Dimensions of box with largest volume

[evilpostingmong]

1. The problem statement, all variables and given/known data
Find the dimensions of a rectangular box with the largest volume with surface area 64cm^2.



2. Relevant equations
area of a rectangular prism 2xy+2yz+2xz


3. The attempt at a solution
took 2xy+2yz+2xz=64
rearranging for z:
z=(xy-32)/(y+x)
partial derivative of z with respect to x
y/(y+x)-(xy-32)/(y+x)^2
partial derivative of z with respect to y
x/(y+x) -(xy-32)/(y+x)^2
setting both derivatives equal to zero to obtain the critical point....
Here's where I hit a wall-
I cannot get a real root for y.
y/(y+x)-(xy-32)/(y+x)^2
y^2+xy=xy-32
y^2=-32
Please only help me up to this point until I ask you to help me further, I
want to challenge myself. Thank you!

[granpa]

what are you maximizing?

[evilpostingmong]

The volume of this box.

[granpa]

you're maximizing xyz?

[evilpostingmong]

Oh wait I'm finding the maximum dimensions that can be used to get the surface area of 64cm^2. So I maximize 2xy+2yz+2xz=64

[granpa]

'maximum dimensions'?

[evilpostingmong]

The problem states that I must find the dimensions of a rectangular box with the largest volume if it's total surface area is 64cm^2. What we don't know is how long those dimensions are.

[granpa]

first you should find the maximum volume of a box with surface area 64cm^2.

[evilpostingmong]

Okay! I'll take the problem from there and tell you how I made out.