a farmer is constructing a rectangular pen with two additional fences across it, separating it in three…

Explanation:

Step1: Define variables

Let the length of the pen parallel to the additional fences be $x$ and the width be $y$. The total length of the fence is $4x + 2y=800$, so $y = 400 - 2x$.

Step2: Express the area function

The area $A$ of the rectangular pen is $A=3xy$. Substitute $y = 400 - 2x$ into the area - formula: $A(x)=3x(400 - 2x)=1200x-6x^{2}$.

Step3: Find the derivative of the area function

Differentiate $A(x)$ with respect to $x$. $A^\prime(x)=\frac{d}{dx}(1200x - 6x^{2})=1200-12x$.

Step4: Find the critical points

Set $A^\prime(x)=0$. Then $1200 - 12x = 0$, which gives $x = 100$.

Step5: Check the second - derivative

Differentiate $A^\prime(x)$ to get the second - derivative $A^{\prime\prime}(x)=\frac{d}{dx}(1200 - 12x)=-12\lt0$. So when $x = 100$, the area function $A(x)$ has a maximum.

Step6: Find the value of $y$

Substitute $x = 100$ into $y = 400 - 2x$, we get $y=400-2\times100 = 200$.

Step7: Calculate the maximum area

$A = 3xy$, substituting $x = 100$ and $y = 200$, we have $A=3\times100\times200 = 60000$.

Answer:

$60000$