An open top box is to be constructed so that its base is twice as long as it is wide. Its volume is to be 2400cm cubed. How do you find the dimensions that will minimize the amount of cardboard requiblack?

valtricotinevh 2022-07-17 Answered
An open top box is to be constructed so that its base is twice as long as it is wide. Its volume is to be 2400cm cubed. How do you find the dimensions that will minimize the amount of cardboard requiblack?
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Answers (1)

Jamarion Roth
Answered 2022-07-18 Author has 13 answers
Secondary Equation : w × ( 2 w ) × h = 2400
Solve for h:
h = 1200 w 2
Primary Equation for Surface Area (with open top) :
f ( w ) = 2 w 2 + 2 w h + 2 ( 2 w ) h
Now use the Secondary Equation by substituting for h:
f ( w ) = 2 w 2 + ( 2 w ) ( 1200 w 2 ) + ( 4 w ) ( 1200 w 2 )
Simplify:
f ( w ) = 2 w 2 + 7200 w
Now find the derivative and set equal to zero:
f = 4 w - 7200 w 2 = 0
w 3 = 1800
w = ( 1800 ) 1 3 12.1644
[I'll leave it for you to prove that this is a minimum]
Finally, the dimensions that minimize the surface area are:
w = ( 1800 ) 1 3
l = 2 ( 1800 ) 1 3
h = 1200 w 2 = 1200 ( 1800 ) 2 3
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