# 3.2. Small Problems to Try¶

With exactly 2700 square inches of cardboard, we wish to construct a box (width x depth x height (that can contain a volume V. We require the width to be double its depth. We would like to maximize the volume the box can hold. Which values of width, depth, and height fulfill our objective?

A cylindrical can is to hold 20m.^{3} The material for the top and bottom costs $10/m.^{2} and material for the side costs $8/m.^{2} Find the radius r and height h of the most economical can.

A farmer wants to customize his fertilizer for his current crop. He can buy plant food mix A and plant food mix B. Each cubic yard of food A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen and 5 pounds of potash. Each cubic yard of food B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen and 10 pounds of potash. He requires a minimum of 460 pounds of phosphoric acid, 9060 pounds of nitrogen and 220 pounds of potash. If food A costs $30 per cubic yard and food B costs $35 per cubic yard, how many cubic yards of each food should the farmer blend to meet the minimum chemical requirements at a minimal cost? What is this cost?

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.