Chapter 3.1, Problem 44E

DIET PLANNING A dietitian whishes to plan a meal around four foods. The meal is to include $600$ mg of phosphorus and $400$ mg of magnesium. The number of units of the nutrients in each ounce of the foods (in milligrams) is summarized in the following table:

	Food A	Food B	Food C	Food D
Phosphorus	$30$	$90$	$30$	$45$
Magnesium	$40$	$20$	$30$	$20$

a. Let $x_1,x_2,x_3$ and $x_4$ denote the amount of Food A, Food B, Food C, and Food D, respectively in a meal. Write a system of linear equations in the four given variables to describe the requirements of the dietitian.

b. Use Gauss-Jordon elimination method to solve the system of part (a) for $x_1$ and $x_2$ in terms of $x_3$ and $x_4$ (see section $2.3$ ).

c. Use the fact that $x_1$ and $x_2$ must be nonnegative to write a system of two linear inequalities in the two variables $x_3$ and $x_4$ and then determine graphically the solution set for the system.

d. Combine the results of parts (b) and (c) to write the solutions to the problem.

e. Suggest three possible meals that meet the nutritional requirements. Include at least one in which all four foods are used. (NOTE: Your answer is not unique.)