Question

1. (a) an orchard owner has maintained records that show that, if 25 apple trees are planted in one acre, then each tree yields an average of 500 apples. the yield decreases by 10 apples per tree for each additional tree that is planted. how many trees should be planted for maximum total yield?

Explanation:

Step1: Define variables

Let $x$ be the number of additional trees planted. The number of trees is $n = 25 + x$. The yield per tree is $y=500 - 10x$.

Step2: Find the total - yield function

The total - yield function $T(x)$ is the product of the number of trees and the yield per tree. So $T(x)=(25 + x)(500 - 10x)$.
Expand the function:
\[

$$\begin{align*} T(x)&=25\times500-25\times10x+500x-10x^{2}\\ &=12500-250x + 500x-10x^{2}\\ &=- 10x^{2}+250x + 12500 \end{align*}$$

\]

Step3: Find the vertex of the quadratic function

For a quadratic function $y = ax^{2}+bx + c$ ($a=-10$, $b = 250$, $c = 12500$), the $x$ - coordinate of the vertex is given by $x=-\frac{b}{2a}$.
\[
x=-\frac{250}{2\times(-10)}=\frac{250}{20}=12.5
\]

Step4: Find the number of trees for maximum yield

The number of trees $n=25 + x$. Substitute $x = 12.5$ into the formula for $n$. Since we can't plant half - a tree, we can check $x = 12$ and $x = 13$.
When $x = 12$, $n=25 + 12=37$, and $T(12)=-10\times12^{2}+250\times12 + 12500=-10\times144 + 3000+12500=-1440+3000 + 12500=14060$.
When $x = 13$, $n=25 + 13=38$, and $T(13)=-10\times13^{2}+250\times13 + 12500=-10\times169+3250 + 12500=-1690+3250+12500=14060$.

Answer:

37 or 38 trees