Chapter 4.5, Problem 36E

Problems 31-36 involve minimization with mixed constraints.

Chemical mixture A chemical storage tank has a capacity of 200 tons. Currently, the tank contains 50 tons of a mixture that has 10% of a certain active chemical and 1.8% of other inert ingredients. The owners of the tank want to replenish the supply in the tank and will purchase some combination of two available mixes. Mix 1 contains 70% of the active chemical and 3% of the inert ingredients; its cost is $100 per ton. Mix 2 contains 30% of the active chemical and 1% of the inert ingredients; its cost is $40 per ton. The desired final mixture should have at least 40% of the active chemical and at most 2% of the inert ingredients. How many tons of each mix should be purchased to obtain the desired final mixture at minimum cost? Find the minimum cost. Note that at least 40% of the active chemical means

$\text{70\% }\left(\text{mix 1}\right)\text{ + 30\% }\left(\text{mix 2}\right)\text{ + 10\% }\left(\text{mix on hand}\right)\ge \text{ 40\% }\left(\text{mix 1 + mix 2 + mix on hand}\right)$