## Task B-2.3.

Dino has inherited an orange grove with 50 mandarin trees. However, Dino would like to leave his children a certain number of trees that he has planted. If it is the birth year, each tree will yield 800 mandarins. For each additional tree planted in this grove, the yield per tree will decrease by 10 mandarins. How many new trees should Dino plant in the grove to maximize the total yield of mandarins? What is the maximum yield?
## Solution.

Let $x$ be the additional number of trees that Dino will plant in the orchard. We want to maximize the total yield $U$ of tangerines.

$$
\begin{array}{ll}
U=(\text { number of trees })(\text { number of tangerines per tree })=(50+x)(800-10 x) \\
\qquad U=U(x)=40000+300 x-10 x^{2} . & 3 \text { points }
\end{array}
$$

Since the total yield of tangerines is a quadratic function of the variable $x$, its maximum will be at

$$
x=\frac{-b}{2 a}=\frac{300}{20}=15 \text { trees. }
$$

The maximum yield is $U(x)=40000+300 \cdot 15-10 \cdot 15^{2}=42250$ tangerines.

1 point