Post 2 for holidays.

Since some people seem to be having trouble with optimization, I'm going to help by presenting some steps and various example problems.

Steps:
1. Read each problem slowly and carefully. Read the problem at least three times before trying to solve it. Sometimes words are tricky and unimportant. Find out exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly (coming from my experience...).
2. If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts.
3. Define variables to be used and carefully label your picture or diagram with these variables.
4. Write down all equations which are related to your problem or diagram. Clearly label the equation which you are asked to maximize or minimize. In most problems, you are given an equation that you're optimizing and an equation that you solve for one variable.
Before differentiating, make sure that the optimization equation is a function of only one variable.

Example 1: Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen ?

Solution 1: Where x is the width and y the length
The total amount of fencing is given to be

500 = 5 (width) + 2 (length) = 5x + 2y ,

so that

2y = 500 - 5x

or

y = 250 - (5/2)x .

We wish to MAXIMIZE the total AREA of the pen

A = (width) (length) = x y .

However, before we differentiate the right-hand side, we will write it as a function of x only. Substitute for y getting

A = x y

= x ( 250 - (5/2)x)

= 250x - (5/2)x2 .

Now differentiate this equation, getting

A' = 250 - (5/2) 2x

= 250 - 5x

= 5 (50 - x )

= 0

for

x=50 .

Now since we have the x, we must plug 50 into the equation in which we solved for the y giving y = 125.

So, If

x=50 ft. and y=125 ft. ,

then

A = 6250 ft.2

is the largest possible area of the pen.

OK!!! That was a lot and I dont feel like doing another example...SO, that's my math!!

SEE YA!