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

Answer 1

let the size of the cut-out square be x cm by x cm
then after bending,
the base of the box will be 30-2x by 30-2x and x cm high
Volume = x(30-2x)^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
(x-5)(x-15) = 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.