Wolfram Demonstrations Project

A farmer has an

foot corner wall and

feet of fence. He wants to use the fence to construct an

foot rectangular field using the corner wall for one corner, and part or all of its sides. What should

and

be to maximize the area

of the field?

For the demonstration,

, and

. There are four cases relating

and

, depending on whether

and

, or

. Vary

to see the possible field boundaries.

The largest field is shown in green, and as a "double square" when it is a 2 by 1 or 1 by 2 rectangle. Study its shape as

(or

) varies, in the cases

and

. It can be a square, a 1x2 or 2x1 rectangle, or an "intermediate" rectangle. The possible shapes and a formula for the dimensions of the largest field are derived in the Details.

Contributed by: Roger B. Kirchner

Maximizing the area of a rectangle with given perimeter, and maximizing the area of a rectangular field bordering a river with a given amount of fence, are special cases (

, and

). In these problems,

is maximum when the rectangle is a square, and a 2 by 1 rectangle, respectively.

The case

=0 and

is a problem of both V. L. Klee, Jr., and J. L. Walsh. In this problem, the max is when

, where

does not exist.

The "Corner Wall Problem" is interesting because there can be one, two, or three critical points, and the largest rectangle can be described geometrically.

There are four cases:

Case 1:

and

Case 2:

and

Case 3:

and

Case 4:

and

In general,

is a continuous, piecewise linear function of

, for

is zero at the endpoints and positive in between. A positive maximum exists.

The maximum occurs where

or where

does not exist, i.e. where

. There can be up to three critical points.

The maximum can be found by 1) determining the domain for

for each case, 2) determining whether

at a point interior to the domain for some case, 3) computing and comparing the values of

where

and at the endpoints for each case. The maximum is the largest of these values.

This method is straightforward, given particular values of

, and

, and is the method used in the Demonstration to find the maximum and where it occurs. But, it is not easy to carry this method out for arbitrary values of

, and

A formula for the dimensions of the maximum rectangle in terms of

, and

can be determined as follows.

Note the graph of

is concave downward because

when it exists. Thus there is a unique maximum.

The maximum occurs either where

, or where

and

, when

does not exist. That is because, at these points,

is increasing from the left and decreasing to the right.

Since

In the four cases,

, and

, and these formulas give the left and right derivatives when

Suppose

. In cases 1 and 4, we see

is increased by making the rectangle more square. In cases 2 and 3, A is increased by making it more 2×1 or more 1×2.

This tells us the largest rectangle is either a square, a 1×2 or 2×1 rectangle, or an "intermediate" rectangle with side

or side

We first consider the non-boundary cases,

and

is maximum when

Case 1:

and

. In this case

. A is max when

. Thus,

and

Case 2:

and

. In this case,

is max when

. Thus,

and

Case 3:

and

. In this case,

is max when

and

Case 4:

and

. In this case,

. A is max when

. Thus,

and

Now the boundary cases, where

. A is maximum when

and

Case 12:

and

. In this case,

, and

is max when

and

Case 34:

and

. In this case,

, and

is max when

and

Case 13:

and

. In this case

, and

is max when

and

Case 24:

and

. In this case

, and

is max when

and

Case 1234:

and

. In this case,

, and

is max when

and

Solving the inequalities in each of these cases, we determine when and where the maximum can occur. The inequalities were solved graphically. An example is given for each case.

is a maximum, with

, when

(1)

, where

. E.g.,

(2)

and

, where

and

. E.g.,

(3)

and

, where

and

. E.g.,

(4)

, where

. E.g.,

is a maximum, with

, when

(12)

and

, where

and

. E.g.,

, and

(34)

and

, or

and

, where

and

. E.g.,

, and

and

(13)

and

, or

and

, where

and

. E.g.,

, or

(24)

and

, or

and

, where

and

. E.g.,

, or

(1234)

and

, where

and

. E.g.,

SUMMARY. We express these results as inequalities for

, given

and

When

, the maximum rectangle is square with

, has

, is 1x2 with

, has

, and is square with

E. g.,

. The maximum rectangle is square if

, has

, is 1x2 if

, has

, and is square if

When

, the maximum rectangle is square with

, has

, and is square with

E. g.,

. The maximum rectangle is square if

, has

, and is square if

When

, the maximum rectangle is square with

, has

, is 2x1 with

, has

, and is square with

E. g.,

. The maximum rectangle is square if

, has

, is 2x1 if

, has

, and is square if

When

, the maximum rectangle is square with

, has

, and is square with

E. g.,

. The maximum rectangle is square if

, has

, and is square if

When

, and the wall a staight wall, the maximum rectangle is 2x1 with

, has

, and is a square with

Thus, for the Walsh/Klee problem, where

and

, the maximum rectangle has

and

This Demonstration can check, and also suggest these results. When the (green) maximum area rectangle is 1×2 or 2×1, it is drawn as two squares, to emphasize this fact.

All possibilites can be examined by choosing

and

, and varying

For reference, a function maxcornerformula[a,b,L], which implements the summary formula, is included in the Initializiation Code section. Changing its name to maxcorner will cause it to replace the function which finds the maximum rectangle by comparing values of A at critical points.

This problem is a generalization of a problem of J. L. Walsh, former Professor of Mathematics at Harvard:

A farmer wishes to enclose a rectangular field of largest area. He has already erected a straight fence of length 100 feet, and has at his disposal 200 additional feet of fencing. What is the largest rectangular field he can fence in, by making use of all or part of the fence already standing?

J. L. Walsh, A Rigorous Treatment of Maximum-Minimum Problems in the Calculus, Heath, 1962.

Also, C. O. Oakley credits the same problem to V. L. Klee, Jr., but when the fance makes use of all of the existing fence.

C. O. Oakley, "End-Point Maxima and Minima, The American Mathematical Monthly, Vol. 54, No. 7, Part 1 (Aug. - Sep., 1947), pp. 407-409.

Pierre Malraison studied the general case of the "Wall Problem" in a film and slides produced at a 1974 NSF Workshop at Carleton College, Computer Graphics for Learning Mathematics.

"Maximum Area Field with a Corner Wall" from the Wolfram Demonstrations Project
http://demonstrations.wolfram.com/MaximumAreaFieldWithACornerWall/

Contributed by: Roger B. Kirchner