An open box is to be made out of a 8inch by 14inch piece of

An open box is to be made out of a 8-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four comers and bending up the sides. Find the dimensions of the resulting box that has the largest volume. Dimensions of the bottom of the box: Height of the box:

Solution

l = 8, b = 14
let x length of square is cut from four corner
resulting box has length, l = 8-x, breadth, b = 14-x, height, h = x
V = lbh
V = (8-x)(14-x)x
V = x(x^2 - 22x + 112)
V = x^3 - 22x^2 + 112x
dV/dx = 3x^2 - 44x + 112
for maximum volume, put dV/dx = 0
3x^2 - 44x + 112 = 0
x = [44 - (44^2 - 4*3*112]/6 or [44 + (44^2 - 4*3*112]/6
x = [44 - 592]/6 or [44 + 592]/6
x must be
x = [44 - 592]/6 =  [44-24.33]/6
x = 3.28
l = 8-3.28 = 4.72
b = 14-3.28 =10.72
hence dimesion of bottom = 4.72*10.72
height, h = 3.28