Help with minimizing the material used for a box...?

Question

A box with a square base and open top must have a volume of 4000cm^3. Find the dimensions of the box that minimizes the amount of material used.

Answer 1

Ok, here is how i think you do it

First you make the equation of volume, since you say the base is going to be square, then you do this

x^2*h = 4000 cm^3

now you get the total area of the box

x^2+4*h*x = Total Area

where x = sides of the base and h = height

Now you solve the first equation for h

h = 4000/x^2

and you substitute in the second equation

x^2+4*4000*x/x^2 = Total Area

now you simplify

x^2+16000*x^-1 = total area

and you derive for x

2*x - 16000*x^-2 = 0

you equal it to 0 because that means that the slope of the graph will be completely horizontal. It only happens with maximum or minimum values.

so you solve for x

x = 20cm

and now you can get h

h = 10cm

hope this helps.

Answer 2

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