Open boxes are being designed to hold mail for the post office. The boxes will be made to have the greatest possible volume. Each box will be made from a sheet of metal that measures 15 inches by 12 inches. They will be formed by cutting and removing a square from each corner. The sides will then be folded up and welded together along each corner. What is the maximum volume?

Answer 1

Let each side of the square to be cut out be x inches
length of base = 15-2x
width of base = 12-2x
height of box = x

where 0 < x < 6 or else the dimensions make no sense

Volume = x(15-2x)(12-2x)
= 180x - 54x^2 + 4x^3
d(Volume)/dx = 180 - 108x + 12x^2
= 0 for a max/min of Volume

12x^2 - 108x + 180 = 0
x^2 - 9x + 15 = 0
x = (9 ± √21)/2
x = 2.2087 or x = 20.62 , but x < 6

so x = 2.2087
Maximum Volume = 2.2087(10.5826)(7.5826) = 177.2340895 cubic inches

check:
let x = 2.2
Volume = 2.2(10.6)(7.6) = 177.232 , a bit smaller
let x = 2.21
Volume = 2.21(10.58)(7.58) = 177.234044 a tiny bit smaller