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Problem*: Marcus is creating a low-fat pie crust recipe for his pie shop. Butter has six grams of saturated fat and one gram of polyunsaturated fat per tablespoon. Vegetable shortening has one gram of saturated fat and four grams of polyunsaturated fat per tablespoon. In the recipe, the butter and vegetable shortening will not be more than 25 tablespoons. The butter and vegetable shortening combine for at least 34 grams of saturated fat and at least 44 grams of polyunsaturated fat. Minimize the number of calories in the recipe if butter has 100 calories per tablespoon and vegetable shortening has 115 calories per tablespoon.
Solution
If we read carefully, we see that the problem requires us to find the number of tablespoons of butter and of shortening that Marcus should use in order to minimize the calories in the recipe. So we let these be our variables:
Let x=tablespoons of butter and let y=tablespoons of shortening.
This is a real problem so our 1st and 2nd constraints are: x ≥ 0 andy ≥ 0
Grams of saturated fat are form 3rd constraint: 6x+y ≥ 34.
Grams of polyunsatured fat form our 4th constraint: x+4y ≥ 44and
Tablespoons of butter and shortening form our 5th constraint: x+y ≤25
Finally, our objective function which we need to minimize is for calories: C=100x+115y.
Let's solve this problem using GeoGebra! Watch the mathcast, study the interactivity below and then do it yourself.
Example 2
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