starting with a 140 - foot - long stone wall, a farmer would like to construct a rectangular enclosure by…

Explanation:

Step1: Set up the perimeter equation

Let the length parallel to the stone - wall be $w$ and the other two equal - length sides be $x$. The total length of the fencing is 980 feet and we have the equation $2x + w=980$, so $w = 980 - 2x$.

Step2: Set up the area function

The area $A$ of a rectangle is $A=x\times w$. Substitute $w = 980 - 2x$ into the area formula, we get $A(x)=x(980 - 2x)=980x-2x^{2}$.

Step3: Find the derivative of the area function

The derivative of $A(x)$ with respect to $x$ is $A^\prime(x)=\frac{d}{dx}(980x - 2x^{2})=980-4x$.

Step4: Find the critical points

Set $A^\prime(x) = 0$, then $980-4x = 0$. Solving for $x$ gives $4x=980$, so $x = 245$.

Step5: Find the second - derivative of the area function

The second - derivative $A^{\prime\prime}(x)=\frac{d}{dx}(980 - 4x)=-4<0$. Since $A^{\prime\prime}(x)<0$ when $x = 245$, the area function $A(x)$ has a maximum at $x = 245$.

Step6: Find the value of $w$

Substitute $x = 245$ into the equation $w = 980 - 2x$. Then $w=980-2\times245=980 - 490 = 490$.

Answer:

$x = 245$ ft $w = 490$ ft