A square-based rectangular prism has a total surface area of 54cm^2. Determine the side lengths if the volume is a maximum and hence find the volume.
I have done a number of these and I am getting really annoyed because I always get all the sides being equal no matter what I try. I have absolutely no clue what I am doing wrong but i need to know!!!
SA=2l^2 + 4lh
The Attempt at a Solution
Ok So what i did was SA= 2l^2 +4lh=54 and rearranged for h so that h=(54-2l^2)/4l
then I subbed into the volume equation to make it v=l^2((54-2l^2)/4l)=( 54l^2-2l^4)/4l=
(54l-2l^3)/4= 54l/4-l^3/2, then i derived to make it 54/4 - 3l^2/2= 0. then i rearranged to find l... 3l^2/2=54/4, 3l^2/2= 13.5, l^2= 13.5/1.5, l^2=9, l=3. then i proved it was a max, i'm not going to show that here. anyway then i subbed l into h=(54-2/^2)/4l and ended up with 3 for h
that means l and h are equal to 3 but that wouldn't make it a rectangular prism it would be a cube... what am i doing wrong? I am so frustrated by this!!!