Find the dimensions of the resulting box

Question

An open box is to be made out of a 6-inch by 18-inch piece of cardboard by cutting out squares of equal size from the four corners 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:

Answer 1

(6-2x)(18-2x)x = Volume = V

= 4x^2 -48x + 108)x = V

V' = = 12x^2 -96x + 108 = 0

x^2 - 8x + 9 = 0

use the quadratic formula

x= 8/2 + or - .5sqr(65 -36)= 4 - .5sqr29 ignore the + square root

x = about 1.35425 inches = height of the box

6-2(1.35425 inches = width of the box

14 -2(1.35425 inches = length of the box

max V = 1.35425(4.64575)(17.64575) = about 108.73 cubic inches cutting about 1.4 inch off each corner and folding the sides up

had you cut 1 inch out the volume = 1x4x16 =64 in^3

cut 2 inches out the V = 2x2x14 = 56

both less than the 108.73 citting off about 1.4 inches