Determine the dimensions of a rectangular solid (with a square base) with maximum volume.

Question

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 253.5 square centimeters.

 x=

x=

x=

Answer 1

253.5 =2x² +4xh where h =height of the box and x = side of the base. solving for h gives (253.5 - 2x²)/4x

So V= x²h V= x²(253.5 -2 x²)/4x = 253.5x/4 -x³/2 then dV/dx = 253.5/4 -3x²/2 = 0 solving for x gives

x²= 253.5/6 =42.25 and x = 6.5 cm plugging this into the equation for h gives h = 6.5 cm and this does maximize the volume since the second derivative is negative. So x= h=6.5 cm.