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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.”

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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