4.4 Optimization Buffalo Bill’s Ranch, North Platte, Nebraska Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1999 With additional examples from:

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose? There must be a local maximum here, since the derivative is 0.

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose? There must be a local maximum here, since the derivative is 0.

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

Box Building We are going to construct a box whose base length is 3 times the base width. The material used to build the top and bottom cost $10/ft 2 and the material used to build the sides cost $6/ft 2. If the box must have a volume of 50ft 3 determine the dimensions that will minimize the cost to build the box.

Getting set up…

Solve the constraint for h:

Plug this into the cost:

Solve the constraint for h: Plug this into the cost:

Solve the constraint for h: Plug this into the cost: Take the derivative:

There are two critical points. The first is obvious, w=0, and it’s also just as obvious that this will not be the correct one. We are building a box here and it makes no sense to talk about a box with zero width.

There are two critical points. The first is obvious, w=0, and it’s also just as obvious that this will not be the correct one. We are building a box here and it makes no sense to talk about a box with zero width. The next critical point will come from determining where the numerator is zero.

There are two critical points. The first is obvious, w=0, and it’s also just as obvious that this will not be the correct one. We are building a box here and it makes no sense to talk about a box with zero width. The next critical point will come from determining where the numerator is zero.

There are two critical points. The first is obvious, w=0, and it’s also just as obvious that this will not be the correct one. We are building a box here and it makes no sense to talk about a box with zero width. The next critical point will come from determining where the numerator is zero. Find the remaining dimensions:

There are two critical points. The first is obvious, w=0, and it’s also just as obvious that this will not be the correct one. We are building a box here and it makes no sense to talk about a box with zero width. The next critical point will come from determining where the numerator is zero. Find the remaining dimensions:

Your dream of becoming a hamster breeder has finally come true. You are constructing a set of rectangular pens in which to breed your furry friends. The overall area you are working with is 60 square feet, and you want to divide the area up into six pens of equal size as shown below. The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. You wish to minimize the cost of the fencing. Find the exact dimensions of each pen that will minimize the cost of the breeding ground. What is the total cost?

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet

The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. 60 square feet