Box Mini Project

Question

Paper dimensions are 8.5 inches by 11 inches

You will now have to determine the length, width, and height to make an open-top box with the maximum volume possible. To do this you will have to cut the corners of the paper to fold up (but don't do this yet). Assume the height of the box will be x, which means you will need to subtract x from both sides of the width and length. Use these three dimensions to create a volume equation which you can derive to find the optimal value for x leading to the maximum dimensions for your box.

Answer 1

V= (8.5-2x)(11-2x)x

= (93.5 -22x - 17x +4x^2)x

= 93.5x - 39x^2 + 4x^3

take the derivative and set = 0

V' = 93.5 - 78x +12x^2 = 0

use the quadratic formula

x= 78/24 + or - (1/24)sqr(78^2 -4(12)(93.5)

x = about 78/24 + or - 39.95/24

x = about 1.6 or 4.9 ignore 4.9

8.5 - 3.2 = 5.3

11- 3.2 = 7.8

V=5.3(7.8)(1.6) = about 66.1 in^3 = maximum volume

if x were 1.5

4.5(8)(1.5) = 54

if x were 1.7

5.1(7.6)(1.7) = 65.9

cut 1.6 inches off to get the maximum volume

a little more or little less than 1.6 and the volume is less