A farm needs to purchase two types of feed mix from a neighboring farm for the nutrition of its animals. The first type of feed costs $30 \mathrm{Ft}$ per sack. It contains $10 \mathrm{~kg}$ of component A and $10 \mathrm{~kg}$ of component B. The second type of feed costs $50 \mathrm{Ft}$ per sack. It contains $10 \mathrm{~kg}$ of component A, $20 \mathrm{~kg}$ of component B, and $5 \mathrm{~kg}$ of component C. It has been determined that for the healthy development of the animals, the farm needs at least $45 \mathrm{~kg}$ of component A, $60 \mathrm{~kg}$ of component B, and $5 \mathrm{~kg}$ of component C daily. How much of each feed mix should they purchase to ensure they can raise healthy animals at the lowest cost?
If the tsz uses $x$ sacks of the first type and $y$ sacks of the second type daily, the animals receive $(10 x + 10 y) \mathrm{kg}$ of component A. As determined, the condition for healthy development is that this amount must be at least $45 \mathrm{~kg}$:

$$
10 x + 10 y \geq 45
$$

From the necessary amounts of the other two components, we get the conditions:

$$
\begin{aligned}
10 x + 20 y & \geq 60 \\
5 y & \geq 5
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_bb4fcb73e5f147b5513dg-1.jpg?height=615&width=805&top_left_y=614&top_left_x=649)

First, we find the points in the Cartesian coordinate system that satisfy the condition

(a)

$$
10 x + 10 y = 45
$$

From this, we can obtain the region (A) characterized by the condition. As is known, (a) is the equation of a straight line, and (A) is the upper half-plane defined by this line. Similarly, by finding the regions (B) and (C), we get that among the points satisfying the obvious conditions

$$
x \geq 0, y \geq 0
$$

those for which (A), (B), and (C) all hold are in the region whose boundary is formed by the segments $PQ$, $QR$, and the positive direction of the $y$-axis from $P$, and the horizontal ray from $R$ parallel to the $x$-axis. Our task is to determine the smallest constant $k$ for which the function representing the daily cost of the feed

$$
30 x + 50 y = k
$$

has a point in common with our feed. For a given $k$, (D) is the equation of a straight line, and the lines corresponding to different $k$ constants are parallel to each other. It is easy to plot, for example, the line passing through the point $S(0; 3)$, which corresponds to the constant $k = 30 \cdot 0 + 50 \cdot 3 = 150$. This line does not yet have a point in common with our region, but starting from this situation and moving it upwards, it soon reaches the region, specifically at $Q$. Here, the value of $k$ is $30 \cdot 3 + 50 \cdot 1.5 = 165$, and plotting the line corresponding to this (D) we can see that $Q$ is its only point satisfying the conditions (A), (B), and (C). Therefore, the tsz should buy 3 sacks of the first type and 1.5 sacks of the second type of feed daily.