Anyone willing to help me with an optimization problem?

Question

Here is the problem:
A box has a square base of side x and height y. Find the dimensions for which the surface area is 20 and the volume is as large as possible.
When I solved the problem, I came up with one of the sides equaling the square root of 10/3, but once I tried it in the first derivative test, it fails. Can someone help me out, my book is bad and I don't know where else but here to look.

Answer 1

If the box has a top your answer is correct. It would be in the shape of a cube (you will have to derive this). And you would have:

Surface area = 6x² = 20
x² = 20/6 = 10/3
x = √(10/3)

However, if the box does NOT have a top you will get a different answer. And from reading what you wrote I suspect this is the case. You would have:

x = length base
x = width base
y = height
S = surface area
V = volume

V = x²y
S = x² + 4xy = 20

Try working with this and see what you get.

Answer 2

Vol = x²y
Area = 2x² + 4xy = 20

Solve for y in the area equation. Substitute it in the first equation, then do dV/dx = 0.

If you do it correctly, you will get x = 1.8257

Plug it into the area equation to get y, which should also equal 1.8257, because the minimum area for a maximum volume is a cube, as you would expect.