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Optimization

Optimization: Problems and Solutions

We will solve every Calculus Optimization problem using the same Problem Solving Strategy time and again. You can see an overview of that strategy here (link will open in a new tab).

We use that strategy to solve the problems below.

1. Least expensive open-topped can

An open-topped cylindrical can must contain V cm$^3$ of liquid. (A typical can of soda, for example, has V = 355 cm$^3$.) What dimensions will minimize the cost of metal to construct the can?

Click to view full Calculus solution to “Least expensive can.”

To access many more problems and solutions, including AP-style multiple choice questions, log in for free with your Google, Apple or Facebook account, or create a dedicated Matheno in 60 seconds. You’ll also then be able to mark the problems you want to be sure to review before your exams. Simply use the log-in area at the bottom of this screen.

2. Garden fence

Sam wants to build a garden fence to protect a rectangular 400 square-foot planting area. His next-door neighbor agrees to pay for half of the fence that borders her property; Sam will pay the rest of the cost. What are the dimensions of the planting area that will minimize Sam’s cost to build the fence?

Click to view full Calculus solution to “Garden fence.”


3. Printed poster

What are the dimensions of the poster with the smallest total area?

A poster must have a printed area of 320 cm$^2$. It will have top and bottom margins that are 5 cm each, and side margins that are 4 cm. What are the dimensions of the poster with the smallest total area?

Click to view full Calculus solution to “Printed poster.”


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Want access to all of our Calculus problems and solutions? Sign in for free with your Google, Facebook or Apple account, or with your dedicated Matheno account (which you can create in 60 seconds). Then visit our Calculus Home screen.

And if you have a Calculus question you could use some help with, please pop over to our Forum and post.  Optimization problems in particular can be tricky to get started on.  If you’re stuck, please post and we’ll be happy to assist!


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