Optimization

Lessons

1. Find two numbers whose sum is 10 and whose product is a maximum.
2. A farmer has 2000 ft of fencing and wants to enclose a rectangular field that borders a straight river. He needs no fence along the river. What dimensions will maximize the enclosed area?
3. A rectangular storage container with an open top is to have a volume of 10 m³. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter.
   1. Find the dimensions of the container that minimize the cost of materials.
   2. Find the cost of materials for the cheapest such container.