# 3.3. Small Problems To Try¶

1. Imagine you are a start-up company focused on reducing the rate of material leftover for the environment, and you were contracted by a shipping company to calculate the perfect box size ratio to reduce the cost for them. With exactly 2700 square inches of cardboard, we wish to construct a box width 2x, depth x, height 2x. We would like to maximize the volume, V, the box can hold. Which values of width, depth, and height fulfill our objective?

2. This time you have been contracted to redesign a soda can for a startup company in your region. They are extremely mindful of the costs, and want to minimize the price of the production. A cylindrical can is to hold 20m3. The material for the top and bottom costs $10/m2, and material for the side costs $8/m2. Find the radius r and height h of the most economical can.

3. A chemistry lab wants to customize their formula for an ingredient from raw mixtures. They could buy a mixture A and a mixture B. Each cubic yard of A has 20 pounds of oils & fats, 30 pounds of wax and 5 pounds of a thickening agent. Each cubic yard of B has 10 pounds of oils & fats, 30 pounds of wax and 10 pounds of a thickening agent. The minimum requirements are: 460 pounds of oils & fats, 960 pounds of wax and 220 pounds of a thickening agent. If A prizes at $30 per cubic yard and B at $35 per cubic yard, how many cubic yards of each should the chemistry lab mix to meet the requirement at a minimal cost? What is this cost?

4. “Travelling salesman problem”: Create a matrix of distances and then a row of nodes to go to and from. Use index to lookup the distance, and minimize the total distance.