Question

a person wants to enclose a rectangular field with three internal partitions as shown. if there is 11000 feet available for fencing, what dimensions will produce the greatest area? note that the internal partitions also require fencing. the width should be 1100.00 x feet and the length should be 2750.00 feet. (round your answers to two decimal places) question help: message instructor

Explanation:

Step1: Set up the perimeter equation

Let the width of the rectangle be $x$ and the length be $y$. The total length of the fencing is $4x + 2y=11000$, so $y = 5500 - 2x$.

Step2: Set up the area function

The area of the rectangle $A=xy=x(5500 - 2x)=5500x-2x^{2}$.

Step3: Find the derivative of the area function

$A'(x)=\frac{d}{dx}(5500x - 2x^{2})=5500-4x$.

Step4: Find the critical - point

Set $A'(x) = 0$, then $5500-4x=0$. Solving for $x$ gives $x=\frac{5500}{4}=1375$.

Step5: Find the second - derivative of the area function

$A''(x)=\frac{d}{dx}(5500 - 4x)=-4<0$, so when $x = 1375$, the area is maximized.

Step6: Find the length

Substitute $x = 1375$ into the equation for $y$: $y=5500-2\times1375=5500 - 2750 = 2750$.

Answer:

The width should be $1375.00$ feet and the length should be $2750.00$ feet.