AP Calculus Slides November 29, 2007

Optimization Problems

how far do you want to go

Solving Optimization Problems - A 6 Step Guide

Step 6 Plug the value of that variable into the first two equations to find all dimensions, including the final goal, such as the amount of material, cost, or volume.

Step 5 Set the derivative to 0, and solve for the value of the remaining variable.

Step 4 Take the derivative of your two-variable equation.

Step 3c In the original equation that you are trying to minimize or maximize, replace the variable you isolated in step 3b.

Step 3b Isolate one of the two variables in the equation drawn in step 3a.

Step 3a To do step 3, you will often have to create a second equation from additional information given in the problem. This may require ingenuity, but it should become natural.

Step 3 Try to get the equation into a two variable form, so you can take the derivative.

Step 2 Write an equation for it. Use V = for volume and M = for material, A for area, etc. I must insist on using descriptive variables, because in optimization, if you are sloppy, you lose track of what's going on.

Step 1 Find what you are trying to maximize or minimize. This will be stated explicitly.

Solving Optimization Problems - A 6 Step Example

Step 1 Find what you are trying to maximize or minimize. This will be stated explicitly.

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Find the size of the corner square that will produce a box having the largest possible volume.

Step 2 Write an equation for it. Use V = for volume and M = for material, A for area, etc. I must insist on using descriptive variables, because in optimization, if you are sloppy, you lose track of what's going on.

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Find the size of the corner square that will produce a box having the largest possible volume.

Solving Optimization Problems - A 6 Step Example

Step 3c In the original equation that you are trying to minimize or maximize, replace the variable you isolated in step 3b.

Step 3b Isolate one of the two variables in the equation drawn in step 3a.

Step 3a To do step 3, you will often have to create a second equation from additional information given in the problem. This may require ingenuity, but it should become natural.

Step 3 Try to get the equation into a two variable form, so you can take the derivative.

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Find the size of the corner square that will produce a box having the largest possible volume.

Solving Optimization Problems - A 6 Step Example

Step 4 Take the derivative of your two-variable equation.

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Find the size of the corner square that will produce a box having the largest possible volume.

Solving Optimization Problems - A 6 Step Example

Step 5 Set the derivative to 0, and solve for the value of the remaining variable.

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Find the size of the corner square that will produce a box having the largest possible volume.

Solving Optimization Problems - A 6 Step Example

Step 6 Plug the value of that variable into the first two equations to find all dimensions, including the final goal, such as the amount of material, cost, or volume.

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Find the size of the corner square that will produce a box having the largest possible volume.

Solving Optimization Problems - A 6 Step Example

Now you try one ...A circular cylindrical metal container, open at the top, is to have a capacity of 24π cubic inches. The cost of the material for the bottom of the container is 15 cents per square inches, and that of the material used for the curved part is 5 cents per square inches. If there is no waste of material, find the dimensions that will minimize the cost of the material.