A machine produces open top boxes by starting with square sh

A machine produces open top boxes by starting with square sheets of aluminium. The machine first cuts equal squares out of each corner, 3 centimeters by 3 centimeters, then folds the sides up. In the end the base of the box is a square. The box needs to have a volume of 1728cm^2| (which is 1728mL) What is the original length of one side for the original sheet of aluminium to create this volume? In Centimeters

Solution

Let the length of each side of the original sheet be x cm, x>0 .

As coreners of 3 cm each are cut out, the total length cut out = 2*3 cm = 6 cm on each side.

Hence, the length of each side of the base = (x-6) cm

Now, volume of the box = l*b*h = (x-6)(x-6)(3) = 3(x²-12x+36)

Hence, 3(x²-12x+36) =1728

Therefore, x²-12x+36 = 576

=> x²-12x-540 = 0

=> x² - 30x +18x - 540=0

=> x(x-30)+18(x-30)=0

=> (x-30)(x+18) = 0

=> x = 30 or x = -18

As length of the sheet has to be positive, x = 30

Hence, the original length of the side is 30 cm.