Problem 35. For the fattening of animals, two types of feed I and II are used. Each kilogram of feed I contains 5 units of nutrient $\boldsymbol{A}$ and 2.5 units of nutrient B, while each kilogram of feed II contains 3 units of nutrient A and 3 units of nutrient B. Experimental data has shown that the fattening of animals will be economically beneficial when each animal receives at least 30 units of nutrient $A$ and at least 22.5 units of nutrient B in their daily diet. It is known that the cost of 1 kg of feed I and 1 kg of feed II is the same and equals 1 monetary unit. What should be the daily consumption of each type of feed to minimize the cost of feeding while meeting the above nutritional requirements?
## S o l u t i o n.

1. Let's construct a mathematical model of this problem. Let $x$ and $y$ be the number of kilograms of feed of types I and II, respectively, consumed daily. Then the system of constraints is:

$$
\left\{\begin{array}{l}
5 x+3 y \geqslant 30 \quad(a) \\
2.5 x+3 y \geqslant 22.5 \\
x \geqslant 0, y \geqslant 0
\end{array}\right.
$$

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The objective function to be minimized is $F=x+y$.

2. Let's construct the set of feasible plans (on Fig. 33, this set is shaded). It is clear that $F$ achieves its minimum value at point $M$.

To find its coordinates, we solve the system

$$
\left\{\begin{array}{l}
5 x+3 y=30 \\
2.5 x+3 y=22.5
\end{array}\right.
$$

We get $(3 ; 5) ; F_{\text {min }}=8$ (monetary units).