The Archer family raises cattle on their farm in West Midlands

Problem 4

The Archer family raises cattle on their farm in West Midlands. They also have a large garden in which they grow ingredients for making two types of relish - SauceA and SauceB. These they sell at local stores.

The profit per kilogram of SauceA is £2.5 and the profit per kilogram of SauceB is £4.25. The ingredients in each relish are cabbage, tomatoes, onions, and oil. One kilogram of SauceA must contain at least 60% but no more than 70% cabbage, and at least 7% onion, and at least 9% oil.

One kilogram of SauceB must contain at least 60% but no more than 75% tomatoes, and at least 5% onion, and at least 5% oil. Both relishes contain no more than 11% onion and no more than 12% oil.

The family has enough time to make no more than 640 kilograms of relish. They know also that they will sell at least 30% more SauceA than SauceB. They will have this year 400 kilograms of cabbage, 380 kilograms of tomatoes, and 200 kilograms of onion. They can use any amount of oil needed.

What is the maximal profit that the family can gain by producing and selling the relish?