Question

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?

Answer 1

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

Explanation:

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`