# Thread: Maximize Dimensions of an open box

1. ## Maximize Dimensions of an open box

A box with a square base and open top must have a volume of 442368 cm3. We wish to find the dimensions of the box that minimize the amount of material used.

First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base.
[Hint: use the volume formula to express the height of the box in terms of x.]
Simplify your formula as much as possible.

can anyone tell me what this equation is? );

2. Originally Posted by tyuolio
A box with a square base and open top must have a volume of 442368 cm3. We wish to find the dimensions of the box that minimize the amount of material used.

First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base.
[Hint: use the volume formula to express the height of the box in terms of x.]
Simplify your formula as much as possible.

can anyone tell me what this equation is? );
This is using calculus

Ok, so it is always a good idea to draw a diagram when we can. see the diagram below.

let the length of each side of the base of the box be x
let the height of the box be y

then the volume is given by:

V = length*width*height = x^2 * y
we want the volume to be 442368

so we have x^2 * y = 442368
we want everything in terms of x, so let's solve for y
=> y = 442368/(x^2)

Now the surface area is the area of all the faces (the top not included since it's an open top). so the surface area is given by:

S = x^2 + xy + xy + xy + xy = x^2 + 4xy
=> S = x^2 + 4x(442368/(x^2))
=> S = x^2 + 1769472/x

we want S to be a minimum.
for minimum we find S' and set it to zero:

S' = 2x -1769472x^-2

for min, set S' = 0
=> 2x - 1769472x^-2 = 0
=> 2x^3 - 1769472 = 0
=> 2x^3 = 1769472
=> x^3 = 1769472/2 = 884736
=> x = cuberoot(884736)
=> x = 96

so the base of the box must be 96 for minimum surface area. what must the height be?

y = 442368/(x^2) = 442368/(96^2) = 48

so the dimensions of the box for minimum surface area are:
width = length = 96 and height = 48

3. thank you so much
yes, i'm in calculus. that was what i needed.