Optimization Chris and Dexter 3/11/11 2 nd Period

What is Optimization? Optimization is the process of finding out the best possible solution to a problem and to have the optimal answering for the constraints that are given. There are 6 fairly simple steps to solving an Optimization problem.

Step 1: Read The Problem This step is very obvious but often over looked. Read the problem slowly and pick out the useful information for solving the problem.

Step 2: Draw A Picture Step 2 is drawing a picture, or diagram for the problem. This is arguably the most important step in the process because it lays out all your information onto a picture where you can visualize it and see what is going on.

Step 3: Write One Equation For Every Variable You Have Look through your picture or diagram you drew in Step 2 and write and equation for every variable in the problem.

Step 4: Take The Equation With Numbers In It And Solve For 1 Variable After Step 3 you should come out with one equation that has a number in it. You should solve for one variable in the problem, the answer you get will be used in Step 5.

Step 5: Substitute That Value In The Other Equation After you solve your equation with numbers in it for one variable, you take what you got and plug it into the other equation you have from Step 3.

Step 6: Set The Derivative Equal To Zero After you find your answer in Step 5 you set the derivative of your equation equal to zero and solve.

Step 7: Solve For Both Variable After you solve Step 6 you should be left with simple algebra and that’s when you finish solving the problem and come out with your optimized answer.

Example Problem A farmer is fencing in his rectangular shaped field. The fencing for the vertical part of the field cost $4 per running foot. The fencing for the horizontal part of the field cost $6 per running foot. One part of the horizontal field runs across a river and does not need to be fenced in. Find the dimensions of the field of largest possible area that can be enclosed with $1800 worth of fence.

Step 1: Read The Problem The important information in this problem is that the vertical fencing cost $4 per foot, the horizontal fencing cost $6 per foot, and that you don’t need fencing on one of the horizontal parts of the field.

Step 2: Draw A Picture

Step 3: Write One Equation With Numbers For Every Variable You Have These two equations will be your equations for the problem.

Step 4: Take The Equation With Numbers In It And Solve For 1 Variable This is the algebra work you should get when solving for y in this equation.

Step 5: Substitute That Value In The Other Equation This is the new equation you will get when you sub the new equation in for y

Step 6: Set The Derivative Equal To Zero After you take the derivative of the problem and set it equal to zero you should get x = 112.5

Step 7: Solve For Both Variables After you find x you go back to your formula where you had it set up to find y, you plug in the x and find find your y

Example #2 Find the dimensions of the right circular cylinder of the greatest volume which can be inscribed in a right circular cone with a radius of.5 inches and a height of 12 inches

Step 1: Read The Problem The important information in this problem is that the right circular cylinder is inscribe in the right circular cone and the cylinder has a radius of.5 inches and a height of 12 inches

Step 2: Draw A Picture

Step 3: Write One Equation With Numbers For Every Variable You Have These are the two equations we are going to use to solve this problem.

Step 4: Take The Equation With Numbers In It And Solve For 1 Variable We will use this equation for H in the next few steps

Step 5: Substitute That Value In The Other Equation This is the equation we will use to take the derivative in the next step

Step 6: Set The Derivative Equal To Zero This is the derivative we get when we set it equal to zero and take the derivative.

Step 7: Solve For Both Variables The optimized height for this problem is 4

Try Me! A box with no top is being made and is 16 inches wide and 21 inches long. The box is in the shape of a square and is cut out of a rectangular piece of paper. To make it a square, squares are cut out of the corners to make it this size. Find the size of each corner square that will produce the biggest box volume.

Useful Formulas For Optimization (Volume)

Useful Formulas For Optimization (Surface Area)

Bibliography Mrs. Autrey’s Notes