Superlesson
Project 3-2
Advanced Algebra
Chapter 3, Solving Systems of Linear Equations and Inequalities
Systems of linear inequalities can be used to determine which combinations of products will fulfill a specific set of needs.
Part A, Systems of Inequalities
1. Suppose that, to be sure you are getting the Recommended Daily Allowance (RDA) of Vitamins A and C, you make a mango and watermelon salad.
a. What percentage of the RDA of Vitamins A and C does 1 serving of mangoes contain?
b. What percentage of the RDA of Vitamins A and C does 1 serving of watermelon contain?
c. In order for the salad to supply sufficient Vitamin A, it must contain at least 100% of the RDA of Vitamin A. Write an inequality expressing this condition, assuming the salad contains m servings of mangoes and w servings of watermelon .
d. Write an inequality stating that the salad contains at least 100% of the RDA of Vitamin C.
e. Graph your system of inequalities.
f. Give three examples of the numbers of servings of each fruit that will satisfy the requirements.
Part C, Solving Linear Programming Problems
2. Like humans, animals have daily nutritional requirements.
a. Look in Table II to find the estimated daily amounts of calcium and potassium that dairy cattle need to maintain their health.
b. A dairy farmer feeds his cows two types of mineral supplements per day.
Type Cost/kg Amount Calcium Amount Potassium A $0.90 200 g per kg of supplement 115 g per kg of supplement B $1.00 100 g per kg of supplement 230 g per kg of supplement
Let x represent the number of kilograms of Type A supplement a cow receives daily and let y represent the number of kilograms of Type B supplement a cow receives daily. Write an inequality giving the number of kilograms of calcium each cow receives daily, assuming the cow receives at least the "estimated daily amount" (Exercise 2a). Write an inequality giving the number of kilograms of potassium each cow receives daily.c. Write inequalities stating that each cow is to receive a combined total of no less than 1 kg of supplements per day and no more than 2 kg.
d. Graph the inequalities, showing the feasible region and vertex points.
e. What is the daily per-cow cost function?
f. Find the optimal solution, the least amount the farmer can spend daily per cow while still meeting the cow's nutritional needs.