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  1. Find two numbers whose sum is 10 and whose product is a maximum.
    1. 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?
      1. 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.
      Topic Notes
      We do not learn how to find extreme values merely for school. From profit maximization for businesses to trip planning for travelers, it has very pragmatic usages in many facets of our daily life. Extreme values hold answers to questions like how one can maximize the profit while minimizing the cost and how to maximize/minimize the product by using the same amount of material. In this section on optimization, we are going to apply the knowledge to tackle problems on area, volume and profit maximization as well as area and cost minimization.