Cutting corner of rectangle to make box and max volume

Question

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 12 in. long and 8 in. wide, find the dimensions of the box that will yield the maximum volume. (Round your answers to two decimal places.)

___in (smallest value)

___in

___in (largest value)

Answer 1

(12-2x)(8-2x)x = volume of the box

4(6-x)(4-x)x = V

V=4(24x-10x^2 +x^3)

take the first derivative and set equal to zero

V'=4(24-20x+3x^2) = 0

3x^2 -20x +24 = 0

x = 20/6 + or - (1/6)sqr(400-288)

x =10/3 + or - (1/6)sqr112

x=3 1/3 + or - (1/3)sqr28

x= 3.3 + or - 5.3/3

x = 3.3 + or - 1.8

x = 1.5 or 5.1

dimensions of the box are 12-3 x 8-3 x 1.5 = 9x5x1.5. V=67.5 cubic feet

or 12-10.2 x 8-10.2 x 5.1 which has a negative width & can be disregarded.

Maximum volume is when 1.5 feet are cut from each corner leaving

a box 9 feet by 5 feet by 1 1/2 feet

Volume of zero would result from cutting corners of 4 feet, leaving one side = zero length, with volume 4x0x4=0 feet