math 151 question

Question

A piece of cardboard measuring 10 inches by 13 inches is formed into an open-top box by cutting squares with side length  from each corner and folding up the sides.

Find a formula for the volume of the box in terms of  =  

Find the value for  that will maximize the volume of the box (round your answer to the nearest hundredth place). = 

Answer 1

10 by 13 inches

will have (10-2x)(13-2x)x = V(x) = the Volume of the box after corners are folded up by cutting x by x from the 4 corners

V(x) = (130-46x +4x^2)x= 130x -46x^2 +4x^3

V'(x) = 130-82x +12x^2 = 0

6x^2 -41x + 65 = 0

x = 41/12 +/-(1/12)sqr(41^2 -24(65))

x= (41 +/-11)/12 = 52/12 or 30/12

x = 13/3 or 5/2

x = 2 1/2 inches

box is 5 in by 8 in by 2.5 in

max volume = 5(8)(2.5) = 100 in ^3

the x=13/3 = 4 1/3 would give

5 2/3 by 7 2/3 by 4 1/3

with V= 17/3(16/3)(13/3) = about 131 in^3

graph V(x), 3 x intercepts: 0, 2.5 and 4 1/3

x=0 gives minimum box volume = 0

x= 2.5 gives a local maximum volume = 100 in^3

x = 4 1/3 gives a global maximum volume = about 131 in^3

closer to a cube with L=W=H, the larger the volume