System Design for Optimum Performance

Maximizing Area of a Fenced in Region

1. Understand the purpose of the problem.

· You are trying to fence in the largest rectangle possible with a certain limited amount of fencing.

· One of your sides is along a straight river. You want to take advantage of the fact that a fence is not necessary along the river.

· You have a total of 1000 feet of fencing to make the other three sides.

· What should the dimensions of the fencing be?

2. Collect the known information. Realize that some of it might be discovered later to be unnecessary.

· We have 1000 feet of fencing.

· We must stick to a rectangular area.

· We can use the river as one side of our rectangle.

3. Determine what information you must find.

· We must find the optimum dimensions of our fenced in region to yield the most area with our 1000 feet of fencing.

4. Simplify the problem only enough to allow the required information to be obtained. State any assumptions you make.

· Restricting ourselves to rectangular areas has made this problem easier. No further simplifications are necessary.

5. Draw a sketch and label any necessary variables.

6. Determine which of the fundamental principles are applicable.

· Area = Length x Width

A=L1 x L2

· Total fence length L = 1000 feet (we want to use it all)

L=2 x L1 + L2

L2=L – 2 x L1

7. Think about your proposed solution approach in general and consider other approaches before proceeding with the details.

· We could try to mathematically optimize using calculus (take the derivative and set it equal to zero…)

· Use a spreadsheet?

· Nah, MATLAB is the easiest way!

8. Label each step of the solution process.

· Set up an array of possible lengths L1 to test (in the range 0 < L1 <500)

· Find the L2 value for each L1, so that we use all 1000 feet

· Compute the area for each L1, L2 combination

· Search for the L1 that yields the largest area. This can be done by plotting the computed areas vs. the corresponging L1s.

9. Solve the problem. If you are solving the problem with a program:

· State the problem concisely.

Maximize A=L1xL2 as a function of L1, assuming L2=L-2xL1.

· Specify the data to be used by the program. This is the “input”.

L=1000 feet (total length available)

· Specify the information to be generated by the program. This is the “output”.

A plot of area vs. L1

The numerical value of L1, which yields the maximum area

· Work through the solution steps by hand or with a calculator; use a simpler data set if necessary.

Here is a partial solution to help reality check our program output:

When L1=0 (L2=1000), Area=0

When L1=500 (L2=0), Area=0

When L1=100 (L2=800), Area=100x800=80,000 ft²

I don’t know which L1 gives maximum area, so this is all I can do by hand.

· Write and run the program.

Let’s do it by writing an m-file!

%

% Script file: fence.m

% Area Optimization Problem

%

% Author:

% Date:

% Put your code here…

Hint: you probably want to use the linspace( ) command and the max( ) command.

Check the output of the program with your hand solution.

Does it check out?

· Run the program with your input data (if different from your simple test data) and perform a reality check on the output.

Do the range of areas look reasonable? Do the hand check points match up?

· If the program will be used as a general tool in the future, test it by running it for a range of reasonable data values; perform a reality check on the results.

Might be interesting to try different values for L.

10. Present your results.

Do not state the answer with any greater precision than is justified by any of the following:

· The precision of the given information.

· The simplifying assumptions.

· The requirements of the problem.