5. a rectangular area is enclosed by a fence and divided by another section of fence parallel to two of its…

Explanation:

Step1: Define variables

Let the length of the rectangle be $x$ (the side parallel to the dividing - fence) and the width be $y$. The total length of the fence is given by $3x + 2y=600$, so $y=\frac{600 - 3x}{2}=300-\frac{3}{2}x$.

Step2: Express the area function

The area of the rectangle $A=xy$. Substitute $y = 300-\frac{3}{2}x$ into the area formula: $A(x)=x\left(300-\frac{3}{2}x\right)=300x-\frac{3}{2}x^{2}$.

Step3: Find the derivative of the area function

Differentiate $A(x)$ with respect to $x$. Using the power - rule, $A^\prime(x)=\frac{d}{dx}\left(300x-\frac{3}{2}x^{2}\right)=300 - 3x$.

Step4: Find the critical points

Set $A^\prime(x)=0$ to find the critical points. So, $300 - 3x = 0$. Solving for $x$ gives $3x=300$, and $x = 100$.

Step5: Find the second - derivative of the area function

Differentiate $A^\prime(x)$ with respect to $x$ to get the second - derivative $A^{\prime\prime}(x)=\frac{d}{dx}(300 - 3x)=-3$. Since $A^{\prime\prime}(x)=-3\lt0$, when $x = 100$, the area function $A(x)$ has a maximum.

Step6: Find the value of $y$

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

Answer:

The dimensions of the enclosure are length $x = 100$m and width $y = 150$m.