a rancher has 600 feet of fencing to put around a rectangular field and then subdivide the field into 2…

Explanation:

Step1: Define variables

Let the length of the field be $l$ (the longer - side) and the width be $w$ (the shorter - side). The total length of the fence is given by $3w + 2l=600$. So, we can express $l$ in terms of $w$ as $l=\frac{600 - 3w}{2}=300-\frac{3}{2}w$.

Step2: Set up the area function

The area of the rectangular field $A=l\times w$. Substitute $l = 300-\frac{3}{2}w$ into the area formula, we get $A(w)=(300-\frac{3}{2}w)w=300w-\frac{3}{2}w^{2}$.

Step3: Find the derivative of the area function

The derivative of $A(w)$ with respect to $w$ is $A^\prime(w)=\frac{d}{dw}(300w-\frac{3}{2}w^{2})=300 - 3w$.

Step4: Find the critical points

Set $A^\prime(w)=0$, then $300 - 3w = 0$. Solving for $w$ gives $3w=300$, so $w = 100$.

Step5: Find the second - derivative of the area function

The second - derivative $A^{\prime\prime}(w)=\frac{d}{dw}(300 - 3w)=-3$. Since $A^{\prime\prime}(w)<0$, when $w = 100$, the area function $A(w)$ has a maximum.

Step6: Find the value of $l$

Substitute $w = 100$ into the equation for $l$: $l=300-\frac{3}{2}\times100=300 - 150 = 150$.

Answer:

The dimensions of the field are length $l = 150$ feet and width $w = 100$ feet.