Q:

Determine the quantities of food that meet the nutritional requirements at least cost?

Solve the following linear programming problem using graphical method.
A dietitian has learned from a nutritional book that his family needs at least 300 grams of protein and at least 60 milligrams iron per day for sound health. These nutrients can be obtained from meat and vegetable products. Each kilogram of meat cost an average of ₱90 and contains an average of 150 grams of protein and 15 milligrams of iron, while each kilogram of vegetables costs ₱22.50 and has 10 grams of protein and 5 milligrams of iron. He wants to determine the quantities of food that meet the nutritional requirements at least cost.

A:

Variables
Let
$$
\begin{split}
x &= \text{ kg of meat to buy;}\\
y &= \text{ kg of vegetable to buy.} \\
\end{split}
$$
Constraints
Of course the variables are non negative and, since we want at least $300$ mg of proteins and $60$ mg of iron, the constraints are
$$
\begin{split}
 150 x + 10 y &\ge 300 \\
 15 x + 5 y &\ge 60 \\
 x,y &\ge  0
\end{split}
$$
Objective function
We want to minimize the total cost of buying food, hence the objective function is
$$
90x + 22.50 y
$$
Therefore the problem can be rewrite as
$$
\begin{split}
&\min \ \ \, 90x + 22.50y \\
&\text{ s.t.}\quad 150x +10y \ge 300 \\
&\phantom{\text{ s.t.}} \quad 15 x + 5y \ge 60\\
&\phantom{\text{ s.t.}} \quad x,y \ge 0\\
\end{split}
$$
Can you continue from here? Which method did you use to solve this? Simplex method?

