math
posted by irma .
a square sheet of tin 30cm on a side is to be used to make an open  top box by cutting a small square of tin from each corner and bending up the sides. how large should be the square cut from each corner to make the box's volume as large as possible

let the size of the cutout square be x cm by x cm
then after bending,
the base of the box will be 302x by 302x and x cm high
Volume = x(302x)^2
= x(900  120x + 4x^2)
= 4x^3  120x^2 + 900x
d(Volume)/dx = 12x^2  240x + 900
= 0 for a max/min of Volume
12x^2  240x + 900 = 0
x^2  20x + 75 = 0
(x5)(x15) = 0
x = 5 or x = 15 , (clearly x = 15 yields a minimum volume of 0)
So the size of the cut should be squares of 5 cm by 5 cm.