Find the volume.

Question

find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 31 inches by 17 inches by cutting congruent squares from the corners and cutting up the sides. find the volume.

Answer 1

This can best be understood by drawing the figure.

 

Say, the side of square cot =x inches

 

So the length of the rectangular box=(31-2x)

 

          width                                   = (17-2x)

 

           height.                                 = x

 

So, the volume V= (31-2x)*(17-2x)*x

 

dV/dx= (31-2x)(17-4x)-2x(17-2x)=0

 

On simplifying the above we get the following quadratic equation----

 

12(x^2)-192x+527=0

 

Solving the quadratic equation

 

x= 3.5 inches

 

So, the volume =(32-7)*(17-7)*3.5

                       =25*10*3.5

                        =875 cu inches