A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of material used.

Question

A box with a square base and open top must have a volume of 4,000 cm3. Find the dimensions of the box that minimize the amount of material used.

Answer 1

VOLUME = 4,000 

 

box's material consists of the bottom and the sides only (no top)

if we let the bottom square of each side = X, and the height = H

we have 1 bottom (X²) area, and 4 sides of  dimen = XH 

 

so the material (M) we want to minimize is then

 

M =  X² + 4XH       (Bottom and 4 sides)   <--- NOTE: this equation we want to minimize

VOLUME = 4,000 =  X²H      (Bottom area x height)

 

put H in terms of X from the volume equation, or 

 

H = 4,000/X²

 

Substitute for H in Material equation, to be minimized

 

M = X² + 4X(4,000/X²)      and simplify to:  

M = X² + 16,000/X     or

 

M = X² + 16,000*X^-1         M is now just in terms of X

 

minimize Material (M) by taking the derivative (with respect to X) and setting = 0

 

DM/DX = 2X  -  16,000 X^-2    = 0    or  

 

2X(X²/X²) - 16,000/X²   or    (2X³  - 16,000) / X²   or    2(X³ - 8,000) / X² = 0

 

For this equation to equal zero,  all we need is   X³ - 8,000 = 0

 

X³ = 8,000   or  X = 20

 

The Volume (X²H) is then  400H  = 4,000  , So H = 10  

 

the dimensions are  20 x 20 x 10 which minimizes material. 

 

the steps are

1) find the equation you need to minimize (or maximize). 

2) use the other information to substitute variables so the equation t minimize is in terms of one variable (X)

3) Take the derivative and set = 0

4) solve for X.