Make-up #6

Optimization

Optimization can be used for anything from finding the maximum amount of fencing to make a pen to finding the least amount of volume for a cylindrical cone. This concept is used commonly throughout the world and needs to be mastered for college level mathematics.

Steps in order to optimize anything:

1. Identify primary and secondary equations. Primary deals with the variable that is being maximized or minimized. The secondary equation is usually the other equation that ties in all the information given in the problem.

2. Solve the secondary equation for one variable and then plug that variable back into the primary. If the primary equation only have one variable you can skip this step.

3. Take the derivative of the primary equation after plugging in the variable, set it equal to zero, and then solve for the variable.

4. Plug that variable back into the secondary equation in order to solve for the last missing variable. Check endpoint if necessary to find the maximum or minimum answers.

Example:

I have an enclosure against a barn for my pigs that has an area of 180,000 ft². What dimensions of my pen would maximize my perimeter?

1. My two equations are P=2l+w (my barn takes up 1 side) and A=lw.

2. I will solve for w because i have one using my secondary equation.

180,000=lw 180,000/l=w

3. Plug in to the primary then differentiate.

2l+(180,000/l)=P 2+(180,000/l²)=0 180,000=2l² 90,000=l² l=300m

4. Plug back into the secondary equation.

(300)w=180,000 w=(180,000/300)=600m

l=300m
w=600m