how to calculate the size of square in a box?

Question

given rectangular sheet of cardboard which is 30 cm by 50 cm

a small square of the same size is cut from each corner and the cuts folded to form a box.

what size of the square would preduce a box of volume 3000 cm cubed

Answer 1

Let x = the side of the square

L = 50 - 2x

W = 30 - 2x

V= x(50 - 2x)(30 -2x) = 1500x - 160x² + 4x³ = 3000

Collect and divide by 4

x³ - 40x² +375x -3000 = 0

Graph this equation and you will see that 10 is a root, i.e. x = 10 and the sides are 30 and 10 which works

There are 2 other roots which may work as well but they are irrational roots which you can find by dividing out the root 10 to get a quadratic which can be solved by formula.

Answer 2

V = lwd = 3000 cm^3

3000 cm^3 = (10)(30)(10)

For example: the square cut from the corner could be 10 cm on a side.

50 - 20 = 30 length

30 - 20 = 10 width

height = 10

But, depending on height, there are an infinite number of solutions:

Let square cut out be 5 cm by 5 cm

Length = 50 - 10 = 40 cm

Width = 30 - 10 = 20 cm

3000/(40)(20) = 3000/800 = 30/8 = 15/4 = 3.75 cm height

And so on...