Bill wrote:

A farmer has 20 feet of fencing and wishing to build a rectangular
pen against his barn (by adding 3 sides), maximizing the area. What
should the dimensions be? Note: It makes the kids smile if I
include a little animal in the diagram.
------------------------------------------------------
Not enough information to develop 3 independent equations.

You can use Taylor's theorem of approximation to determine
that a 5 ft x 10 ft pen with the 10 ft side parallel to barn wall will
yield max area.
-----------------------------------------------------------------
W = Width
L = Length
A = Area

2W + L = 20

A = WL
-------------------------------------------------------------
WHEN W = 7, THEN L = 6
A = 42

WHEN W = 6, THEN L = 8
A = 48

WHEN W = 5, THEN L = 10
A = 50

WHEN W = 4, THEN L = 12
A = 48

WHEN W = 3, THEN L = 14
A = 42

Ye gads you are making me dig.

Haven't used most of this stuff in over 50 years.

Lew