Answer
The number of walnut trees which should be planted per acre to maximize the annual yield for the acre is $25$ and the maximum number of pounds of walnuts per acre is $1250$.
Work Step by Step
The number of pounds of walnuts $N$ depends on the number of walnut trees, x. In particular the number of pounds of walnuts is the original number $60$, minus the number lost to overcrowding.
The number of pounds of walnuts per tree is the original number of pounds of walnuts, $60$, minus the decrease due to overcrowding.
$N\left( x \right)=60-2x$.
The maximum number of walnut trees is $20+x$.
The number of pounds of walnuts $R$ for the agency is the number of pounds of walnuts per tree times the number of walnut trees $20+x$.
$\begin{align}
& R\left( x \right)=\left( 60-2x \right)\left( 20+x \right) \\
& =1200+20x-2{{x}^{2}}
\end{align}$
Thus, the number of pounds of walnut is $R\left( x \right)=-2{{x}^{2}}+20x+1200$.
Compare it with the standard equation of quadratic function $f\left( x \right)=a{{x}^{2}}+bx+c$ .
The value of a is $-2$ , b is $20$ and c is $1200$ .
Recall that if $a<0$ , then f has a maximum that occurs at $x=\frac{-b}{2a}$ and maximum value is $f\left( \frac{-b}{2a} \right)$.
Since, $a=-2$ , $a<0$ , $R$ has a maximum.
Recall that the maximum value is at $x=\frac{-b}{2a}$.
Substitute $-2$ for a and $20$ for b.
$\begin{align}
& x=\frac{-b}{2a} \\
& =\frac{-20}{-4} \\
& =5
\end{align}$
The maximum number of walnut trees is $20+x$.
Substitute $5$ for x.
Thus, the maximum number of walnut trees is $20+x=25$.
And the maximum value is $f\left( \frac{-b}{2a} \right)$
Substitute $5$ for x in $R\left( x \right)=-2{{x}^{2}}+20x+1200$.
$\begin{align}
& R\left( x \right)=-2{{x}^{2}}+20x+1200 \\
& =-2\left( 25 \right)+20\left( 5 \right)+1200 \\
& =1250
\end{align}$
Hence, the maximum number of walnut trees is $25$ and maximum number of pounds of walnut is $1250$.
