How do you find the volume of the largest open box that can be made from a piece of cardboard 35 inches by 19 inches by cutting equal squares from the corners and turning up the sides?

Question

Emilia Aguilar · Accepted Answer

Volume of largest open box is #2167i n^3#

If we make a box from a piece of cardboard #35# inches by #19# inches by cutting equal squares from the corners and turning up the sides (ssee g=figure below),

we get a box, whose height is #x# and length and bradth are #(35-2x)# and #(19-2x)# and its volume #V# is given by

#V=x(35-x)(19-x)=x(665-54x+x^2)=-x^3-54x^2+665x#

Volume maximizes when #(dV)/(dx)=0#

as #(dV)/(dx)=-3x^2-108x+665=0#

or #3x^2+108x-665=0#

i.e. #x=(-108+-(108^2-4*3*(-665)))/6#

= #(-108+-sqrt19644)/6=(-108+-140.157)/6#

and omitting negative values #x=(-108+140.157)/6=32.157/6=5.36# inches

and when #x=5.36# volume is

#V=5.36(35-5.36)(19-5.36)=5.36*29.64*13.64=2167i n^3#

Axel Alvarez · Answer

undefined To find the volume of the largest open box, first, calculate the side length of the squares to be cut. This is done by setting up an equation with the dimensions given: $ V = x(35-2x)(19-2x) $, where $ x $ represents the side length of the squares. Then, find the value of $ x $ that maximizes the volume using calculus or optimization techniques.