Question

A candy box is made from a piece of cardboard that measures 19 by 11 inches. squares of equal size will be cut out of each corner. the sides will then be folded up to form a rectangular box. what size square should be cut from each corner to obtain maximum​ volume?

Answer 1

Let the side length of each of the squares cut out be x inches.

The volume of the box will be:-
V =  x(19 - 2x)(11 - 2x) 
V = x( 209 - 38x - 22x + 4x^2)
V = x(209 - 60x + 4x^2)
V = 209x - 60x^2 + 4x^3
For maximum/minimum  volume we find the derivative and equate to zero:-
dV/dx =  209 - 120x + 12x^2 = 0
solving for x:-

x = 7.75 , 2.25
For  maximum volume  evaluate the second derivative:-
d^2V/dx^2 =  24x - 120

for x = 2.25  second derivative =  24(2.25) - 120 = -66  so x = 2.25 gives a maximum volume.

Answe:-  r the dimensions of the squares  is 2.25 * 2.2.5 inches