4500
h%2Aw%5E2=1500     (1)



Total surface area
S = 3w%5E2%2B3w%5E2%2B3wh%2B3wh%2Bwh%2Bwh = 6w%5E2%2B8wh    (2)


From (1), express h = 1500%2Fw%5E2 and substitute it into (2). You will get

S = highlight_green%286w%5E2%2B12000%2Fw%29

-

  %28d%2Fdw%29%286w%5E2%2B12000%2Fw%29 = 
= 12w%2B12000%28-1%29w%5E%28-2%29
= %2812w%5E3-12000%29%2Fw%5E2
-

d%2Fdw+=+0

12w%5E3-12000=0
w%5E3-1000=0
w%5E3=1000=10%5E3
highlight_green%28w=10%29, width

-

Length, 3w
highlight_green%2830%29

Answer. Length = 30, width = 10, height = 15 inches.